Stable local dynamics: expansion, quasi-conformality and ergodicity
SSTABLE LOCAL DYNAMICS:EXPANSION, QUASI-CONFORMALITY AND ERGODICITY
ABBAS FAKHARI, MEYSAM NASSIRI, AND HESAM RAJABZADEH
Abstract.
In this paper, we study the stable ergodicity of the action of groupsof diffeomorphisms on smooth manifolds. The existence of such actions is knownonly on one-dimensional manifolds. The aim of this paper is to overcome thisrestriction and to give a method for constructing higher-dimensional examples.In particular, we show that every closed manifold admits stably ergodic finitelygenerated group actions by diffeomorphisms of class C α . We also prove thestable ergodicity of certain algebraic actions including the natural action of ageneric pair of matrices near the identity on a sphere of arbitrary dimension.These are consequences of a new local and stable mechanism/phenomenon whichwe call quasi-conformal blender . This tool stably provides quasi-conformal orbitsand yields stable local ergodicity. The quasi-conformal blender is developed in thecontext of pseudo-semigroup actions of locally defined smooth diffeomorphismswhich allows for applications in several different settings. Contents
1. Introduction 12. Preliminary definitions and notations 53. Expanding sequences 74. Quasi-conformal dynamics 155. Quasi-conformal blenders 216. Stably ergodic actions on manifolds 277. Some questions 32References 331.
Introduction
Let G be a subgroup of Diff α ( M ), the space of all diffeomorphisms with α -H¨older derivative on a smooth Riemannian manifold M endowed with the C topol-ogy. The action of G is minimal if every orbit is dense. Also, the action of G is ergodic (w.r.t. Leb.) if every G -invariant set of positive Lebesgue measure in M hasfull measure. Recall that for a map f , a measurable set S ⊆ M is called f -invariant if f ( S ) ⊆ S up to a set of zero Lebesgue measure. This definition of ergodicityconcerns only the class of the Lebesgue measure, and the invariance of the Lebesguemeasure is not assumed. For F ⊆
Diff α ( M ), we say that the action of (cid:104)F (cid:105) , thegroup generated by F , is stably ergodic if the action of the group generated by any C small perturbation of F in Diff α ( M ) is ergodic. a r X i v : . [ m a t h . D S ] F e b A. FAKHARI, M. NASSIRI, AND H. RAJABZADEH
The existence of stably ergodic actions by diffeomorphisms is known only in di-mension one and the aim of this paper is to introduce a mechanism for stableergodicity in every dimension.For one-dimensional manifolds, one has the following classic result.
Theorem 1.1 (Sullivan) . Let G be a group of C α circle diffeomorphisms, with α > . Assume that for all x ∈ S , there exists some g ∈ G such that g (cid:48) ( x ) > . Ifthe action of G is minimal, then it is ergodic. From the proof of this theorem one can deduce that the action of the groupsgenerated by certain finite subsets of G are indeed stably ergodic (cf. [Nav04,SS85]). This provides examples of stably ergodic finitely generated group actions ofdiffeomorphisms in Diff α ( S ).In dimension one, every action is conformal, i.e. balls are mapped to balls. Thisis a crucial fact in the study of group actions on a one dimensional manifold and par-ticularly in Theorem 1.1 above (cf. [DKN18, Nav18] and their references for recentdevelopments). Indeed, generalizations of Theorem 1.1 are proved for conformal ac-tions in higher dimensions (cf. [DK07a, BFMS17]). However, such generalizations donot provide stably ergodic actions, since conformality and even quasi-conformalityof an action are not stable in higher dimensions.On the other hand, for C α expanding endomorphisms one can bypass confor-mality by considering a dynamically defined sequence of partitions with arbitrarilysmall diameters, such that every element of each partition is eventually mapped to aball of uniform size. This idea combined with the techniques from thermodynamicalformalism implies the existence of a unique ergodic invariant measure in the class ofLebesgue measure. In particular, C α expanding endomorphisms are stably ergodic(cf. [VO16, Rue04, Krz78, Sac74]). This approach is adapted and used in differentsettings including under non-uniform expansions (cf. [ABV00]), however, it doesnot seem applicable for the actions of groups of diffeomorphisms.Our approach in this paper leads to the following variant of Theorem 1.1 fornon-conformal actions on arbitrary manifolds of dimension greater than one. Theorem A.
Let G be a group of C α diffeomorphisms on a smooth closed Rie-mannian manifold M , with α > . Assume that for any ( x, v ) ∈ T M , there existssome g ∈ G such that m ( D x g ) > and (cid:107) ˆ D x g | v ⊥ (cid:107) < . If the action of G is minimal,then it is ergodic. Moreover, G contains a finite subset generating a stably ergodicgroup action. Here, ˆ D x ( f ) := σ − /d D x f denotes the normalized derivative, where σ > f at x . Also, m ( . ) is the co-norm of a linear map, and v ⊥ is the linearsubspace orthogonal to a vector v . If dim( M ) = 1, the statement of this theoremis exactly the one of Theorem 1.1, since (cid:107) ˆ D x g | v ⊥ (cid:107) ≡ T M . The conclusionof Theorem A is not sensitive to the choice of Riemannian metric and it suffices toensure its assumptions for a metric on M .Theorem A and its counterpart for the pseudo-groups of locally defined diffeomor-phisms (Theorem 6.4) are obtained from a local and stable mechanism for ergodicityintroduced in Theorem E. It also allows us to prove the following. Theorem B.
Every closed manifold M admits a stably ergodic semigroup actiongenerated by two diffeomorphisms in Diff α ( M ) . TABLE LOCAL ERGODICITY 3
This theorem gives the first example of a stably ergodic finitely generated groupaction in Diff α ( M ) on a manifold M of dimension greater than one. The notion ofstable ergodicity in the space Diff α ( M ) should not be mistaken with the notion ofstable ergodicity within the class of volume-preserving diffeomorphisms Diff α vol ( M ).In fact, on a closed surface S , we observe that the action of a cyclic group inDiff α ( S ) is not stably ergodic, while area-preserving Anosov diffeomorphisms of T are stably ergodic in Diff α vol ( T ). Such observations indicate that the numberof generators in Theorem B is optimal. See Section 7 for further discussion.We should mention the striking results on ergodic theory of groups of surfacediffeomorphisms in [BR17], which give a classification of stationary measures forsmooth group actions in dimension 2 and yield remarkable examples of stably er-godic group actions in the space of area-preserving surface diffeomorphisms [Liu16,Chu20]. On the other hand, the stable ergodicity of finitely generated dense sub-groups of isometries of even dimensional spheres within the class of sufficientlysmooth volume-preserving diffeomorphisms is proved in [DK07b]. Unfortunately,the arguments in these remarkable papers do not work for showing stable ergodicitybeyond the conservative setting.The following algebraic example is a simple consequence of Theorem A. Theorem C.
Let d ≥ and F be a finite subset of SL( d + 1 , R ) . Assume that theclosure of (cid:104)F (cid:105) strictly contains SO( d + 1) . Then, the natural action of (cid:104)F (cid:105) on S d isstably ergodic in Diff α ( S d ) . Moreover, it is C -robustly minimal. Here, the action of A ∈ SL( d + 1 , R ) on S d is defined by x (cid:55)→ Ax | Ax | . Recall thatthe action of (cid:104)F (cid:105) is called C -robustly minimal if the action of the group generatedby every small perturbation of F in the space of all C diffeomorphisms is minimal.It is known that for d ≥
2, every generic pair of elements near the identitygenerates a dense subgroup of SL( d, R ) [Kur51]. Corollary D.
The family F in Theorem C can be a generic pair near the identityelement in SL( d + 1 , R ) . The problem of ergodicity is more subtle for the pseudo-groups of localized dy-namics, where the restriction of the maps to a given domain are considered. Whilethe localized dynamics appears in many setting, such as the return maps or thelocal holonomy of a foliation leaf, it is well-understood only in dimension one. Mostbasic questions in higher dimensions are open, even for the local affine actions orlocal homogeneous actions (cf. [BIS17] for a remarkable development for certainalgebraic actions).1.1.
Quasi-conformal blender.
The next theorem provides a stable and localmechanism for generating quasi-conformal orbits of pseudo-semigroups and to de-duce local ergodicity. As mentioned before, it plays a fundamental role in provingour main results stated above. Moreover, it makes the proof of Theorem B con-structive, as well as flexible.We denote Diff ( M ) := (cid:83) α> Diff α loc ( M ), where Diff s loc ( M ) is the space of all C s diffeomorphisms f : U f → V f such that U f and V f are open subset of M . Toobtain the strongest stability results we consider C topology on this space, i.e.,two elements of Diff s loc ( M ) are C -close if their graphs are C -close submanifolds of M × M . A. FAKHARI, M. NASSIRI, AND H. RAJABZADEH
A diffeomorphism f ∈ Diff ( M ) is called expanding if m ( D x f ) > x in its domain of definition.Let π : E ( M ) → M be the fiber bundle over a Riemannian manifold M of dimen-sion d defined by E ( M ) := (cid:8) ( x, v ) : x ∈ M, v ∈ ( T x M ) d and det( A v ) = 1 (cid:9) , where for v := ( v , . . . , v d ), A v is a d × d matrix with ( i, j )-entries equal to (cid:104) v i , v j (cid:105) x ,the inner product of v i , v j assigned by the Riemannian metric on T x M . Note that E ( M ) is an SL ± ( d, R ) bundle over M , where SL ± ( d, R ) consists of all d × d matriceswith determinant ± E ( M ) inducing the following norm on its fibers, (cid:107) ( x, v ) (cid:107) := (cid:0) d (cid:88) i =1 (cid:104) v i , v i (cid:105) x (cid:1) . In particular, whenever M is an open subset of R d , E ( M ) is isomorphic to thetrivial bundle M × SL ± ( d, R ), endowed with the Hilbert-Schmidt norm on the fibers.For a diffeomorphism f ∈ Diff ( M ) with f : U f → V f , one can naturally definea fiber map ˆ D f : π − ( U f ) → π − ( V f ) defined byˆ D f ( x, v ) := (cid:0) f ( x ) , ˆ D x f ( v ) (cid:1) , where ˆ D x f ( v ) := ( ˆ D x f ( v ) , . . . , ˆ D x f ( v d )) for v = ( v , . . . , v d ) ∈ ( T x M ) d (cf. Re-mark 4.3).Also, we use the notation f ↓ V := f | V ∩ f − ( V ) for the restriction of an invertiblemap f to the set of points in a set V that are mapped to V . Similarly, we denote F ↓ V := { f ↓ V : f ∈ F } for a family of maps localized to V . Theorem E (Quasi-conformal blender) . Let
F ⊆
Diff ( M ) be a family of expand-ing diffeomorphisms between open subsets of a smooth manifold M . Let W ⊆ E ( M ) be an open set with compact closure, and V := π ( W ) . Assume that (1) W ⊆ (cid:91) f ∈F ( ˆ D f ) − ( W ) . Then, there exists a real number r > such that every measurable F ↓ V -invariantset S with Leb( S ∩ V ) > contains a ball of radius r (up to a set of zero measure).Moreover, this property is C -stable with uniform r > . The principal role of the covering condition (1) is to guarantee the existence ofquasi-conformal orbit-branches in the domain V . In dimension one, it is equivalentto V ⊆ F − ( V ).We would like to emphasize on the multiple stability in this theorem. The coveringcondition (1), the domain V and the radius r are all stable under small perturbationsin the C topology, and are independent of the family’s regularity class and itscorresponding norm. This resembles the idea behind the creation of blenders inpartially hyperbolic dynamics. The concept of blender was introduced in the seminalwork of Bonatti and D´ıaz [BD96] as a stable and local mechanism for transitivity.During the last decades, it has been generalized and used in diverse settings (cf.[RRTU11, NP12, ACW17, Ber16] among others).Most results in this paper are proved for the actions of pseudo-semigroups oflocally defined diffeomorphisms. This allows to deal with broader classes of systems. TABLE LOCAL ERGODICITY 5
In particular, one can apply Theorem E to get stably ergodic smooth foliations ofarbitrary codimension, while the only known examples were of codimension one.This will be discussed in a forthcoming paper. One may expect further applicationsof the local tool introduced in Theorem E in smooth ergodic theory.Let say a few words about the proof of Theorem E. The proof of ergodicity isbased on the simplest known method, i.e., by means of expansions. Given a set S ofpositive measure, one has to show that its orbit has full measure. Then, one showsthat the iteration of the infinitesimal neighborhood of a Lebesgue density point of S gives large open sets that are mostly contained in the orbit of S . To realize this ideafor the locally defined diffeomorphisms (or for the action of diffeomorphisms thateach one is expanding at some regions) one needs to control the shapes of balls toremain almost round through the expanding iterations. This requires new techniqueswhich are the main ingredients of this paper. Two main steps are involved. First, weshow that covering condition (1) implies that for some κ >
1, the pseudo-semigroupgenerated by
F ↓ V (or its perturbations) has a κ -conformal orbit-branch at everypoint of V (Theorem 4.1). Second, we obtain a good control of the shapes of thesmall balls under iteration of sequences of maps satisfying infinitesimal assumptionsof expansion and quasi-conformality (Theorem 3.1). The C α regularity of thesequence is essential in this analysis. This type of shape control together withthe classical distortion control argument allows us to obtain the local ergodicity.Further analysis leads to uniform size of radius r , independent of modulus of H¨olderregularity of derivatives or their norms. Organization.
The paper is organized as follows. In Section 2, we introduce somedefinitions, the precise setting, and notations. Section 3 contains a crucial technicalstep in the proof of the main theorems (Theorem 3.1).Section 4 is devoted to quasi-conformality, where we show that the existence ofquasi-conformal orbit-branches is equivalent to condition (1) for some bounded W .Also, two methods are given to verify that condition. They will be used in the proofof Theorems A and B. Section 5 contains the proof of Theorem E and its variants.In Section 6,we prove Theorems A, B and C. In Section 7, some questions related to the mainresults are discussed.2. Preliminary definitions and notations
Let M be a boundaryless smooth manifold of dimension d endowed with a Rie-mannian metric. We denote by | . | the norm induced by this metric on the tangentspace. We also denote the measure induced from this metric by Leb . , and call it theLebesgue measure. Furthermore, we denote the ball of radius r with center x ∈ M by B ( x, r ). For k ∈ N and α ∈ (0 , f : M → M is of class C k + α , whenever f is C k and its k -th derivative is α -H¨older continuous.For a real number s > C s means C k + α , where s = k + α and k is its integerpart. For f ∈ Diff α loc ( R d ), the C α norm of f , denoted by (cid:107) f (cid:107) C α is defined by (cid:107) f (cid:107) C α := (cid:107) f (cid:107) C + sup (cid:8) (cid:107) D x f − D y f (cid:107)| x − y | α : x, y ∈ Dom( f ) , < | x − y | < (cid:9) . One can use local charts and define C α norm for the smooth maps between opensubsets of manifolds. A. FAKHARI, M. NASSIRI, AND H. RAJABZADEH
We use Diff s ( M ) to denote the group of C s diffeomorphisms of M . Also, Diff ( M )is the union of all Diff s ( M ) with s >
1. Throughout the paper, we usually considerthe C topology on Diff s ( M ). Denote by Diff s loc ( M ) the space of all C s diffeomor-phisms f : Dom( f ) → Im( f ) such that Dom( f ) and Im( f ) are open subset of M .Two elements of Diff s loc ( M ) are C l -close if their graphs are C l -close submanifolds of M × M , for l ≤ s . Similarly, Diff ( M ) is the union of of all Diff s loc ( M ) with s > x ∈ M , U ⊆ M and families of maps F , G ⊆
Diff ( M ), we denote F ( U ) := (cid:83) f ∈F f ( U ), F ( x ) := F ( { x } ), and F ◦ G := { f ◦ g : f ∈ F , g ∈ G} . Also, put F = { Id } and for k ∈ N , denote F k := F k − ◦ F . We use (cid:104)F (cid:105) + (resp. (cid:104)F (cid:105) ) for the semigroup (resp. the group) generated by F . By IFS ( F ), we mean theiterated function system generated by F , that is the action of (cid:104)F (cid:105) + on M . Givena finite family F = { f , . . . , f k } ⊆ Diff α loc ( M ), the (cid:15) -neighbourhood of F in the C topology is the set of all families ˜ F = { ˜ f , . . . , ˜ f k } ⊆ Diff α loc ( M ) such that f i , ˜ f i are (cid:15) -close in the C topology, for any i = 1 , . . . , k . Similarly, one can define (cid:15) -neighbourhood of an infinite family of maps in Diff α loc ( M ).We say a property P holds C -stably for F in Diff α ( M ), if P holds for every˜ F in a C -open neighbourhood of F in Diff α ( M ). Also, we say a property Pholds C -robustly for F , if P holds for every ˜ F in a C -open neighbourhood of F in Diff ( M ). Clearly, by the definition, C -robustness is stronger than C -stability.Similarly, C stability and robustness are defined for Diff ( M ), Diff ( M ) andDiff s loc ( M ), s > Localized dynamics.
For an open set V ⊆ M and f ∈ Diff loc ( M ) with f : Dom( f ) → Im( f ), we define the localization of f to V , by f ↓ V := f | V ∩ f − ( V ) .Clearly, the domain and the image of f ↓ V are Dom( f ↓ V ) = V ∩ f − ( V ∩ Im( V )),and Im( f ↓ V ) = f ( V ∩ Dom( f )) ∩ V , respectively, and f ↓ V : Dom f ↓ V → Im f ↓ V isa bijective map.For a family F of invertible maps, we denote the pseudo-semigroup (resp. pseudo-group) generated by localization of elements of F to V by (cid:104)F ↓ V (cid:105) + (resp. (cid:104)F ↓ V (cid:105) ) andby IFS( F ↓ V ) the action of this pseudo-semigroup. A finite orbit-branch of IFS(
F ↓ V ) at x is a sequence { x i } ni =0 in V such that x = x and for any 1 ≤ i ≤ n , there exists f i ∈ F with x i − ∈ Dom( f i ↓ V ) and f i ( x i − ) = x i . Infinite orbit-branches aredefined similarly. The orbit of IFS( F ↓ V ) at x ∈ V , denoted by (cid:104)F ↓ V (cid:105) + ( x ), is theset of all points in finite orbit-branches at x . For S ⊆ V , define (cid:104)F ↓ V (cid:105) + ( S ) := ∪ x ∈ S (cid:104)F ↓ V (cid:105) + ( x ).In this paper, we deal with two basic dynamical concepts, namely, minimalityand ergodicity. IFS( F ↓ V ) is called minimal if for any x ∈ V , (cid:104)F ↓ V (cid:105) + ( x ) is densein V . Fixing a measure µ on V , we say a measurable map f ↓ V is non-singular with respect to µ , if f ∗ (cid:0) µ | Dom( f ) (cid:1) is absolutely continuous with respect to µ | Im( f ) .When both f, f − are measurable and non-singular with respect to µ , we say that µ is quasi-invariant for f . A measurable set S ⊆ V is called F ↓ V -invariant , if (cid:104)F ↓ V (cid:105) + ( S ) ⊆ S up to a set of measure zero. Moreover, IFS( F ↓ V ) is called ergodic with respect to µ , if µ is quasi-invariant for all the elements of F , and there is nomeasurable F ↓ V -invariant set S with 0 < µ ( S ∩ V ) < µ ( V ). TABLE LOCAL ERGODICITY 7
Throughout the paper, the sets we are localizing the dynamics on are open subsetsof smooth manifolds. We fix the Lebesgue measure on the manifolds and all thestatements regarding ergodicity are with respect to the Lebesgue measure.2.2.
Expanding maps.
For a linear map D , we denote its operator norm by (cid:107) D (cid:107) ,and its co-norm by m ( D ) := inf {| D ( v ) | : | v | = 1 } . If D is invertible, then m ( D ) = (cid:107) D − (cid:107) − .A diffeomorphism f ∈ Diff ( M ) is called expanding, if there exists η > m ( D x f ) > η for every x ∈ Dom( f ). Clearly, the expanding property is C -robust. Definition 2.1.
For η > N ∈ N ∪ {∞} , we say a sequence { f i } Ni =1 inDiff ( M ) is η -expanding at x ∈ M , if for any integer i ∈ [1 , N ], x i − ∈ Dom( f i )and m ( D x i − f i ) > η , where x i = f i ◦ · · · ◦ f ( x ). Furthermore, the sequence is expanding at x if it is η -expanding for some η > Quasi-conformality.
For a real number κ ≥
1, a matrix D ∈ GL( d, R ) is κ -conformal , if (cid:107) D (cid:107) /m ( D ) = (cid:107) D (cid:107) · (cid:107) D − (cid:107) ≤ κ and a sequence { D i } Ni =1 in GL( d, R )is κ -conformal , if for any integer n ∈ [1 , N ], D n D n − · · · D is κ -conformal. Here, N can be finite or infinite. It follows immediately from definition that for D , D ∈ GL( d, R ), if D i is κ i -conformal ( i = 1 , D D is κ κ -conformal, and D − is κ -conformal. These in particular imply that for a κ -conformal sequence { D i } Ni =1 ,all the products of the form D j D j − · · · D i for 1 ≤ i < j ≤ N are κ -conformal. Asderivatives of smooth maps are linear maps between tangent spaces, one can definesimilar notions for them. Definition 2.2.
For κ ≥ N ∈ N ∪{∞} , we say a sequence { f i } Ni =1 in Diff ( M )is κ -conformal at x ∈ M , if for any integer n ∈ [1 , N ], x ∈ Dom( f n ) and the linearmap D x f n is κ -conformal, where f n = f n ◦ · · · ◦ f . Furthermore, the sequence is quasi-conformal at x if it is κ -conformal at x for some κ ≥ Expanding sequences
Here, we prove two technical results which will be used for showing ergodicityof quasi-conformal blenders. The first one provides a precise distortion control ofshapes under quasi-conformal expanding sequences. The second one is the standardbounded distortion lemma adapted to our setting of expanding sequences of localmaps.Throughout the section, M is a closed manifold of dimension d . For a sequence { f i } ∞ i =1 in Diff α loc ( M ), denote f i := f i ◦ · · · ◦ f and f := Id. We denote the openball of radius of r > R d with B r (0).3.1. Control of shapes.
Our main goal in this subsection is to prove the followingtheorem.
Theorem 3.1.
Let { f i } ∞ i =1 be a sequence in Diff α loc ( M ) with bounded C α norm.Let also x ∈ M and ρ > be such that for any n ∈ N , f n is defined on B ( f n − ( x ) , ρ ) .If the sequence is quasi-conformal expanding at x , then there exist ξ > and θ > such that for any ξ ∈ (0 , ξ ] and n ∈ N , f n ( B ( x, r n )) ⊆ B ( f n ( x ) , ξ ) ⊆ f n ( B ( x, θr n )) , for some r n > . A. FAKHARI, M. NASSIRI, AND H. RAJABZADEH
In other words, one obtains a control of the shape of the iteration of a ball fromcertain assumptions on the derivatives at the center. The passage from linear tononlinear follows from precise estimates on pseudo-orbits of corresponding productof matrices. The proof of this theorem occupies the entire subsection. As it is alocal statement, we prove Theorem 3.1 by showing similar statements on Euclideanspace and for uniformly contracting sequences of local diffeomorphisms.Fix
R, C > , κ ≥ > λ > λ > α ∈ (0 , N ∈ N ∪ {∞} , we considerthe following hypotheses for the sequence { h n } Nn =1 .(H0) h n : B R (0) → h n ( B R (0)) is a C α diffeomorphism fixing the origin,(H1) (cid:13)(cid:13) h n (cid:13)(cid:13) C α < C ,(H2) for any y ∈ B R (0), λ < m ( D y h n ) ≤ (cid:107) D y h n (cid:107) < λ ,(H3) h n is κ -conformal at the origin.Note that, it follows from (H2) that h n ( B R (0)) ⊆ B R (0). Fix
R, C > , > > > ↵ (0 , N N [ {1} , we considerthe following hypotheses for the sequence { h n } Nn =1 .(H0) h n : B R (0) ! h n ( B R (0)) is a C ↵ di↵eomorphism fixing the origin,(H1) h n C ↵ < C ,(H2) for any y B R (0), < m ( D y h n ) k D y h n k < ,(H3) h n is -conformal at the origin.Note that, it follows from (H2) that h n ( B R (0)) ✓ B R (0). Figure 1.
Sequence of maps satisfying (H0)-(H3) in Lemma 3.2.
Lemma 3.2.
Let d N , R, C > , > > > and ↵ (0 , . Then,there exist ⇠ (0 , R ) and , ✓ > such that for any ⇠ (0 , ⇠ ] , any n N and anysequence { h j } nj =1 of maps satisfying (H0)-(H3), (2) B r/✓ (0) ✓ h n ( B ⇠ (0)) ✓ B ✓r (0) , for r = ⇠ | det D h n | /d . Moreover, for any | x | ⇠ , (3) | h n ( x ) D h n ( x ) | < | det D h n | /d | x | ↵ . This lemma follows from the next one on sequences of linear maps satisfyingconditions (H2)-(H3). It establishes precise estimates on the di↵erence between theorbits and certain pseudo orbits.Given a sequence { D i } i =1 in GL( d, R ), for a pair i < j of positive integers denote D j,i := D j D j · · · D i . Lemma 3.3 (Key Lemma) . Let
C > , > > > and ↵ (0 , . Then,there exist ⇠ , > such that for any n N , any sequence { D i } ni =1 of matrices in GL( d, R ) satisfying (C1) Contraction: For any i n , m ( D i ) k D i k < , (C2) Quasi-conformality: For any i n , k D i, k m ( D i, ) ,and any sequence { y i } ni =0 in R d with | y | ⇠ and (4) | y i +1 D i +1 y i | < C | y i | ↵ , the following holds (5) | y n D n, y | < | det D n, | /d | y | ↵ . Figure 1.
Sequence of maps satisfying (H0)-(H3) in Lemma 3.2.
Lemma 3.2.
Let d ∈ N , R, C > , κ ≥ > λ > λ > and α ∈ (0 , . Then,there exist ξ ∈ (0 , R ) and γ, θ > such that for any ξ ∈ (0 , ξ ] , any n ∈ N and anysequence { h j } nj =1 of maps satisfying (H0)-(H3), (2) B r/θ (0) ⊆ h n ( B ξ (0)) ⊆ B θr (0) , for r = ξ | det D h n | /d . Moreover, for any | x | ≤ ξ , (3) | h n ( x ) − D h n ( x ) | < γ | det D h n | /d | x | α . This lemma follows from the next one on sequences of linear maps satisfyingconditions (H2)-(H3). It establishes precise estimates on the difference between theorbits and certain pseudo-orbits.Given a sequence { D i } ∞ i =1 in GL( d, R ), for a pair i < j of positive integers denote D j,i := D j D j − · · · D i . Lemma 3.3 (Key Lemma) . Let
C > , κ ≥ > λ > λ > and α ∈ (0 , . Then,there exist ξ , γ > such that for any n ∈ N , any sequence { D i } ni =1 of matrices in GL( d, R ) satisfying (C1) Contraction: For any ≤ i ≤ n , λ ≤ m ( D i ) ≤ (cid:107) D i (cid:107) ≤ λ < , (C2) Quasi-conformality: For any ≤ i ≤ n , (cid:107) D i, (cid:107) m ( D i, ) ≤ κ , TABLE LOCAL ERGODICITY 9 and any sequence { y i } ni =0 in R d with | y | ≤ ξ and (4) | y i +1 − D i +1 y i | < C | y i | α , the following holds (5) | y n − D n, y | < γ | det D n, | /d | y | α . Proof.
The proof of the lemma consists of several steps. First, a rough a priori upperbound for the norms of the terms in the sequence is given. Second, one observesthat if the matrices in the sequence satisfy an extra assumption between the normand the co-norm, a more accurate estimate holds which leads to the statement of thelemma. The extra assumption is not restrictive as it can be verified if one replacesthe sequence by its large blocks of compositions.For the proof, fix
C, κ, λ, λ, α > { D i } ni =1 , { y i } ni =0 as inthe lemma. It follows from (C2) that for any j > i , D j,i is κ -conformal. Since m ( D j,i ) ≤ | det D j,i | /d ≤ (cid:107) D j,i (cid:107) , one gets (cid:107) D j,i (cid:107)| det D j,i | /d ≤ (cid:107) D j,i (cid:107) m ( D j,i ) ≤ κ , and | det D j,i | /d m ( D j,i ) ≤ (cid:107) D j,i (cid:107) m ( D j,i ) ≤ κ , So the condition (C2) implies the following.(C2) (cid:48)
For any j > i ≥ κ − | det D j,i | /d ≤ m ( D j,i ) ≤ (cid:107) D j,i (cid:107) ≤ κ | det D j,i | /d . Claim 1.
For any α (cid:48) > , there exist K = K ( κ, d, λ, λ, α (cid:48) ) ∈ N and τ > such thatfor any i ≥ , (6) (cid:0) (cid:107) D i + K,i +1 (cid:107) + τ (cid:1) α (cid:48) ≤ | det D i + K,i +1 | /d . Proof.
For t ∈ R + , denote ϕ ( t ) := t α (cid:48) − κ t . Since 1 + α (cid:48) >
1, there exists T = T ( α (cid:48) , κ ) >
0, such that ϕ ( t ) is positive and increasing on (0 , T ). Let K ∈ N be large enough such that λ K < T and τ := ϕ ( κ − λ K ). Now, the conclusion easilyfollows from (C2) (cid:48) and (C1). (cid:3) Note that from (4), | y i +1 | < C | y i | α + (cid:107) D i +1 (cid:107)| y i | . Therefore, if | y | ≤ ξ (cid:48) := C − α (1 − λ ) α , then for any i ≥ C | y i | α + (cid:107) D i +1 (cid:107) ≤ | y i +1 | ≤ | y i | . Define (cid:15) i := y i +1 − D i +1 y i ∈ R d . For any pair i, k ≥
0, one obtains an explicitformula for y i + k in terms of y i and { (cid:15) j : i ≤ j < i + k } , y i + k = D i + k y i + k − + (cid:15) i + k − = D i + k ( D i + k − y i + k − + (cid:15) i + k − ) + (cid:15) i + k − = · · · = D i + k,i +1 y i + i + k (cid:88) j = i +2 D i + k,j (cid:15) j − + (cid:15) i + k − . (8) Claim 2. If | y | ≤ ξ (cid:48) , then for any i ≥ and k ≥ , | D i + k,i +1 y i − y i + k | < Ck | y i | α . Proof.
By (8) and since (cid:107) D j (cid:107) < | y i + k − D i + k,i +1 y i | ≤ i + k − (cid:88) j = i | (cid:15) j | . On the other hand, (4) and (7) imply that for any j = i, . . . , i + k − | (cid:15) j |
There exist ξ , τ (cid:48) > with τ (cid:48) < min { − λ K , τ } and ξ ≤ ξ (cid:48) such that if | y i | ≤ ξ , then for any p ≥ , (9) | y pK + i | ≤ p − (cid:89) j =0 (cid:0) (cid:107) D ( j +1) K + i,jK + i +1 (cid:107) + τ (cid:48) (cid:1) | y i | ≤ | y i | . Proof.
It suffices to prove (9) for p = 1. The general case follows immediatelyfrom induction on p . Let τ (cid:48) , ξ > τ (cid:48) < min { − λ K , τ } and ξ ≤ min { ( C − K − τ (cid:48) ) α , ξ (cid:48) } . Then, by Claim 2, | y i + K || y i | < (cid:107) D i + K,i +1 (cid:107) + CK | y i | α ≤ (cid:107) D i + K,i +1 (cid:107) + τ (cid:48) ≤ λ K + τ (cid:48) < . This finishes the proof of Claim 3. (cid:3)
Claim 4.
There exists γ (cid:48) > such that for any p ∈ N and q ≥ , | y pK + q − D pK + q,q +1 y q | < γ (cid:48) | det D pK + q,q +1 | /d | y q | α , provided that | y q | ≤ ξ .Proof. Define α (cid:48)(cid:48) := α − α (cid:48) >
0. For simplicity, for a fixed q and 1 ≤ i ≤ p , wewrite D (cid:48) i := D iK + q, ( i − K + q +1 , λ (cid:48) i := (cid:107) D (cid:48) i (cid:107) , λ (cid:48) i := m ( D (cid:48) i ). Moreover, for 0 ≤ i ≤ p ,let y (cid:48) i := y iK + q . Also, denote (cid:15) (cid:48) i := y (cid:48) i +1 − D (cid:48) i +1 y (cid:48) i . Similar to (8), one gets, y (cid:48) p = D (cid:48) p, y (cid:48) + D p, (cid:15) (cid:48) + · · · + D (cid:48) p,p − (cid:15) (cid:48) p − + D (cid:48) p (cid:15) (cid:48) p − + (cid:15) (cid:48) p − , and so,(10) | y (cid:48) p − D (cid:48) p, y (cid:48) | ≤ | D (cid:48) p, (cid:15) (cid:48) | + · · · + | D (cid:48) p (cid:15) (cid:48) p − | + | (cid:15) (cid:48) p − | . Let λ (cid:48) := λ K and λ (cid:48) := λ K . Claim 1 and τ (cid:48) ≤ τ imply(11) ( λ (cid:48) j + τ (cid:48) ) α = ( λ (cid:48) j + τ (cid:48) ) α (cid:48) ( λ (cid:48) j + τ (cid:48) ) α (cid:48)(cid:48) ≤ δ (cid:48) j ( λ (cid:48) + τ (cid:48) ) α (cid:48)(cid:48) . Hence, by Claims 2 and 3, | (cid:15) (cid:48) k | ≤ CK | y (cid:48) k | α ≤ CK | y (cid:48) | α k (cid:89) j =1 ( λ (cid:48) j + τ (cid:48) ) α ≤ CK ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) δ (cid:48) k · · · δ (cid:48) | y (cid:48) | α TABLE LOCAL ERGODICITY 11
By (C2) (cid:48) , for p ≥ k + 2, | D (cid:48) p,k +2 (cid:15) (cid:48) k | ≤ κ δ (cid:48) p · · · δ (cid:48) k +2 | (cid:15) (cid:48) k |≤ CKκ δ (cid:48) p · · · δ (cid:48) k +2 ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) δ (cid:48) k · · · δ (cid:48) | y (cid:48) | α ≤ CKκ λ (cid:48) ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) δ (cid:48) p · · · δ (cid:48) | y (cid:48) | α , where the last inequality follows from δ (cid:48) k +1 ≥ λ (cid:48) k +1 . Now, from (10), | y (cid:48) p − D (cid:48) p, y (cid:48) | ≤ CKκ λ (cid:48) (cid:16) p − (cid:88) k =0 ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) (cid:17) δ (cid:48) p · · · δ (cid:48) | y (cid:48) | α . On the other hand, λ (cid:48) + τ (cid:48) <
1. Consequently, p − (cid:88) k =0 ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) ≤ C (cid:48) := ∞ (cid:88) k =0 ( λ (cid:48) + τ (cid:48) ) kα (cid:48)(cid:48) < ∞ . Therefore, taking γ (cid:48) := κ CKC (cid:48) λ − K finishes the proof. (cid:3) Claim 5.
There exists γ > such that if | y | ≤ ξ , | y n − D n, y | < γ | det D n, | /d | y | α . Proof.
Let δ i := | det D i | /d and write n = pK + q for 0 ≤ q < K . Since | y q | ≤ ξ from Claim 4, | y pK + q − D pK + q,q +1 y q | ≤ γ (cid:48) δ pK + q · · · δ q +1 | y q | α . Meanwhile, from (C2) (cid:48) and Claim 2, it follows that | D pK + q,q +1 y q − D pK + q, y | ≤ (cid:107) D pK + q,q +1 (cid:107) . | y q − D q, y |≤ κ δ pK + q · · · δ q +1 Cq | y | α . Finally, by Claim 4 and since δ i ≥ λ , one has | y pK + q − D pK + q, y | ≤ δ pK + q · · · δ q +1 | y | α ( γ (cid:48) + κ Cq ) ≤ δ pK + q · · · δ q +1 δ q · · · δ | y | α λ − q ( γ (cid:48) + κ Cq ) . So, the conclusion holds for γ := λ − K ( γ (cid:48) + κ CK ). (cid:3) Claim 5 completes the proof of Lemma 3.3. (cid:3)
Now, we are ready to complete the proof of Lemma 3.2 and Theorem 3.1.
Proof of Lemma 3.2.
Take x ∈ B R (0). Clearly, if the sequence { h j } nj =1 satisfies(H0)-(H3), then the conditions of Lemma 3.3 are satisfied for D j := D h j , y j := h j ( x ) = h i ( x i − ). So, (3) holds for any | x | ≤ ξ . In order to prove (2), note that (3)in particular implies that(12) | D h n ( x ) | + γ | det D h n ( x ) | /d | x | α < | h n ( x ) | , and(13) | h n ( x ) | < | D h n ( x ) | + γ | det D h n ( x ) | /d | x | α . On the other hand, by (C2) (cid:48) , κ − | det D h n ( x ) | /d | x | ≤ | D h n ( x ) | ≤ κ | det D h n ( x ) | /d | x | . This combined with (12) and (13), implies that κ − | x | − γ | x | α < | h n ( x ) || det D h n ( x ) | /d < κ | x | + γ | x | α . Therefore, for any ξ with κ − γξ α > B r ( ξ ) (0) ⊆ h n ( B ξ (0)) ⊆ B r ( ξ ) (0), providedthat r ( ξ ) := ( κ − − γξ α ) ξ | det D h n | /d , r ( ξ ) := ( κ + γξ α ) ξ | det D h n | /d . To prove (2), take θ > κ . Then, for sufficiently small ξ > κ + γξ α < θ and κ − − γξ α > θ − . (cid:3) Proof of Theorem 3.1.
Let R > M and x j := f j ( x ). Suppose that the sequence is κ -conformal η -expanding at x . Since the sequence has bounded C α norm, there is ρ (cid:48) ∈ (0 , ρ )and η, η > i ∈ N and y ∈ B ( x i − , ρ (cid:48) ), η < m ( D y f i ) ≤ (cid:107) D y f i (cid:107) < η. In fact, one has sup i ∈ N (cid:107) Df i | B ( x i − ,ρ ) (cid:107) < ∞ , and if C > sup i ∈ N (cid:107) f i (cid:107) C α , | m ( D x i − f i ) − m ( D y f i ) | < C | x i − − y | α . So, m ( D y f i ) > η for any y ∈ B ( x i − , ρ (cid:48) ), provided that ρ (cid:48) < ( C − ( η − η )) α , .For R < R := ( η ) − min { R , ρ (cid:48) } and i ∈ N , the map ˜ f i := exp − x i ◦ f i ◦ exp x i − is defined on B R (0) ⊆ T x i ( M ) and is a diffeomorphism onto its image. After anisometric identification of the tangent spaces with R d , one can consider the sequence { ˜ f i } ∞ i =1 as a sequence of expanding maps defined on B R (0) ⊆ R d . By uniformexpansion of the maps, B R (0) ⊆ ˜ f i ( B R (0)).Next, fix n ∈ N . The sequence { h j } nj =1 defined by h j := ˜ f − n +1 − j | B R (0) satisfieshypotheses (H0)-(H3) with constants independent of the choice of n . So, by Lemma3.2, there are θ > ξ > ξ ≤ ξ and for some r n > h n ) − ( B r n (0)) ⊆ B ξ (0) ⊆ ( h n ) − ( B θr n (0)) ⊆ B R (0) . This finishes the proof, since ( h n ) − = exp − x n ◦ f n ◦ exp x and for small r >
0, thefunction exp x : T x M → M , maps B r (0) ⊆ T x M to B ( x, r ) ⊆ M . (cid:3) Bounded distortion.
In this subsection, we present a bounded distortionlemma for a sequence of contracting maps which permits us to control the growthof measure of iterations of measurable sets. The proof here is an adaptation ofthe classical argument to our setting. Let R > M . Lemma 3.4.
Let α, λ ∈ (0 , and C > . Then, there exists L > such that forany R < R , any n ∈ N , any sequence { x j } nj =0 in M , and any sequence { h j } nj =1 in Diff α loc ( M ) with h j : B ( x j − , R ) → h j ( B ( x j − , R )) satisfying h j ( x j − ) = x j , (cid:107) Dh j (cid:107) < λ , (cid:107) h j (cid:107) C α < C , and every pair of measurable sets S , S ⊆ B ( x , R ) ofpositive Lebesgue measure, L − Leb( S )Leb( S ) < Leb( h n ( S ))Leb( h n ( S )) < L Leb( S )Leb( S ) . Recall that h n := h n ◦ · · · ◦ h . TABLE LOCAL ERGODICITY 13
Proof.
Since there is an upper bound for the C α norm of the derivative of the expfunction on the balls of radius R on the whole manifold M , by replacing h j withexp x j ◦ h j ◦ exp − x j − : B R (0) → B R (0), one can assume that the maps are definedbetween open sets of R d . Now, it is enough to show that there exists L > S ⊆ B R (0),(14) L − | det D h n | Leb( S ) ≤ Leb( h n ( S )) ≤ L | det D h n | Leb( S ) . Since the sequence has bounded C α norm, the maps z (cid:55)→ log | det D z h j | are α -H¨older on B R (0) with some uniform constant, that is, there exists L (cid:48) > j ) such that for any j ≥ z, z (cid:48) ∈ B R (0), (cid:12)(cid:12)(cid:12) log (cid:12)(cid:12) det D z h j (cid:12)(cid:12) − log (cid:12)(cid:12) det D z (cid:48) h j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < L (cid:48) | z − z (cid:48) | α . For x ∈ B R (0) and j ≤ n , denote x j := h j ( x ). From the contraction property, | x j | ≤ (cid:16) sup z ∈ B R (0) (cid:107) D z h j (cid:107) (cid:17) | x | ≤ λ j | x | ≤ λ j R. Therefore, (cid:12)(cid:12)(cid:12) log | det D x h n || det D h n | (cid:12)(cid:12)(cid:12) = n − (cid:88) j =0 (cid:12)(cid:12)(cid:12) log | det D x j h j +1 | − | det D h j +1 | (cid:12)(cid:12)(cid:12) < L (cid:48) n − (cid:88) j =0 | x j | α ≤ L (cid:48) R α n − (cid:88) j =0 λ jα . Now, since λ < L := exp (cid:0) L (cid:48) R α ∞ (cid:80) j =0 λ jα (cid:1) < ∞ and so(15) L − | det D h n | ≤ | det D x h n | ≤ L | det D h n | . By the change of variable formula, (cid:90) S | det D x h n | dLeb( x ) ≤ L | det D h n | Leb( S ) . The proof of the other inequality in (14) is similar. (cid:3)
Infiltrated quasi-conformality.
In this subsection, we show that under hy-potheses (H0)-(H3), quasi-conformality of the derivatives at the origin leads to thequasi-conformality in a neighbourhood. Informally, the idea is that the contractingassumption forces the derivatives of long blocks in the nearby points to imitate thebehaviour of derivatives at the origin. The results of this subsection will not be usedin other parts of the paper and is included here for its own interest.
Theorem 3.5.
Let x ∈ M , { f i } ∞ i =1 be a sequence in Diff α ( M ) with bounded C α norm and quasi-conformal expanding at x . Then, there exist R > and θ > suchthat for any n ∈ N and any ball B ( y, r ) ⊆ B ( f n ( x ) , R ) , B ( f − n y, r n ) ⊆ f − n ( B ( y, r )) ⊆ B ( f − n ( y ) , θr n ) , where f − n := ( f n ◦ · · · ◦ f ) − and r n = rθ − | det D x f − n | /d . To prove Theorem 3.5, we first prove the following proposition.
Proposition 3.6.
Let
R, C > , κ ≥ > λ > λ > and α ∈ (0 , . Then, thereexist ξ > , κ > such that any sequence { h j } ∞ j =1 of maps satisfying hypotheses(H0)-(H3) is κ -conformal at every point of B ξ (0) .Proof. The proposition follows from a refinement of the proof of Lemma 3.3. For α <
1, one should take blocks of compositions and repeat the claims of the proofof Lemma 3.3. To avoid repeating the arguments, here we give a proof for C regularity, that is, for α = 1. For x ∈ B R (0), denote x i := h i ( x ), D i := D h i , δ i := | det D i | /d , ˜ D i := D x i − h i and E i := D i − ˜ D i . Observe that for n ∈ N , D h n − D x h n = D n · · · D − ˜ D n · · · ˜ D = n (cid:88) i =1 D n · · · D i +1 E i ˜ D i − · · · ˜ D . Hence,(16) (cid:107) D x h n − D h n (cid:107) ≤ n (cid:88) i =1 (cid:107) D n · · · D i +1 (cid:107) · (cid:107)E i (cid:107) · (cid:107) ˜ D i − · · · ˜ D (cid:107) . Since D n · · · D i +1 is κ -conformal, in view of (C2) (cid:48) , one gets that (cid:107) D n · · · D i +1 (cid:107) ≤ κ δ n · · · δ i +1 . On the other hand, (H1) implies (cid:107) ˜ D i − · · · ˜ D (cid:107) ≤ λ i − . Now, for θ > | x i − | ≤ θδ i − · · · δ | x | and so, by (H3), (cid:107)E i (cid:107) = | D x i − h i − D h i | ≤ Cθ | x i − | ≤ CθRδ i − · · · δ . Using (16), (cid:107) D x h n − D h n (cid:107) ≤ n (cid:88) i =1 CRκ δ n · · · δ i +1 δ i − · · · δ λ i − ≤ CRκ λ − (cid:16) n (cid:88) i =1 λ i − (cid:17) δ n · · · δ . Now, by the convergence of the series (cid:80) ∞ i =0 λ i − , there exists C > n ≥ x ∈ B R (0),(17) (cid:107) D x h n (cid:107) ≤ (cid:107) D h n (cid:107) + (cid:107) D x h n − D h n (cid:107) < C | det D h n | /d . By (15), one obtains that (cid:107) D x h n (cid:107) ≤ C L /d | det D x h n | /d . Thus, there exists κ = κ ( d, C L /d ) > D x h n is κ -conformal, as claimed. (cid:3) Proof of Theorem 3.5.
The proof is similar to the one of Theorem 3.1. Take x i , η, η, R as in the proof of that theorem. Define { y i } ni =0 by y n := y and y i − := f − i ( y i ). Byuniform expansion, one obtains that for any j < n , f − j +1 ◦ · · · ◦ f − n ( B ( y n , r )) ⊆ B ( y j , r ) ∩ B ( x j , R ) . Now, define ˆ f i := (exp − y i ◦ f i ◦ exp y i − ) | B r (0) and ˆ h i := ˆ f − n +1 − i | B r (0) . So, the conclu-sion follows from Lemma 3.2 for this sequence. Indeed, condition (H3) is guaranteedby Proposition 3.6. (cid:3) Remark . By Proposition 3.6, Lemma 3.4 and the results in the theory of quasi-conformal and quasi-symmetric maps, one can give another proof for Theorem 3.5.In fact, if a map is κ -conformal on its domain, then there exists a bound for theratio between outer and inner radii of the image of a ball (see [HK98, Section 4] and TABLE LOCAL ERGODICITY 15 [V¨ai89]). Then, the inner and outer radii can be estimated by means of estimatingthe volume and the bounded distortion lemma (Lemma 3.4).4.
Quasi-conformal dynamics
This section is devoted to the notion of covering property for derivatives. We dis-cuss its consequences in providing quasi-conformal orbit-branches and also sufficientconditions to ensure it.Throughout the section, M is a boundaryless Riemannian manifold of dimen-sion d . We consider the fiber bundle π : E ( M ) → M . Recall that for w =( x, ( v , . . . , v d )) ∈ E ( M ), (cid:107) w (cid:107) is defined by (cid:107) w (cid:107) = ( (cid:80) i | v i | ) , where | . | is thenorm induced by the Riemannian metric on T M .Furthermore, for a linear map T : W → W between finite dimensional vectorspaces endowed with inner products, we denote the Hilbert-Schmidt norm of T by (cid:107) T (cid:107) , defined by (cid:107) T (cid:107) := (cid:0) (cid:88) i | T e i | (cid:1) , where { e i } i is an orthonormal basis for W . In particular, for f ∈ Diff ( M ),and w = ( x, ( e , . . . , e d )) ∈ E ( M ), where x ∈ Dom( f ) and { e , . . . , e d } form anorthonormal basis for T x M , it follows that (cid:107) ˆ D x f (cid:107) = (cid:107) ˆ D f ( w ) (cid:107) .For linear isomorphisms T : W → W and S : W → W between d -dimensionalvector spaces, we will use the following properties. (cid:107) T (cid:107) ≤ (cid:107) T (cid:107) ≤ √ d (cid:107) T (cid:107) . (18) (cid:107) S ◦ T (cid:107) ≤ (cid:107) S (cid:107) . (cid:107) T (cid:107) and (cid:107) S ◦ T (cid:107) ≤ (cid:107) S (cid:107) . (cid:107) T (cid:107) . (19) If | det T | = 1 , then (cid:107) T − (cid:107) ≤ (cid:107) T (cid:107) d − . Therefore, T is (cid:107) T (cid:107) d -conformal . (20)4.1. Stable quasi-conformality.
In this subsection, we present a criterion for theexistence of quasi-conformal orbit-branches in pseudo-semigroup actions.A subset
W ⊆ E ( M ) is called bounded , if sup w ∈W (cid:107) w (cid:107) < ∞ . Theorem 4.1 (Quasi-conformality criterion) . Let V ⊆ M be an open set and F ⊆
Diff ( M ) . Then, the following are equivalent (a) There exists κ > such that for every x ∈ V , the pseudo-semigroup gener-ated by F ↓ V has a κ -conformal orbit-branch at x . (b) There exists a bounded subset
W ⊆ E ( M ) such that π ( W ) = V , and (21) W ⊆ (cid:91) f ∈F ( ˆ D f ) − ( W ) . Proof.
For the implication (a) ⇒ (b). For every x ∈ V , let { f i,x } ∞ i =1 be a sequencedefining a κ -conformal orbit-branch of F ↓ V at x . Consider an arbitrary orthonormalbasis { e i } i for T x M and let w x = ( x, e ) ∈ π − ( x ) be such that e = ( e , . . . , e d ). Notethat for any x ∈ V , (cid:107) w x (cid:107) = √ d . Then, the set W defined by W := (cid:91) x ∈ V (cid:91) i ≥ ˆ D f ix ( w x ) , satisfies the desired properties, where f ix = f i,x ◦ f i − ,x ◦ · · · ◦ f ,x . Indeed, π ( W ) = V and for any x ∈ V and i ∈ N , ˆ D x f ix is κ -conformal and so by (19), (cid:107) ˆ D f ix ( w x ) (cid:107) ≤ (cid:107) ˆ D x f ix (cid:107) . (cid:107) w x (cid:107) ≤ κ √ d. Therefore, sup w ∈W (cid:107) w (cid:107) ≤ κ √ d and so W is bounded. On the other hand, coveringproperty (21) follows immediately from the definition of W , since every element of W is of the following formˆ D f ix ( w x ) = ( ˆ D f i +1 ,x ) − ( ˆ D f i +1 x ( w x )) ∈ ( ˆ D f i +1 ,x ) − ( W ) . Next, in order to show (b) ⇒ (a), first denote H := sup w ∈W (cid:107) w (cid:107) . Fix w =( x, ( v , . . . , v d )) ∈ W . We first claim that there exists a sequence { f i } ∞ i =1 in F suchthat for any n ≥
1, ˆ D f n ( w ) ∈ W . The proof is by induction, assume that f , . . . , f n are defined satisfying the properties. Then,ˆ D f n ( w ) ∈ W ⊆ (cid:91) f ∈F ( ˆ D f ) − ( W ) . So, there is f n +1 ∈ F with ˆ D f n ( w ) ∈ ( ˆ D f n +1 ) − ( W ). Consequently, by the chainrule, ˆ D f n +1 ( w ) ∈ W . This finishes the proof of the claim.Consider an orthonormal basis { e , . . . , e d } for T x M and let T : T x M → T x M be the linear map with T ( e i ) = v i . Clearly, | det T | = 1, (cid:107) T (cid:107) = (cid:107) w (cid:107) ≤ H , andfor any n ≥ (cid:107) ˆ D x f n ◦ T (cid:107) = (cid:107) ˆ D f n ( w ) (cid:107) ≤ H . The last inequality holds sinceˆ D f n ( w ) ∈ W . Using (18) and (20), one obtains (cid:107) ˆ D x f n (cid:107) ≤ (cid:107) ˆ D x f n ◦ T (cid:107) . (cid:107) T − (cid:107) ≤ (cid:107) ˆ D x f n ◦ T (cid:107) . (cid:107) T (cid:107) d − ≤ H d . Again, by (20), this implies that the sequence { f i } ∞ i =1 is κ -conformal at x for κ = H d . (cid:3) Corollary 4.2 (Stable quasi-conformality) . Let V ⊆ M be an open set and F ⊆
Diff ( M ) . Assume that W is an open subset of E ( M ) with compact closure satis-fying π ( W ) = V and (22) W ⊆ (cid:91) f ∈F ( ˆ D f ) − ( W ) . Then, there exists κ > such that for any family ˜ F in a C neighbourhood of F , IFS( ˜
F ↓ V ) has a κ -conformal orbit-branch at every point of V .Proof. By the compactness of W and the openness of W , the same covering property(22) holds for any family ˜ F in a small C neighbourhood of F . Hence, the conclusionfollows from Theorem 4.1. Moreover, the proof of Theorem 4.1 shows that κ =(sup w ∈W (cid:107)W(cid:107) ) d only depends on W . (cid:3) Remark . For oriented manifolds and orientation-preserving maps, one can workwith the following alternative fiber bundle instead of E ( M ), { ( x, v ) : x ∈ M, v ∈ ( T x M ) d , and ω | x ( v ) = 1 } , where ω is the volume form on M induced from the Riemannian metric compatiblewith the orientation, and ω | x is its restriction to T x M . This defines a SL( d, R ) fiberbundle over M which is a quotient of E ( M ) by an involution. TABLE LOCAL ERGODICITY 17 xV Figure 2.
Covering condition (22): diversity of maps correspondingto x ∈ V , i.e. maps in the subfamily F x . Reinterpretation of the covering property.
Let us discuss the meaning of coveringcondition (22). Indeed, it is equivalent to the following (see Figure 2): • For any x ∈ V , there exists F x ⊆ F such that (i) f ( x ) ∈ V , for f ∈ F x , (ii) W x ⊆ (cid:83) f ∈F x ( ˆ D f ) − ( W f ( x ) ), where W x := π − ( x ) ∩ W .Roughly speaking, this means that over every point x there are several maps in thefamily F with diverse directions of contraction and expansion for the normalizedderivative that allows to obtain covering (ii) which yields the quasi-conformalityalong an orbit-branch of x .In the case of M = R d , E ( R d ) is isomorphic to the trivial fiber bundle R d × SL ± ( d, R ) and the action of ˆ D f on fibers in nothing but the product of matrices inSL ± ( d, R ). More precisely, for f ∈ Diff ( R d ), ˆ D f maps ( x, A ) ∈ R d × SL ± ( d, R ) to( f ( x ) , A x A ) where A x := ˆ D x f ∈ SL ± ( d, R ). In other words, the covering condition(22) will be reduced to finding a map f ∈ F such that A x A is in the boundedset W f ( x ) . Observe that R d × SL( d, R ) is invariant under ˆ D x f for an orientation-preserving f ∈ Diff ( R d ). In particular, it is enough to satisfy the covering condi-tion for W ⊆ R d × SL( d, R ). The next subsection gives a method of doing that.4.2. Sufficient conditions for covering: algebraic method.
Following the dis-cussion above, we investigate the covering property for the action of SL( d, R ) onitself.Recall that the sequence { D i } ∞ i =1 in SL( d, R ) is quasi-conformal if and only if theset { D n · · · D : n ∈ N } is bounded. Also, the sequence { D i } ∞ i =1 is κ -conformal, iffor any n ∈ N , D n · · · D is κ -conformal. A A A A Figure 3.
A quasi-conformal sequence of matrices
Definition 4.4.
For κ ≥ D ⊆
SL( d, R ), we say (cid:104)D(cid:105) + has a κ -conformalbranch, if there exists a κ -conformal sequence in D . In addition, when D is finite,we say it robustly has κ -conformal branches, if for every ˜ D in a neighbourhood of D , (cid:104) ˜ D(cid:105) + has κ -conformal branches.For a finite subset D ⊆
SL( d, R ), if (cid:104)D(cid:105) + is not compact, typical branches arenot quasi-conformal in a probabilistic sense. More precisely, by assigning positiveprobabilities to the elements of D , almost every branch with respect to the productmeasure on D N is not κ -conformal for any κ > n ∈ N , for almost every n -tuple D ∈
SL( d, R ) n (w.r.t the natural measure), the Lyapunov spectrum associated to therandom product of elements of D is non-degenerate provided that we assign positiveweights to the elements of D . In such case, for almost every branch with respectto the product measure on D N , the norm of the products diverges exponentially toinfinity (cf. [Via14]).Nevertheless, the following lemma which is an analogue of Theorem 4.1 for the ac-tion of SL( d, R ) on itself expresses that the covering condition leads to the existenceof bounded branches for a finitely generated semigroup. Moreover, Corollary 4.6 forthis special case evidently implies that given κ >
1, any open neighbourhood U ofthe identity in SL( d, R ) with compact closure has a finite subset D with robustly κ -conformal branches. Lemma 4.5.
Let
D ⊆
SL( d, R ) be a finite set. Then, there exists κ > such that (cid:104)D(cid:105) + has a κ -conformal branch if and only if U ⊆ D − U for some U ⊆
SL( d, R ) with compact closure. Moreover, if U is open, (cid:104)D(cid:105) + robustly has a κ -conformalbranch for some κ > .Proof. Let V = R d and W = R d × U . Considering the natural action of SL( d, R )on R d , D can be seen as a family in Diff( R d ). Then, the first part of the lemmafollows from Theorem 4.1. The second part is similar. Note that U ⊆ D − U impliesthat for any family ˜ D sufficiently close to D , U ⊆ ˜ D − U holds and so the secondconclusion is again a consequence of Theorem 4.1. (cid:3) We can deduce the following corollary from Lemma 4.5.
Corollary 4.6.
For any κ > and any open set U ⊆
SL( d, R ) containing theidentity, there is a finite set D ⊆ U such that (cid:104)D(cid:105) + robustly has a κ -conformalbranch.Proof. Consider small open neighbourhood V of the identity with compact closuresuch that every element of V is κ -conformal. Clearly, V ⊆ U − V = (cid:83) u ∈U u − V . Bythe compactness of V , one can choose a finite set D ⊆ U with
V ⊆ D − V . Thus, theconclusion follows from Lemma 4.5. (cid:3) Remark . Lemma 4.5 and Corollary 4.6 can be stated for the existence of boundedbranches in an abstract setting for more general topological groups. However, dueto the applications for the derivatives of smooth maps, in this paper the discussionis restricted to the special cases of SL( d, R ) and R d . Explicit construction for covering.
Corollary 4.6 is existential and does not introduceelements of D explicitly and even does not give any estimate for the cardinality ofthis set. The next lemma guaranties the covering of small open sets with d elements. TABLE LOCAL ERGODICITY 19
Lemma 4.8.
For any neighbourhood U of the identity in SL( d, R ) , there exists anopen set U ⊆ U and a finite set D ⊆ U with d elements such that U ⊆ D − U . Denote by sl ( d, R ) the Lie algebra of SL( d, R ) which consists of all d × d real ma-trices whose traces are equal to zero. Furthermore, here exp denotes the exponentialfunction from a neighbourhood of the zero matrix in sl ( d, R ) to a neighbourhoodof the identity in SL( d, R ) which verifies the Baker-Campbell-Hausdorff formula. Itwill be used in the proof of the next lemma. Lemma 4.9.
For any neighbourhood U of the zero matrix in sl ( d, R ) , there existan open subset U and a finite subset D = { w , . . . , w d } of U such that exp ( U ) ⊆ exp ( D ) − exp ( U ) . For the proof, we use the following notation. For a connected open set U ⊆ R N and small t >
0, denote(23) U t := { x ∈ U : d ( x, ∂U ) > t } . In addition, the following observation will be used in the proof of Lemma 4.9.
Lemma 4.10.
Let U ⊆ R N be an open set with U homeomorphic to the closed unitdisk. Suppose that ϕ , ϕ are two continuous maps defined on a neighbourhood of U and are homeomorphisms onto their images. If there exists t > such that for any x ∈ U , | ϕ ( x ) − ϕ ( x ) | < t , then (cid:0) ϕ ( U ) (cid:1) t ⊆ ϕ ( U ) . The proof of this lemma is based on considering an affine homotopy between ϕ (cid:12)(cid:12) ∂U and ϕ (cid:12)(cid:12) ∂U , then showing that the image of homotopy does not intersect (cid:0) φ ( U ) (cid:1) t .Further details are left to the reader. Proof of Lemma 4.9.
Suppose v , . . . , v N are N points in R N − with | v i | = 1 suchthat the origin is contained in their convex hull. Denote the interior of their convexhull by ∆. Clearly, for any small positive t ∈ R , ∆ ⊆ (cid:83) Ni =1 (∆ + tv i ). Since ∆is compact and ∆ + tv j ’s are all open, one can find sufficiently small c > ⊆ (cid:83) Ni =1 (∆ + tv i ) c (following the notation introduced in (23)). As the wholeconstruction is invariant under homothety, for any r > r ∆ = r ∆ ⊆ N (cid:91) i =1 ( r ∆ + trv i ) cr . By identification of sl ( d, R ) with R N − for N = d , v i ’s can be seen as d × d matrices with zero trace. For any 1 ≤ j ≤ N , and any sufficiently small r >
0, wedefine u ( r ) j : r ∆ → sl ( d, R ) as the following u ( r ) j ( x ) = exp − (cid:0) exp ( trv j ) exp ( x ) (cid:1) . By the Baker-Campbell-Hausdorff formula for the Lie groups (see for instance [Ros06]), exp − (cid:0) exp ( trv j ) exp ( x ) (cid:1) = x + trv j + ε j ( x, tr ) , where ε j satisfies | ε j ( x, s ) | < E j | x | s for some E j >
0. Thus, whenever r < min ≤ j ≤ N { cE j t } , for 1 ≤ j ≤ N one has,sup x ∈ r ∆ (cid:12)(cid:12) u ( r ) j ( x ) − ( x + trv j ) (cid:12)(cid:12) = sup x ∈ r ∆ | ε j ( x, rt ) | ≤ E j rt sup x ∈ r ∆ | x | ≤ E j r t < cr. By Lemma 4.10, for every 1 ≤ j ≤ N , r ∆ + trv j ⊆ h ( r ) j ( r ∆) and so by (24), r ∆ ⊆ N (cid:91) j =1 u ( r ) j ( r ∆) . Finally, as the exponential map is a diffeomorphism on u ( r ) j ( r ∆) and on r ∆,(25) exp ( r ∆) = exp ( r ∆) ⊆ exp (cid:16) N (cid:91) j =1 u ( r ) j ( r ∆) (cid:17) = N (cid:91) j =1 (cid:16) exp (cid:0) u ( r ) j ( r ∆) (cid:1)(cid:17) . Meanwhile, by the definition of u ( r ) j , we have exp ( u ( r ) j ( r ∆)) = exp ( trv j ) exp ( r ∆) . Thus, the conclusion of Lemma 4.9 follows from (25) by taking w j := − trv j and U := r ∆ for sufficiently small r . (cid:3) Remark . One can start with any simplex ∆ in sl ( d, R ) (cid:39) R d − containing theorigin in the interior to provide an explicit formula for D in Lemma 4.9. Proof of Lemma 4.8.
Consider an open neighbourhood U of the zero matrix in sl ( d, R ) such that the exp function is a diffeomorphism in a neighbourhood of U and exp( U ) ⊆ U . Now, applying Lemma 4.9 to U , one can get U , D . Then, U := exp ( U ) and D := exp ( D ) satisfy the conditions of Lemma 4.8 and the proof isfinished. (cid:3) Sufficient conditions for covering: analytic method.
This subsectionstates another approach to derive a sufficient condition leading to the covering prop-erty with respect to a bounded subset of E ( M ). Lemma 4.12.
Let
F ⊆
Diff ( M ) and V ⊆ M be an open set with compact closure.Assume that for any ( x, v ) ∈ T M with x ∈ V , there exists f ∈ F satisfying f ( x ) ∈ V and (cid:107) ˆ D x f | v ⊥ (cid:107) < . Then, there exists an open set W ⊆ E ( M ) withcompact closure such that π ( W ) = V and W ⊆ (cid:83) f ∈F ( ˆ D f ) − ( W ) .Proof. Since the set { ( x, v ) ∈ T M : x ∈ V } is compact, one can find a finite subset F ⊆ F and (cid:15) > x, v ) ∈ T M with x ∈ V , there exists f ∈ F satisfying f ( x ) ∈ V and (cid:107) ˆ D x f | v ⊥ (cid:107) < − (cid:15) .Denote Θ := max {(cid:107) ˆ D x f (cid:107) : f ∈ F , x ∈ V } . Let H ≥ Θ (cid:112) d/(cid:15) be a real number.Then, consider W ⊆ E ( M ), defined by(26) W := { w ∈ E ( M ) : π ( w ) ∈ V and (cid:107) w (cid:107) < H } . Clearly, W is an open subset with compact closure and π ( W ) = V . It is enoughto prove that (22) holds for the subfamily F and W defined by (26). Consider w = ( x, ( v , . . . , v d )) ∈ W . So, (cid:107) w (cid:107) ≤ H . If (cid:107) w (cid:107) < Θ − H and f ∈ F with f ( x ) ∈ V , then by (19), (cid:107) ˆ D f ( w ) (cid:107) = (cid:0) d (cid:88) i =1 | ˆ D x f ( v i ) | (cid:1) ≤ (cid:107) ˆ D x f (cid:107) . (cid:107) w (cid:107) < H. Thus, ˆ D f ( w ) ∈ W and consequently, w ∈ (cid:83) f ∈F ( ˆ D f ) − ( W ). Now, assume that (cid:107) w (cid:107) ∈ [Θ − H, H ]. Without loss of generality, assume that | v | ≥ · · · ≥ | v d | .Clearly, | v d | ≤
1. So, | v d | Θ (cid:112) d/(cid:15) ≤ Θ (cid:112) d/(cid:15) ≤ Θ − H ≤ (cid:107) w (cid:107) = (cid:112) | v | + · · · + | v d | ≤ √ d | v | . TABLE LOCAL ERGODICITY 21
This implies that Θ | v d | ≤ (cid:15) | v | . Let v be a vector perpendicular to v , . . . , v d − .By assumption, one can choose f ∈ F with f ( x ) ∈ V and (cid:107) ˆ D x f | v ⊥ (cid:107) < − (cid:15) . Then, (cid:107) ˆ D f ( w ) (cid:107) = d (cid:88) i =1 | ˆ D x f ( v i ) | < (1 − (cid:15) ) ( | v | + · · · + | v d − | ) + Θ | v d | ≤ (1 − (cid:15) ) ( | v | + · · · + | v d − | ) + (cid:15) | v | < (cid:107) w (cid:107) ≤ H . Therefore, w ∈ (cid:83) f ∈F ( ˆ D f ) − ( W ). (cid:3) Quasi-conformal blenders
This section is devoted to a new mechanism/phenomenon that we call quasi-conformal blender. For pseudo-semigroup actions, the quasi-conformal blender guar-antees the existence of quasi-conformal expanding orbit-branches at every point insome region which leads to the stable local ergodicity. Here, we present the proofof Theorem E and its variants using the results of the previous sections.Throughout the section, M is a boundaryless, not necessarily compact, smoothRiemannian manifold of dimension d . Recall that x ∈ M is a (Lebesgue) densitypoint of a measurable set S ⊆ M iflim r → Leb (cid:0) S ∩ B ( x, r ) (cid:1) Leb (cid:0) B ( x, r ) (cid:1) = 1 . It is well-known that a measurable set S ⊆ M is equal to the set of its Lebesguedensity points, up to a zero measure set. Moreover, for f ∈ Diff ( M ), f ( x ) is adensity point of f ( S ), provided that x ∈ Dom( f ) is a density point of S . Definition 5.1 ( ρ -ergodic) . Let V ⊆ M be an open set with compact closure. For ρ > F ⊆
Diff ( M ), we say IFS( F ↓ V ) is ρ -ergodic (w.r.t. Leb.), if the setof density points of every measurable F ↓ V -invariant subset of V either is empty orcontains a ball of radius ρ .Clearly, this definition is equivalent to say that every measurable F ↓ V -invariantset of positive measure in V contains a ball of radius ρ , up to a set of zero Lebesguemeasure.5.1. From quasi-conformal expansion to local ergodicity.
In this subsection,we state a technical lemma about the quasi-conformal expanding sequences. It willbe used in both local and global settings for proving local ergodicity.
Lemma 5.2 (Local ergodicity) . Let η, κ > and ρ, α, C > . Also, let { x j } ∞ j =0 bea sequence in M and { f j } ∞ j =1 with f j : B ( x j − , ρ ) → f j ( B ( x j − , ρ )) be a sequencein Diff α loc ( M ) satisfying m ( Df j ) > η and f j ( x j − ) = x j . Then, (a) for any open neighbourhood U ⊆ B ( x , ρ ) of x , there exists n ∈ N with B ( x n , ρ ) ⊆ f n ( U ) . (b) If in addition, the set { x , x , . . . } is bounded, the sequence { f j } ∞ j =1 is κ -conformal at x , and (cid:107) Df j (cid:107) C α < C , then for any measurable set S ⊆ B ( x , ρ ) with density point at x , there exists n ∈ N such that (cid:83) i ∈ N f i ( S ) contains B ( x n , ρ ) , up to a set of zero Lebesgue measure. Proof.
We prove assertions (a) and (b) separately.
Proof of (a) . For every i ≥
1, let s i be the largest positive number in (0 , ρ ] satisfying B ( x i , s i ) ⊆ f i ( U ). Since m ( Df i | B ( x i − ,ρ ) ) > η , if for some i ≥ s i < ρη − , then s i +1 > ηs i , and if s i ≥ ρη − , then s i +1 = ρ . Hence, there is n ≥ s n = ρ . Thisfinishes the proof of part (a). Note that for this part of the lemma, we only needthe sequence { f i } to be C -regular. Proof of (b). By applying Theorem 3.1 to this sequence, one can find ξ > θ > j ∈ N , there is r j > f j ( B ( x , r j )) ⊆ B ( x j , ξ ) ⊆ f j ( B ( x , θr j )) , and lim j →∞ r j = 0. For the rest of the proof, fix ξ and assume that ξ < ρ . Denoteˆ S := (cid:83) i ≥ f i ( S ). Since x is a density point of S , for any (cid:15) >
0, there exists j ∈ N such that whenever j > j , Leb (cid:0) B ( x , θr j ) \ ˆ S (cid:1) Leb (cid:0) B ( x , θr j ) (cid:1) < (cid:15). There exists σ > r, r (cid:48) ∈ (0 , ρ ),Leb( B ( x , r ))Leb( B ( x , r (cid:48) )) ≤ σ ( rr (cid:48) ) d . Denote f − j := f − ◦ · · · ◦ f − j . Then, by (27),Leb (cid:0) f − j (cid:0) B ( x j , ξ ) \ ˆ S ) (cid:1) Leb (cid:0) f − j ( B ( x j , ξ )) (cid:1) ≤ Leb (cid:0) B ( x , θr j ) \ ˆ S (cid:1) Leb (cid:0) B ( x , θr j ) (cid:1) · Leb (cid:0) B ( x , θr j ) (cid:1) Leb (cid:0) f − j ( B ( x j , ξ )) (cid:1) ≤ Leb (cid:0) B ( x , θr j ) \ ˆ S (cid:1) Leb (cid:0) B ( x , θr j ) (cid:1) · Leb (cid:0) B ( x , θr j ) (cid:1) Leb( B ( x , r j ) (cid:1) < (cid:15)σθ d . It follows from Lemma 3.4 that for some
L > (cid:0) B ( x j , ξ ) \ ˆ S (cid:1) Leb (cid:0) B ( x j , ξ ) (cid:1) < L Leb (cid:0) f − j ( B ( x j , ξ ) \ ˆ S ) (cid:1) Leb (cid:0) f − j ( B ( x j , ξ )) (cid:1) <(cid:15)σLθ d . Thus, for any j > j ,(28) Leb (cid:0) ˆ S ∩ B ( x j , ξ ) (cid:1) Leb (cid:0) B ( x j , ξ ) (cid:1) > − (cid:15)σLθ d . Now, since (cid:15) was arbitrary, by (28), the density of ˆ S in B ( x j , ξ ) tends to 1 (as j → ∞ ). Then, for each accumulation point y of the bounded sequence { x i } ∞ i =1 , ˆ S contains B ( y , ξ ) up to a set of zero Lebesgue measure. Next, take a sufficientlylarge i such that x i ∈ B ( y , ξ ). This implies that ˆ S contains an open neighbourhood U of x i , up to a set of zero Lebesgue measure. Then, by part (a), there exists n > i such that B ( x n , ρ ) ⊆ f n − i ( U ). Finally, since diffeomorphisms maps sets of zeroLebesgue measure to sets of zero Lebesgue measure, ˆ S contains B ( x n , ρ ), up to aset of zero Lebesgue measure. (cid:3) Lemma 5.2 has the following global consequence which can be seen as a general-ization of Theorem 1.1.
TABLE LOCAL ERGODICITY 23
Theorem 5.3.
Let M be a closed manifold and F ⊆
Diff α ( M ) be finite. Supposethat there exist η, κ > such that IFS( F ) has a κ -conformal η -expanding orbit-branch at every point. Then, IFS( F ) is ρ -ergodic for some ρ > . In particular, theaction of a group G ⊆ Diff ( M ) is ergodic, if it is minimal and F ⊆ G .Proof. Since F is finite, there exist ρ > η (cid:48) ∈ (1 , η ) such that whenever m ( D x f ) > η , for some x ∈ M and f ∈ F , then m ( Df | B ( x,ρ ) ) > η (cid:48) . Indeed, if C := max f ∈F (cid:107) f (cid:107) C α , then for any x, y ∈ M , | m ( D x f ) − m ( D y f ) | ≤ Cd ( x, y ) α . where d ( ., . ) denotes the distance on M . So, m ( D x f ) > η implies that m ( D y f ) > η (cid:48) ,provided that d ( x, y ) α < ρ := C − ( η − η (cid:48) ).For f ∈ F , denote U f := { x ∈ M : m ( D x f ) > η (cid:48) } and ˆ f := f | U f . Also, letˆ F := { ˆ f : f ∈ F } . Consider a measurable ˜ F ↓ V -invariant subset S of positivemeasure and pick x to be a density point of S . Then, the κ -conformal η -expandingorbit-branch of IFS( ˆ F ) at x , provides a sequence of maps satisfying the assumptionsof Lemma 5.2 and the conclusion follows from this lemma.For the second part, denote the set of density points of S by S • . Since F ⊆ G , itfollows from the first part that S • contains an open ball B . We claim that S • = M .Indeed, Leb( B \ S ) = 0 implies that for every g ∈ G , Leb( g ( B ) \ g ( S )) = 0. Then,by the invariance of S , Leb( g ( B ) \ S ) = 0 and in particular, g ( B ) ⊆ S • . On theother hand, by the minimality assumption, (cid:83) g ∈ G g ( B ) = M . This proves the claimand finishes the proof of the theorem. (cid:3) Proof of Theorem E.
We will prove the following theorem which in particularimplies Theorem E.
Theorem 5.4 (Quasi-conformal blender) . Let
F ⊆
Diff ( M ) . Let W ⊆ E ( M ) bean open set with compact closure and V := π ( W ) . Assume that for any w ∈ W ,there exists f ∈ F satisfying (i) ˆ D f ( w ) ∈ W , (ii) m ( D x f ) > , where x = π ( w ) .Then, there exist real numbers ρ > and κ > such that for every ˜ F ⊆
Diff ( M ) in a C -neighbourhood of F , (a) for any x ∈ V , IFS( ˜
F ↓ V ) has an orbit-branch which is κ -conformal at x , (b) IFS( ˜ F ↓ V ) is ρ -ergodic.In addition, if M is compact, IFS( F ) and IFS( F − ) are minimal, then IFS( F ) isstably ergodic in Diff ( M ) and IFS( F − ) is C -robustly minimal.Proof of Theorem E. It follows from (1) that for every w ∈ W , there exists f ∈ F with ˆ D f ( w ) ∈ W . Now, since every element of F is expanding, one has m ( D x f ) > x = π ( w ). So, Theorem E follows from part (b) of Theorem 5.4. (cid:3) Remark . If a family F and a set W satisfy the assumptions of Theorem 5.4, thenfor w ∈ W , one can get a subfamily F w ⊆ F consisting of all elements satisfying(i)-(ii). Then, for any element f ∈ F w , restrict its domain to a small neighborhoodof w such that the restricted map is expanding. Let F (cid:48) be the family of all theserestricted diffeomorphisms. It is clear that F (cid:48) and W satisfy the assumption of V x Images of a neighbourhood of x One-dimensional localized maps
Figure 4.
Families of expanding maps satisfying the covering con-dition in Theorem E.Theorem E. In other words, one can deduce part (b) in Theorem 5.4 from TheoremE, and vice-versa.
Remark . As a matter of fact, the C stability in Theorem 5.4 and its consequencesin this paper are valid in a substantially stronger form that allows to perturb thefamily at each step of iterations. We do not discuss the details in this paper. Cf.[HN14], where this notion of strong stability has been introduced. Proof of Theorem 5.4.
Since the assumptions (i)-(ii) are stable under small pertur-bation of f , in the C topology, and w ∈ E ( M ), one can deduce that • For every w ∈ W there exist (cid:15) ( w ) > f w ∈ F such that if B w denotesthe open ball of radius (cid:15) ( w ) with center w ∈ E ( M ), then for any ˜ f sufficientlyclose to f w , in the C topology, π ( B w ) ⊆ Dom( ˜ f ) and for any w (cid:48) ∈ B w ,(1) ˆ D ˜ f ( w (cid:48) ) ∈ W ,(2) m ( D x (cid:48) ˜ f ) >
1, where π ( w (cid:48) ) = x (cid:48) . Proof of (a) . Let B (cid:48) w be the open ball of radius of (cid:15) ( w ) with center w . Then,by the compactness of W , there is a finite subset { w , . . . , w k } of W such that W ⊆ (cid:83) i B (cid:48) w i . By Corollary 4.2, this implies the existence of κ > F sufficiently close to F , in the C topology, IFS( ˜ F ↓ V ) has a κ -conformalorbit-branch at every point of V . So, the proof of part (a) is finished. Proof of (b) . Let ρ > (cid:83) i B (cid:48) w i for W . Then, there is η > w ∈ W , there exists 1 ≤ i ≤ k such that for any ˜ f sufficiently close to f w i , in the C topology, B ( x, ρ ) ⊆ Dom( ˜ f ),˜ f ( B ( x, ρ )) ⊆ V , and m ( D ˜ f | B ( x,ρ ) ) > η , where x = π ( w ).Denote ˜ F := { ˜ f w , . . . , ˜ f w k } . Let α > F ⊆ Diff α loc ( M ). Considera measurable ˜ F ↓ V -invariant set S of positive measure and let x be a density pointof S . Family F satisfies the assumptions (i)-(ii) of the theorem, so it follows frompart (a) that IFS( ˜ F | V ) has a κ -conformal orbit-branch { x i } ∞ i =0 at x . Let { ˜ f i } ∞ i =1 be the sequence of maps providing this orbit-branch, namely f i ( x i − ) = x i for every i ∈ N . Next, our aim is to apply Lemma 5.2 to this sequence. The problem is that x may be close to the boundary of V and B ( x , ρ ) (cid:54)⊆ ˜ f ↓ V . To avoid this challenge, TABLE LOCAL ERGODICITY 25 we remove the first term of the sequences and consider { ˜ f i } ∞ i =2 and { x i } ∞ i =1 , whichby means of above arguments satisfy the assumptions of Lemma 5.2. Note that x = ˜ f ( x ) is also a density point of the invariant set S . By part (b) of Lemma5.2, there is ξ > y of { x i } ∞ i =1 such that S contains B ( y , ξ ), up to a set of measure zero. Since for any i ≥ B ( x i , ρ ) ⊆ V and so B ( y , ρ ) ⊆ V . Similarly, an expanding orbit-branch { y i } ∞ i =0 of IFS( ˜ F ↓ V ) at y canbe provided in such a way that for any i ≥ B ( y i , ρ ) ⊆ V . Using part (a) of Lemma5.2 for open set U = B ( y , ξ ), one can conclude that the set of density points of S contains B ( y n , ρ ) for some n ≥
0. This finishes the proof of part (b).Next, we go to the proof of global results. We assume that M is compact,IFS( F ) and IFS( F − ) are minimal on M . By the minimality, (cid:104)F (cid:105) + ( B ( x, ρ (cid:48) )) = (cid:104)F − (cid:105) + ( B ( x, ρ (cid:48) )) = M where ρ (cid:48) < ρ . Suppose that (cid:83) x ∈ X B ( x, ρ (cid:48) ) = M for somefinite set X ∈ M . On the other hand, by the compactness of M , there exists a finiteset F ⊆ (cid:104)F (cid:105) + with F ( B ( x, ρ (cid:48) )) = F − ( B ( x, ρ (cid:48) )) = M , for any x ∈ X . Considersmall perturbation ˜ F of F and denote the elements of ˜ F corresponding to the family F by ˜ F . If the perturbation is sufficiently small, then for every ball B of radius ρ ,(29) ˜ F ( B ) = ˜ F − ( B ) = M. Proof of stable ergodicity.
Let S be a measurable ˜ F -invariant subset of M withpositive measure. Consider an arbitrary ball B ⊆ V of radius ρ . By (29), S ⊆ ˜ F − ( B ) = M and so Leb( S ∩ B ) >
0. Then, by ρ -ergodicity of IFS( ˜ F ↓ V ), theset of density points of S contains some ball B ⊆ M of radius ρ . Finally, by (29),˜ F ( B ) = M and this implies that S contains M , up to a set of measure zero. Proof of robust minimality.
The proof of robust minimality is similar to the oneof stable ergodicity. Consider an open set U ⊆ M , and ball B as above. Let˜ F ⊆
Diff ( M ) be a family in a small C -neighbourhood of F satisfying (29).Again, (29) implies that ˆ U := (cid:104) ˜ F (cid:105) + ( U ) intersects B ⊆ V . Consider z ∈ ˆ U ∩ B .Then, similar to the arguments of part (b), one can find an expanding orbit-branchof IFS( ˜ F ↓ V ) at z and use part (a) of Lemma 5.2 to deduce that ˆ U contains a ball B ⊆ V of radius ρ . Finally, again by (29), M = ˜ F ( B ) ⊆ ˆ U . Since open set U wasarbitrarily chosen, it follows that the orbit of every point under IFS( ˜ F − ) is densein M . This means IFS( ˜ F − ) is minimal. (cid:3) Contracting quasi-conformal blenders.
In this subsection, we present avariant of Theorem E in the setting of contracting maps. This is useful to get localergodicity and local minimality simultaneously.
Theorem 5.7 (Contracting quasi-conformal blender) . Let
G ⊆
Diff ( M ) and W ⊆E ( M ) be a bounded open set with V := π ( W ) . Let U ⊆ M be an open set withcompact closure containing π ( W ) . Also, assume that (i) W ⊆ (cid:83) g ∈G ˆ D g ( W ) , (ii) for every g ∈ G , U ⊆ Dom( g ) and g ( U ) ⊆ U . (iii) every g ∈ G is uniformly contracting on U .Then, for every ˜ G ⊆
Diff ( M ) in a C -neighbourhood of G , any measurable ˜ G↓ U -invariant set of positive measure in U , has full measure in V . Also, for every x ∈ U , (cid:104)G↓ U (cid:105) + ( x ) is dense in V . Proof.
By the compactness of W and the openness of W , one can replace G with a fi-nite subfamily satisfying the assumptions. So, we may assume that G ⊆
Diff α loc ( M ),for some α >
0, is finite.Let Λ := ∩ n G n ( U ) be the Hutchinson attractor of IFS( G| U ). It follows from[Hut81] that for any x ∈ U , (cid:104)G(cid:105) + ( x ) is dense in Λ (cf. [HN14, Theorem 4.2]). Onthe other hand, for any n ∈ N , V ⊆ G ( V ) ⊆ G n ( V ) ⊆ G n ( U ) , and thus V ⊆ Λ. This in particular implies that for every x ∈ U , (cid:104)G↓ U (cid:105) + ( x ) isdense in V and consequently, every G↓ U -invariant subset of positive measure in U ,intersects V in a set of positive measure.Note that the family G − ↓ U is uniformly expanding on V . On the other hand,in view of assumption (i), G − satisfies the covering property (1) for W . Now, let S be a measurable G↓ U -invariant set with Leb( S ) >
0. It follows from above thatLeb( S ∩ V ) >
0. Suppose that Leb( S ∩ V ) < Leb( V ). Then, S (cid:48) := V \ S is G − ↓ V -invariant and Leb( S (cid:48) ) >
0. By applying Theorem 5.4 to the family G − ↓ V , one getsthat the set of density points of S (cid:48) contains an open ball B . Let x be a density pointof S . By above arguments, some element of (cid:104)G↓ U (cid:105) + maps x to B . This contradictsto the invariance of S under G↓ U and shows that S must have full measure in V .Finally, note that after small perturbation of the generators of G , all the assump-tions of the theorem hold. That is, the conclusion follows for the family ˜ G sufficientlyclose to G , following similar arguments above. (cid:3) Remark . It follows from the proof of Theorem 5.7 that one can replace theassumption (i) with V ⊆ (cid:83) g ∈G g ( V ) and deduce the density of (cid:104)G(cid:105) + ( x ) in V for any x ∈ V .5.4. Other statements on quasi-conformal blenders.
In this subsection, westate an analogue of Theorem E. First note that if V ⊆ M is completely inside anopen chart of M , one can use the coordinating map to translate the problem to anopen subset of the Euclidean space. In this case, one can get the following statementin view of Theorem 5.4. Theorem 5.9.
Let
F ⊆
Diff ( R d ) be a family of orientation-preserving maps. Let V ⊆ R d be a bounded open set and U ⊆
SL( d, R ) be a bounded open neighbourhoodof the identity. Assume that for any x ∈ V , there exists F x ⊆ F such that (i) f ( x ) ∈ V , for f ∈ F x , (ii) U ⊆ (cid:83) f ∈F x ( ˆ D x f ) − U , (iii) m ( D x f ) > , for f ∈ F x .Then, there exists ρ > such that for every ˜ F ⊆
Diff ( M ) in a C -neighbourhoodof F , IFS( ˜
F ↓ V ) is ρ -ergodic. Note that V is not necessarily connected in Theorem 5.9. This yields a flexibletool to get a result similar to Theorem E for every relatively compact open set inmanifolds using local charts.Let M be a boundaryless manifold of dimension d with a finite atlas of charts { ( W i , h i ) } i ∈ I on M such that for any i ∈ I , h i : W i → h i ( W i ) is a C diffeomorphismand h i ( W i ) is an open disk in R d . Assume that h i ( W i ) ∩ h j ( W j ) = ∅ if i (cid:54) = j . For i ∈ I , let W (cid:48) i be an open set such that W (cid:48) i ⊆ W i and M = (cid:83) i ∈ I W (cid:48) i . Now, for TABLE LOCAL ERGODICITY 27 an open set V ⊆ M with compact closure, denote V ∗ := (cid:83) i ∈ I h i ( V ∩ W (cid:48) i ). Notethat V ∗ ⊆ R d is an open set with compact closure. Let F ⊆
Diff ( M ) and denote F ∗ ⊆ Diff ( R d ), F ∗ := { h j ◦ f ◦ h − i : f ∈ F ∪ { Id } , i, j ∈ I such that x ∈ W i and f ( x ) ∈ W j } . It is easy to see that the dynamical properties of F are translated to the ones of F ∗ , and vice-versa. Then, we get the following. Theorem 5.10.
Let F , F ∗ , V, V ∗ be as above. Let also U ⊆
SL( d, R ) . If F ∗ , V ∗ , U satisfies the assumptions of Theorem 5.9, then there exists ρ > such that for every ˜ F ⊆
Diff ( M ) in a C -neighbourhood of F , IFS( ˜
F ↓ V ) is ρ -ergodic.In addition, if M is compact, and the action of (cid:104)F (cid:105) is minimal, then it is C -stablyergodic and C -robustly minimal.Proof. The first part is an immediate consequence of Theorem 5.9, since the smoothmaps send the sets of zero Lebesgue measure to the sets of zero Lebesgue measure.Moreover, Theorem 5.9 shows that for every family ˜ F in a C -neighbourhood of F ,IFS( ˜ F ↓ V ) is ρ -ergodic.The second part is a duplication of the last parts of Theorem 5.4, in which mini-mality implies the stable covering of M by the images of arbitrary balls of radius ρ in V , under finitely many elements of (cid:104)F (cid:105) . (cid:3) Stably ergodic actions on manifolds
In this section, we prove Theorems A, B and C using the local results of theprevious sections.6.1.
Ergodic IFS on arbitrary manifold.
In this subsection, we use the resultsof the previous section to construct a pair of diffeomorphisms generating a C -stably ergodic, C -robustly minimal IFS on any closed manifold M of dimension d .Theorem B is a consequence of the following. Theorem 6.1.
Every closed manifold M admits a semigroup generated by twosmooth diffeomorphisms that acts C -stably ergodic and C -robustly minimal in Diff s ( M ) , s ∈ (1 , ∞ ] .Remark . As far as the authors know, this is the first example of a stably ergodicaction on a manifold of dimension greater than one. In [BFMS17] and [Sar15] theexistence of such actions on surfaces is claimed, however, with an argument that onlyworks for conformal actions and mistakenly assumes “conformality of maps withcomplex eigenvalues”. Indeed, it is easy to provide a non-quasiconformal sequenceof matrices all with complex eigenvalues.We will use the following lemma. We omit its proof, since it is very similar to thelast part of the proof of Theorem 5.4.
Lemma 6.3.
Let V be an open subset of compact manifold M and F be a familyin Diff ( M ) . Assume that (cid:104)F (cid:105) + ( V ) = (cid:104)F − (cid:105) + ( V ) = M . If every measurable F -invariant set has either zero or full Lebesgue measure in V , then IFS( F ) is ergodicon M . Similarly, if for every x ∈ M , (cid:104)F (cid:105) + ( x ) is dense in V , then IFS( F ) isminimal. Proof of Theorem 6.1.
The case dim( M ) = 1 is known. It comes from Theorem 1.1and [GI00]. Indeed, Theorem 6.5 provides an explicit example for this case.So, we may assume that dim( M ) ≥
2. The same statement for robust minimality(without the ergodicity) is proved in [HN14, Theorem A]. Here, we carefully modifyits proof and make use of our results in the previous sections to prove both robustminimality and stable ergodicity. First, we establish some local construction onthe Euclidean space and then realize them by C ∞ diffeomorphisms on an arbitrarysmooth manifold. Recall that we work with the C topology on Diff s ( M ). Step 1.
Local construction: finitely many generators on R d . In this step, we construct a family of contracting affine transformations in R d sufficiently close to Id and satisfying the assumptions of Theorem 5.7.Fix (cid:15) > κ < (cid:15) . Consider openneighbourhood U of the identity in SL( d, R ) such that any element D ∈ U is κ -conformal. Consequently, 1 − (cid:15) < m ( D ) and max {(cid:107) D (cid:107) , (cid:107) D − (cid:107)} < (cid:15) . By Lemma4.8, one can find a set D ⊆ U containing d elements, and an open set U ⊆ U with(30) U ⊆ D − U . Denote λ := 1 − (cid:15) and V := B (cid:15) (0). For any D ∈ D and v ∈ R d , define T D,v ( x ) := λD − ( x ) + v . Clearly, for any D ∈ D , (cid:107) D x T D,v (cid:107) = λ (cid:107) D − (cid:107) < − (cid:15) and V (cid:48) := B (cid:15) (1 − (cid:15) ) (0) ⊆ T D, ( V ). Take a finite subset J ⊆ V such that V ⊆ (cid:83) v ∈ J ( V (cid:48) + v ).Hence, for any D ∈ D ,(31) V ⊆ (cid:91) v ∈ J T D,v ( V ) . Note that the number of elements of J can be chosen independent of (cid:15) and dependingonly on d , because (31) is invariant under scaling. Denote the cardinality of J by Q d ∈ N .Next, define G := { T D,v : D ∈ D , v ∈ J } . By (31), one gets that for any D ∈ D and x ∈ V , there exists T ∈ G satisfying y = T − ( x ) ∈ V and ˆ D y T = D − . Thiscombined with (30) implies that for W := V × U ⊆ E ( R d ), W ⊆ (cid:83) T ∈G ˆ D T ( W ). Onthe other hand, one can easily check that for every T ∈ G , T ( B (0)) ⊆ B (0).Therefore, the family G satisfies the assumptions of Theorem 5.7 for open sets U := B (0) ⊆ R d and W . Accordingly, every G↓ U invariant set of positive Lebesguemeasure in U has full measure in V . Moreover, this property is stable under smallperturbations of G , in the C topology. Let ˜ G ⊆
Diff ∞ ( R d ) be a family of diffeomor-phisms obtained by extending all elements of G↓ U to R d such that every T ∈ ˜ G isequal to the identity outside B (0). This is possible, since (cid:15) is small enough. Indeed,we can assume that the diffeomorphisms in ˜ G are close to the identity (of order (cid:15) ).Enumerate the elements of ˜ G by T , . . . , T n , where n := d Q d . Step 2.
Local construction: a pair of generators on M.
We first choose a C ∞ Morse-Smale diffeomorphism f with a unique attractingperiodic orbit O ( p ) of period N > n and attraction rate sufficiently close to 1. Wecan assume that f N is close to the identity. This can be done by deforming the timeone map of the gradient flow of a suitable Morse function with a unique minimumpoint (cf. [HN14] for the details). Denote the set of all other periodic points of f by P f . Clearly, P f is finite. Let U ⊆ M be a small neighbourhoods of p such that- for i = 0 , . . . , N −
1, the sets f i ( U ) are pairwise disjoint and f N ( U ) ⊆ U , TABLE LOCAL ERGODICITY 29
STABLE LOCAL ERGODICITY 29 fff f fV pf N ( V ) f i ( p ) h h Figure 5.
Local constructions of f and h .For any 0 i N , denote U i := f i ( U ) ✓ M . Then, define h Di↵ ( M ) asfollows (see Figure 5)- For 1 i n , h | U i := f i N h i f N i | U i , where h i = T i .- h equals to the identity outside S ni =1 U i .It is clear that h is close to the identity. We claim that every measurable set S invariant under h{ f, h } U i + with Leb( S \ U ) > V = ( V ) ✓ M . In order to prove this, note that the family H := { h i : 1 i n } isconjugate to ˆ H := { T i : 1 i n } and by Step 1, every measurable H U -invariantset of positive measure in U , contains V up to a set of measure zero and the sameholds for every family close to H . On the other hand, for any 1 i n , h i f N | U = ( f N i h f i N ) f N | U f, h } U i + . Now, the family ˜ H := { h i f N | U : 1 i n } is su ciently close to H providedthat f N is su ciently close to the identity. Therefore, the proof of the claim isfinished. Similarly, one can show that for any x U , h f, h i + ( x ) is dense in V .Finally, note that the arguments above show that if one perturbs f, h outside S Ni =0 f i ( U ), the local ergodicity and the local minimality remain true. Step 3.
Global construction: a pair of generators on M . Recall that f Di↵ ( M ) is a Morse-Smale di↵eomorphism with a unique peri-odic orbit O ( p ). The forward orbit of every point under f converges either to O ( p )or an element of P f . Therefore, M = [ q P f W sf ( q ) [ W sf ( O ( p )) , where W s denotes the stable manifold. Since O ( p ) is the unique attracting periodicorbit for f , W sf ( O ( p )) = S f i ( V ) is an open and dense subset of M . On the otherhand, for any q P f , W s ( q ) is nowhere dense in M .Pick Di↵ ( M ) su ciently close to the identity such that ( P f ) [ ( P f ) ✓ W sf ( O ( p )) and equals to the identity outside a small neighbourhood of P f . Let˜ h Di↵ ( M ) be close to h such that Figure 5.
Local constructions of f and h .- the orbit of U under f does not intersect P f .- there exists a diffeomorphism φ : U → R d with B (0) ⊆ φ ( f N ( U )).For any 0 ≤ i ≤ N , denote U i := f i ( U ) ⊆ M . Then, define h ∈ Diff ∞ ( M ) asfollows (see Figure 5)- For 1 ≤ i ≤ n , h | U i := f i − N ◦ h i ◦ f N − i | U i , where h i = φ − ◦ T i ◦ φ .- h equals to the identity outside (cid:83) ni =1 U i .It is clear that h is close to the identity. We claim that every measurable set S invariant under (cid:104){ f, h }↓ U (cid:105) + with Leb( S ∩ U ) > V = φ − ( V ) ⊆ M . In order to prove this, note that the family H := { h i : 1 ≤ i ≤ n } isconjugate to ˆ H := { T i : 1 ≤ i ≤ n } and by Step 1, every measurable H↓ U -invariantset of positive measure in U , contains V up to a set of measure zero and the sameholds for every family close to H . On the other hand, for any 1 ≤ i ≤ n , h i ◦ f N | U = ( f N − i ◦ h ◦ f i − N ) ◦ f N | U ∈ (cid:104){ f, h }↓ U (cid:105) + . Now, the family ˜ H := { h i ◦ f N | U : 1 ≤ i ≤ n } is sufficiently close to H providedthat f N is sufficiently close to the identity. Therefore, the proof of the claim isfinished. Similarly, one can show that for any x ∈ U , (cid:104) f, h (cid:105) + ( x ) is dense in V .Finally, note that the arguments above show that if one perturbs f, h outside (cid:83) Ni =0 f i ( U ), the local ergodicity and the local minimality remain true. Step 3.
Global construction: a pair of generators on M . Recall that f ∈ Diff ∞ ( M ) is a Morse-Smale diffeomorphism with a unique peri-odic orbit O ( p ). The forward orbit of every point under f converges either to O ( p )or an element of P f . Therefore, M = (cid:91) q ∈ P f W sf ( q ) ∪ W sf ( O ( p )) , where W s denotes the stable manifold. Since O ( p ) is the unique attracting periodicorbit for f , W sf ( O ( p )) = (cid:83) f − i ( V ) is an open and dense subset of M . On the otherhand, for any q ∈ P f , W s ( q ) is nowhere dense in M . Pick ψ ∈ Diff ∞ ( M ) sufficiently close to the identity such that ψ ( P f ) ∪ ψ − ( P f ) ⊆ W sf ( O ( p )) and ψ equals to the identity outside a small neighbourhood of P f . Let˜ h ∈ Diff ∞ ( M ) be close to h such that- ˜ h = h on (cid:83) ≤ i ≤ N f i ( U ),- for a small neighbourhood of q ∈ P f , ˜ h = ψ ◦ f − ◦ ψ − ◦ f .Now, define g := ˜ h ◦ f − ∈ Diff ∞ ( M ) and let F := { f, g } . Clearly, ˜ h = g ◦ f ∈(cid:104)F (cid:105) + and so (cid:104) ˜ h, f (cid:105) + ⊆ (cid:104)F (cid:105) + . Moreover, if ˜ h is sufficiently close to the identity, then g is close to f − . So, g is a Morse-Smale diffeomorphism, and since ψ is equal to theidentity near O ( p ), where O ( p ) is the unique repelling periodic orbit of g . Denotethe set of all other periodic points of g by P g .We claim that IFS( F ) is C -stably ergodic and C -robustly minimal on M . Since (cid:104) ˜ h, f (cid:105) + ⊆ (cid:104)F (cid:105) + and ˜ h equals to h near O ( p ), in view of Step 2 and Lemma 6.3,it suffices to prove that (cid:104)F (cid:105) + ( V ) = (cid:10) F − (cid:11) + ( V ) = M . Indeed, for every x ∈ M , g i ( x ) converges to some element of P g ⊆ W sf ( O ( p )) = (cid:83) i ∈ N f − i ( V ). Consequently, (cid:10) F − (cid:11) + ( V ) = M .Similarly, f − i ( x ) converges to some element of P f ⊆ W sg − ( O ( p )), where O ( p ) isthe unique attracting periodic orbit for g − . Thus, (cid:104)F (cid:105) + ( V ) = M . Hence, theproof of Theorem 6.1 is complete. (cid:3) Proof of Theorems A and C.
We present a variant of Theorem A for dif-feomorphisms between open sets of M , i.e. they are not necessarily globally definedon the manifold.In particular, it implies Theorems A and C. Theorem 6.4.
Let M be a closed smooth Riemannian manifold and F ⊆
Diff ( M ) .Assume that for any ( x, v ) ∈ T M there exists f ∈ F such that m ( D x f ) > and (cid:107) ˆ D x f | v ⊥ (cid:107) < . If the action of (cid:104)F (cid:105) is minimal, is stably ergodic in Diff α ( S d ) andis C -robustly minimal.Proof. The proof follows from Lemma 4.12 and Theorem 5.4. By the compactnessof T M , one can find a finite subset F ⊆ F such that for any ( x, v ) ∈ T M thereexists f ∈ F with m ( D x f ) > (cid:107) ˆ D x f | v ⊥ (cid:107) <
1. For any f ∈ F , denote U f := { x ∈ M : m ( D x f ) > } . Now, by Lemma 4.12, the assumptions of Theorem5.4 are satisfied for family ˆ F := { f | U f : f ∈ F } . Therefore, there exists ρ > F ) is stably ρ -ergodic. In particular, IFS( F ) is stably ρ -ergodic.Then, similar to the proof of the last part of Theorem 5.4, we use minimality ofthe action of (cid:104)F (cid:105) to stably cover M by the images of an arbitrary ball of radius ρ under a finite set F ⊆ (cid:104)F (cid:105) . This proves stable ergodicity and robust minimality ofthe action of (cid:104)F (cid:105) . (cid:3) The proof of Theorem A is the same as the proof above. Indeed, one should onlyreplace F by G .We conclude this section with the proof of Theorem C. It is obtained from itsspecial case stated below. Theorem 6.5.
Let d ≥ and { A , . . . , A k } ⊆ SO( d + 1) generates a dense subgroupof SO( d + 1) . Then, for any A ∈ SL( d + 1 , R ) \ SO( d + 1) , the natural action of thegroup generated by { A , . . . , A k } on S d is stably ergodic in Diff α ( S d ) . Moreover,it is C -robustly minimal. TABLE LOCAL ERGODICITY 31
We denote by f A , the action of A ∈ SL( d + 1 , R ) on S d , which is defined by x (cid:55)→ Ax | Ax | . Also, denote the standard orthonormal basis of R d +1 by ( e , . . . , e d +1 ).One can easily check that if the diagonal matrix ˆ A = diag( r , . . . , r d +1 ) , satisfies0 < r d +1 < min ≤ i ≤ d r i , and max , and (cid:107) ˆ D e d +1 f ˆ A | W (cid:107) < , where W is the ( d − T e d +1 S d perpendicular to e . Notethat the inequalities in (32) are stable under perturbations of e d +1 , W and f . Moreprecisely,( ∗ ) There exist λ < (cid:15) > U of e d +1 such that forany C map f sufficiently close to f ˆ A in the C topology, and any ( p, v ) ∈ T S d with p ∈ U and ∠ ( v, e ) < (cid:15) , one has (cid:107) ˆ D p f | v ⊥ (cid:107) < λ and (cid:107) D p f (cid:107) > λ − .We will need the following linear algebraic lemma. Lemma 6.6.
Given any matrix D ∈ SL( d + 1 , R ) \ SO( d + 1) , there exist n > , R , R , . . . , R n ∈ SO( d + 1) , and α , . . . , α n ∈ {− , +1 } such that R D α R D α R . . . R n − D α n R n = diag( r , . . . , r d +1 ) , where < r d +1 < min ≤ i ≤ d r i and max σ of { , . . . , d + 1 } with σ (1) = 1. Then, D := (cid:81) σ ∈ Σ R σ DR − σ = diag( t , . . . , t d +1 ) satisfies | t | > | t | = · · · = | t d +1 | = 1 . Let σ be the permutation on { , . . . , d + 1 } with σ ( i ) = d + 1 − i . Hence, R σ D − R − σ = diag( t − d +1 , . . . , t − ) , and so D = D R σ D − R − σ = diag( r , . . . , r d +1 ) with | r | > > | r d +1 | and | r | = · · · = | r d | = 1. Now, D has positive diagonal entries and satisfies the conditions. (cid:3) Proof of Theorem 6.5.
Denote F := { f A , . . . , f A k } and F := F ∪ { f A } . Note thatergodicity and minimality of the action of (cid:104)F (cid:105) is guaranteed, since (cid:104) A , . . . , A k (cid:105) =SO( d + 1). So, we only need to show the stability under C perturbations. To thisend, we use minimality of the isometric action of (cid:104)F (cid:105) to transfer the properties ofthe derivatives in some open set to the whole manifold. Claim (Reduction to Theorem 6.4) . There exists a finite set F ⊆ (cid:104)F (cid:105) such thatfor any ( x, v ) ∈ T S d , m ( D x g ) > and (cid:107) ˆ D x g | v ⊥ (cid:107) < for some g ∈ F . Proof.
The fact (cid:104) A , . . . , A k (cid:105) = SO( d + 1) combined with Lemma 6.6 for D = A implies that there exists A ∈ (cid:104) A , A , . . . , A k (cid:105) sufficiently close to ˆ A , defined above,such that ( ∗ ) holds for f A and its C perturbations. Fix this A for the rest of theproof and denote f := f A , for simplicity.It follows from the minimality of the action on T S d that for any ( x, v ), thereexists h ∈ (cid:104)F (cid:105) such that for every ( y, w ) in a neighbourhood of ( x, v ), h ( y ) ∈ U and ∠ ( D y h ( w ) , e ) < (cid:15) . Now, by ( ∗ ), g := f ◦ h ∈ (cid:104)F (cid:105) satisfies m ( D y g ) > λ − and (cid:107) ˆ D y g | w ⊥ (cid:107) < λ . Finally, by the compactness of T S d , one can choose a finite subset F of (cid:104)F (cid:105) satisfying these properties. (cid:3) Since the action of F on S d is minimal, this claim combined with Theorem 6.4implies that the action of (cid:104)F (cid:105) is C -stably ergodic and C -robustly minimal inDiff ( S d ). (cid:3) Proof of Theorem C.
Consider a finite family { A , . . . , A k } ⊆ SO( d + 1) generatinga dense subgroup of SO( d + 1). The existence of such elements for d = 1 is trivialand for d ≥
2, is granted by [Kur51], as SO( d + 1) is a semi-simple Lie group.Since (cid:104)F (cid:105) is dense in SO( d + 1), it contains elements ˜ A , . . . , ˜ A k arbitrary close to A , . . . , A k . On the other hand, F has an element A ∈ SL( d + 1 , R ) \ SO( d + 1).So, Theorem 6.5 implies that the natural action of (cid:104)F (cid:105) on S d is stably ergodic androbustly minimal. (cid:3) Some questions
The number of generators for stably ergodic actions.
Theorem B statesthat every manifold M admits a stably ergodic semigroup action generated by twodiffeomorphisms. A natural question to ask is whether or not the number of gener-ators in Theorem B is optimal. In other words, Question 7.1.
Does there exist a manifold M with a stably ergodic (w.r.t. Leb.)diffeomomorphism in Diff α ( M ) ? Two observations support a negative answer to this question. First, no Anosovdiffeomorphism is C -stably ergodic (w.r.t. Leb.) in Diff s ( M ) ( s ≥ ) . Second, somemanifolds do not admit a stably transitive diffeomorphisms. More precisely, there isno C -stably ergodic cyclic group in Diff α ( M ) , if M is either the circle, a closedsurfaces, a 3-manifolds that does not admit partially hyperbolic diffeomorphisms (e.g. S ), or a manifold whose tangent bundle does not split (e.g. S k ). It follows from [GO73] that any C Anosov diffeomorphism which is ergodic withrespect to Lebesgue admits a unique invariant measure in the class of Lebesguemeasure. On the other hand, by [LS72], generic C Anosov diffeomorphism hasno invariant measure in the class of Lebesgue measure. Since every Anosov diffeo-morphism can be approximated by C Anosov diffeomorphisms, it follows that noAnosov diffeomorphism is stably ergodic (w.r.t. Leb.) in Diff α ( M ).Next, it is a consequence of [Ma˜n82], [DPU99] and [BDP03] that some forms ofhyperbolicity (and thus splitting of the tangent bundle) can be obtained from robusttransitivity in Diff ( M ). Moreover, on T , a C -robustly transitive diffeomorphismis indeed an Anosov diffeomorphism. As a matter of fact, the same proofs works ifone considers the C topology in the space Diff α ( M ), as we do here. Thus, none ofthe manifolds listed above (except T ) do admit a stably transitive diffeomorphism. TABLE LOCAL ERGODICITY 33
The claims on S and S k follow from [BBI04] and the obstruction theory in topology[MS05], respectively.These arguments raise the following questions (see also [AB06]). Question 7.2.
Which ergodic partially hyperbolic diffeomorphisms in
Diff ( M ) ad-mit an invariant probability measure equivalent to the Lebesgue measure? Question 7.3.
Is it true that a generic diffemorphism in
Diff ( M ) admits no in-variant measure in the class of Lebesgue measure. Ergodicity vs. quasi-conformality.
In the theory of stable ergodicity inDiff ( M ), usually the ergodicity follows from (some forms of) hyperbolicity usinga Hopf type arguments. Clearly, any form of hyperbolicity obstructs the existenceof quasi-conformal orbits. In contrast, as discussed above, even Anosov diffeomor-phisms are not stably ergodic (w.r.t. Leb.) in Diff α ( M ). Moreover, the existenceof quasi-conformal orbits is a crucial ingredient of our arguments to establish stable(local) ergodicity. One may ask the following. Question 7.4.
Does there exist a C -stably ergodic finitely generated semigroup in Diff ( M ) such that (stably) the set of points having quasi-conformal orbit-brancheshas zero Lebesgue measure? In Theorem A, while the contraction hypothesis for the normalized derivativeon hyper-subspaces obstructs conformality, it allows one to obtain (stably) quasi-conformal orbit-branches at every point. Inspired by the work of [ABY10] on semi-groups of SL(2 , R ) we ask the next question concerning the optimality of this as-sumption in dimension 2. Question 7.5.
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Department of Mathematics, Shahid Beheshti University, Tehran, 19839, Iran;School of Mathematics, Institute for Research in Fundamental Sciences (IPM),P.O. Box 19395-5746, Tehran, Iran
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