Ergodic optimization theory for Axiom A flows
aa r X i v : . [ m a t h . D S ] A ug ERGODIC OPTIMIZATION THEORY FOR AXIOM A FLOWS
WEN HUANG, ZENG LIAN, XIAO MA, LEIYE XU, AND YIWEI ZHANG
Abstract.
In this article, we consider the weighted ergodic optimization problemAxiom A attractors of a C flow on a compact smooth manifold. The main resultobtained in this paper is that for a generic observable from function space C ,α ( α ∈ (0 , C the minimizing measure is unique and is supported on a periodic orbit. Introduction
Context and motivation.
Ergodic optimization theory focuses on the ergodic mea-sures on which a given observable taking an extreme ergodic average (maximum or min-imum), which has strong connection with other fields, such as Anbry-Mather theory[Co2, Ma, CIPP] in Lagrangian Mechanics; ground state theory [BLL] in thermody-namics formalism and multifractal analysis; and controlling chaos [HO1, OGY, SGOY]in control theory.In this paper, we study the typical optimization problem in weighted ergodic op-timization theory for Axiom A attractors of a C flow on a compact smooth mani-fold. For discrete time case, ergodic optimization theory has been developed broadly.Among them, Yuan and Hunt proposed an open problem in [YH, Conjecture 1.1]on 1999, which provides a mathematical mechanism on Hunt and Ott’s experimen-tal and heuristic results in [HO2, HO3] and becomes one of the fundamental ques-tions raised in the field of ergodic optimization theory. A more general form of Yuanand Hunt’s conjecture is now called the Typical Periodic Optimization Conjecture,and has attracted sustained attentions and yielded considerable results, for instances[BZ, Bo1, Bo2, Bo4, Co1, CLT, Mo, QS]. For a more comprehensive survey for theclassical ergodic optimization theory, we refer the reader to Jenkinson [Je1, Je2], toBochi [B], to Baraviera, Leplaideur, Lopes [BLL], and to Garibaldi [Ga] for a historicalperspective of the development in this area. In our recent paper [HLMXZ], we extendthe applicability of the theory both to a broader class of systems and to a broader classof observables, which leads to a positive answer to Yuan and Hunt’s conjecture for C -observable case. To our knowledge, because of difficulties appears on both conceptuallevel and technical level, there is no existing result of ergodic optimization theory for Huang is partially supported by NSF of China (11431012,11731003). Lian is partially supported byNSF of China (11725105,11671279). Xu is partially supported by NSF of China (11801538, 11871188).Zhang is partially supported by NSF of China (117010200,11871262). flows so far, which make the results obtained in the present paper the first achievementon flows towards ergodic optimization theory.On the other hand, as mentioned in [HLMXZ], the reason of adding the non-constantweight ψ mainly lies in the studies on the zero temperature limit (or ground state) ofthe ( u, ψ )-weighted equilibrium state for thermodynamics formalism (for more details,we refer readers to works [BF, BCW, FH]). Summary of the results.
To avoid unnecessary complexity, we only introduce theresult in the framework of standard ergodic optimization theory. Consider a C flowΦ on a compact smooth manifold M . Let Λ be an Axiom A attractor of Φ (detaileddefinition is given by Definition 2.1). For a given observable u : M → R , the ergodicaverages of u on Λ is defined by the integration of u with respect to Φ | Λ -ergodic mea-sures, and the u -maximizing (or minimizing) measure is the measure with respect towhich the ergodic average of u takes maximum (or minimum) value. As a consequenceof the main result (Theorem 2.2) of the present paper, we have the following result: Theorem A:
Let (Λ , Φ) be an Axiom A attractor on a compact smooth Riemannianmanifold M , then for a generic observable u in function space C ,α ( M ) or C ( M ), the u -maximizing (or u -minimizing) measure is unique and is supported on a periodic orbit. Remarks on techniques of the proof.
It seems that the proof given in [HLMXZ]provides a more general mechanism in the study on ergodic optimization problems,which also shed a light on the case of flows for sure. However, the results of the presentpaper depend crucially on the continuous time nature of the system; that is to say,they do not follow from the properties of their time-1 maps. Therefore, we must buildcertain theoretical base and create certain new techniques to address issues raised inthe case of flows.We mention three differences of note between our setting and the existing literaturesat both conceptual level and technical level which pervade the arguments in this pa-per: (1) At conceptual level, the most significant issue is that the gap function of adiscrete time periodic orbit, i.e. the minimum separation of finite isolated points, isnot well defined for continuous periodic orbit. Such a gap function plays a key rolein the proof of [HLMXZ]. (2) The presence of shear , i.e. the sliding of some orbitspast other nearby orbits due to the slightly different speed at which they travel, isa typical phenomenon of continuous time systems, which causes tremendous amountof ”tail estimates” throughout this paper. (3) Several main fundamental theoreticaltools are not existing and need to be rebuilt from the base, such as Anosov ClosingLemma, Ma˜n´e-Conze-Guivarc’h-Bousch’s Lemma and Periodic Approximation Lemma.
Structure of this paper.
In Section 2, we set up the theoretic model and statethe main results; In Section 3, we state (without proofs) some well known propertiesof Axiom A attractors, and some theoretical tools including Anosov Closing Lemma,Ma˜n´e-Conze-Guivarc’h-Bousch’s Lemma and Periodic Approximation Lemma prepar-ing for the proof the main results; In Section 4, we give the proof of Theorem 2.2, ofwhich proving Part I) of the Theorem costs most efforts; As follows, we leave the proofsof all the technical lemmas to Section 5. On one hand, readers may go through the mainproof by assuming the validity of these technical lemmas without extra interruptions;on the other hand, these technical lemmas with their proofs may be of independentinterest. Finally, we discuss the case when observables have higher regularity in Section6 in which only some partial results are presented.2. main setting and results
Let M be a compact smooth Riemannian manifold with Riemannian metric d andΦ = { φ t : M → M } t ∈ R be a C flow on M . Definition 2.1.
For Λ ⊂ M , (Λ , Φ) is called an Axiom A attractor if the followingconditions hold:A1) Λ is a nonempty Φ-invariant compact set.A2) There exists an ǫ > x ∈ M with d ( x, Λ) < ǫ lim t →∞ d ( φ t ( x ) , Λ) = 0 . A3) There exist λ > C > M restricted on Λ, T x M = E ux ⊕ E cx ⊕ E sx ∀ x ∈ Λ, such that the following hold( D M φ t ) x ( E τx ) = E τφ t ( x ) , τ = u, c, s, ∀ t ∈ R and x ∈ Λ , max (cid:8) k ( D M φ − t ) x | E ux k , k ( D M φ t ) x | E sx k (cid:9) ≤ C e − tλ , ∀ t ∈ R + , where ( D M φ t ) x is the derivative of the time- t map φ t on x with respect to spacevariables.A4) inf x ∈ Λ (cid:13)(cid:13)(cid:13) dφ t ( x ) dt (cid:13)(cid:13)(cid:13) > E cx = span (cid:8) ddt φ t ( x ) (cid:9) , ∀ x ∈ Λ.Denote by M (Λ , Φ) the set of all Φ-invariant Borel probability measures on Λ, whichis a non-empty convex and compact topological space with respect to weak ∗ topology.Denote by M e (Λ , Φ) ⊂ M (Λ , Φ) the set of ergodic measures, which is the set of theextremal points of M (Λ , Φ). Let u : M → R and ψ : M → R + be continuous functions.The quantity β ( u ; ψ, Λ , Φ) being defined by β ( u ; ψ, Λ , Φ) := min ν ∈M (Λ , Φ) R udν R ψdν , (2.1) WEN HUANG, ZENG LIAN, XIAO MA, LEIYE XU, AND YIWEI ZHANG is called the ratio minimum ergodic average , and any ν ∈ M (Λ , Φ) satisfying R udν R ψdν = β ( u ; ψ, Λ , Φ)is called a ( u, ψ ) -minimizing measure . Denote that M min ( u ; ψ, Λ , Φ) := (cid:26) ν ∈ M (Λ , Φ) : R udν R ψdν = β ( u ; ψ, Λ , Φ) (cid:27) . By compactness of M (Λ , Φ), and the continuity of the operator R ud ( · ) R ψd ( · ) , it directly fol-lows that M min ( u ; ψ, Λ , Φ) = ∅ , which contains at least one ergodic ( u, ψ )-minimizingmeasure by ergodic decomposition.For α ∈ (0 , C ,α ( M ) be the space of α -H¨older continuous real-valued func-tions on M endowed with the α -H¨older norm k u k α := k u k + [ u ] α , where k u k :=sup x ∈ M | u ( x ) | is the super norm, and [ u ] α := sup x = y | u ( x ) − u ( y ) | ( d ( x,y )) α . Also note that when α = 1, C , ( M ) becomes the collection of all real-valued Lipschitz continuous func-tions, and [ u ] becomes the minimum Lipschitz constant of u . Additionally, denote by C , ( M ) the Banach space of continuous differentiable functions on M endowed withthe standard C -norm.In this paper, we consider the weighted ergodic optimization problem and derive thefollowing result. Theorem 2.2.
Let M be a compact smooth Riemannian manifold with Riemannianmetric d and Φ be a C flow on M . Suppose that (Λ , Φ) is an Axiom A attractor, thenthe following hold: I) For α ∈ (0 , , given a ψ ∈ C ,α ( M ) with inf x ∈ M ψ ( x ) > , then there existsan open and dense set P ⊂ C ,α ( M ) such that for any u ∈ P , the ( u | Λ , ψ | Λ ) -minimizing measure of (Λ , Φ) is unique and is supported on a periodic orbit of Φ . II)
For ψ ∈ C , ( M ) with inf x ∈ M ψ ( x ) > , there exists an open and dense set P ⊂ C , ( M ) such that for any u ∈ P , the ( u | Λ , ψ | Λ ) -minimizing measure of (Λ , Φ) is unique and is supported on a periodic orbit of Φ . We remark here that M, Λ , Φ are assumed to satisfy conditions in Theorem2.2 throughout the rest of this paper. Properties of Axiom A attractors
This section devotes to building theoretic tools as preparations for the proof of The-orem 2.2.
Invariant Manifolds.
For a point x ∈ Λ and ǫ > W sǫ ( x ) = { y ∈ M : d ( φ t ( x ) , φ t ( y )) ≤ ǫ ∀ t ≥ , d ( φ t ( x ) , φ t ( y )) → t → + ∞} ,W uǫ ( x ) = { y ∈ M : d ( φ − t ( x ) , φ − t ( y )) ≤ ǫ ∀ t ≥ , d ( φ − t ( x ) , φ − t ( y )) → t → + ∞} . The following Lemma is a standard result of invariant manifolds in existing literature,of which the proof is omitted.
Lemma 3.1.
For any λ ∈ (0 , λ ) , there exists ǫ > and C ≥ such that for any ǫ ∈ (0 , ǫ ] , the following hold: i) W sǫ ( x ) , W uǫ ( x ) are C embedded discs for all x ∈ Λ with T x W τǫ ( x ) = E τx , τ = u, s ; ii) d ( φ t ( x ) , φ t ( y )) ≤ C e − tλ d ( x, y ) for y ∈ W sǫ ( x ) , t ≥ , and d ( φ − t ( x ) , φ − t ( y )) ≤ C e − tλ d ( x, y ) for y ∈ W uǫ ( x ) , t ≥ ; iii) W sǫ ( x ) , W uǫ ( x ) vary continuously with respect to x (in C topology). By choosing the Riemannian metric, the Axiom A flow in Theorem 2.2 meets thefollowing basic canonical setting : There are positive constants δ, ǫ, β, λ, C with C ≥ δ ≪ ǫ ≪ min { ǫ , ǫ } , where ǫ is as in A2) of the definition of Axiom Aattractors and ǫ is as in Lemma 3.1, such that:(1) For x, y ∈ M with d ( x, y ) ≤ δ , there is a unique time v = v ( x, y ) with | v | ≤ Cd ( x, y ) such that(a) W sǫ ( φ v ( x )) ∩ W uǫ ( y ) is not empty and contains only one element which isnoted by w = w ( x, y ).(b) d ( x, y ) ≥ C − max { d ( φ v ( x ) , w ) , d ( y, w ) , d ( φ v ( x ) , x ) , d ( w, x ) } . (2) For x ∈ M , y ∈ W uǫ ( x ) and t ≥ d ( φ − t x, φ − t y ) ≤ Ce − λt d ( x, y ),For x ∈ M , y ∈ W sǫ ( x ) and t ≥ d ( φ t x, φ t y ) ≤ Ce − λt d ( x, y ).(3) For x, y ∈ M , d ( φ t x, φ t y ) ≤ Ce β | t | d ( x, y ) for all t ∈ R . Remark 3.2.
In our following text, δ, ǫ, λ, β, C are the positive constants as above.Additionally, for convenience, we assume C ≫ , < δ ≪ ǫ ≪
1. Otherwise, we seta positive constant ǫ ′ such that ǫ ′ ≪ ǫC e β . We set another positive constant δ ′ with δ ′ ≪ δ such that for any x, y ∈ M with d ( x, y ) ≤ C e β +10 λ e λ − δ ′ , there is an unique time v = v ( x, y ) with | v | ≤ Cd ( x, y ) such that W sǫ ′ ( φ v ( x )) ∩ W uǫ ′ ( y ) is not empty and containsonly one element. Remark 3.3.
For proofs and more details of Lemma 3.1 and the basic canonicalsetting , we refer readers to [PSh], [Bowen], and [BR]. The only property which is notappearing in the above references is the following inequality | v ( x, y ) | ≤ Cd ( x, y ) (3.1)appearing in (1) of basic canonical setting . We remark here that this inequalityholds when Φ is C . When Φ is C α for some α ∈ (0 , WEN HUANG, ZENG LIAN, XIAO MA, LEIYE XU, AND YIWEI ZHANG be replaced by | v ( x, y ) | ≤ Cd α ( x, y ) which is still sufficient for the proof of this paper(although necessary modifications are required). This concludes that the Theorem 2.2still holds for C α flows.Finally, to our knowledge, there is no explicit statement equivalent to (3.1) in existingliterature. Nevertheless, (3.1) can be proved by combining Lemma 6, Proposition 8,Proposition 9 and Lemma 13 from [LY]. Since (3.1) is intuitively natural but at thesame time the proof involves considerable technical complexity, we decide not to putthe detailed proof in this paper for the sake of simplicity.3.2. Anosov Closing Lemma.
Let δ ′ , ǫ ′ , δ, ǫ , λ, β, C be the constants as in Remark3.2. Then we have the following Lemma. Lemma 3.4.
Given η ≤ C e β +10 λ e λ − δ ′ and T > , if x, y ∈ Λ and continuous function s : R → R with s (0) = 0 satisfy d ( φ t + s ( t ) ( y ) , φ t ( x )) ≤ η for t ∈ [0 , T ] , then for all t ∈ [0 , T ] , the following hold: ASh1) | s ( t ) | ≤ Cη ; ASh2) d ( φ t φ v ( y,x ) ( y ) , φ t ( x )) ≤ C e − λ min( t,T − t ) ( d ( y, x ) + d ( φ T ( y ) , φ T ( x ))) , where v ( y, x ) is as in Remark 3.2 satisfying | v ( y, x ) | ≤ Cd ( x, y ) .Especially, one has that (1) If d ( φ t + s ( t ) ( y ) , φ t ( x )) ≤ η for all t ≥ , then d ( φ t φ v ( y,x ) ( y ) , φ t ( x )) → as t → + ∞ . (2) If d ( φ t + s ( t ) ( y ) , φ t ( x )) ≤ η for all t ∈ R , then φ v ( y,x ) ( y ) = x. A segment of Φ is a curve S : [ a, b ] → M : t → φ t ( x ) for some x ∈ M and realnumbers a ≤ b . We denote the left endpoint of S by S L = φ a ( x ), the right endpoint of S by S R = φ b ( x ) and the length of S by |S| = b − a . By a segment S , if S L = S R , wesay S is a periodic segment. We have the following version of Anosov Closing Lemma. Lemma 3.5 (Anosov Closing Lemma) . There are positive constants L and K dependingon the system constants only such that if segment S of Φ | Λ satisfy (a) |S| ≥ K ; (b) d ( S L , S R ) ≤ δ ′ .Then, there is a periodic segment O such that ||S| − |O|| ≤ Ld ( S L , S R ) and d ( φ t ( O L ) , φ t ( S L )) ≤ Ld ( S L , S R ) for all ≤ t ≤ max( |S| , |O| ) . Remark 3.6.
In the following text, we also use S (so is O and Q ) to represnet thecollection of points φ t ( S L ) , ≤ t ≤ |S| as no confusion being caused. By Lemma 3.4and the choices of ǫ and δ , O clearly belongs to Λ.3.3. Ma˜n´e-Conze-Guivarc’h-Bousch’s Lemma.
For γ ∈ R \ { } and continuousfunction u : M → R , define that u γ ( x ) := 1 γ Z γ u ( φ t ( x )) dt. (3.2) Lemma 3.7 (Ma˜n´e-Conze-Guivarc’h-Bousch’s Lemma) . For < α ≤ and N > ,there exists a positive constant γ = γ ( α ) > N such that if u ∈ C ,α ( M ) satisfies β ( u ; 1 , Λ , Φ | Λ ) ≥ , then there is a v ∈ C ,α (Λ) such that u γ | Λ ≥ v ◦ φ γ | Λ − v. Remark 3.8.
We remark that the key point of Lemma 3.7 lies in the fact that v ischosen with the same H¨older exponent as u . Indeed, there were a number of weakversions of Lemma 3.7 in the setting of smooth Anosov flows without fixed points, orcertain expansive non-Anosov geodesic flows, where v is still H¨older, but the H¨olderexponent is less than α (for details, see [LRR, LT, PR]).By using Lemma 3.7, we have the following Lemma. Lemma 3.9.
For < α ≤ , there exists large γ = γ ( α ) such that, for u ∈ C ,α ( M ) and strictly positive ψ ∈ C ,α ( M ) , there is a v ∈ C ,α (Λ) such that (1) u γ | Λ − v ◦ φ γ | Λ + v − β ( u ; ψ, Λ , Φ) ψ γ | Λ ≥ Z u,ψ ⊂ { x ∈ Λ : ( u γ | Λ + v ◦ φ γ | Λ − v − β ( u ; ψ, Λ , Φ) ψ γ | Λ ) ( x ) = 0 } , where Z u,ψ = ∪ µ ∈M min ( u ; ψ, Λ , Φ) supp ( µ ) . Remark 3.10.
For convenience, in the following text, if we need to use Lemma 3.9,we use ¯ u to represent u γ | Λ + v ◦ φ γ | Λ − v − β ( u ; ψ, Λ , Φ) ψ γ | Λ for short. Then, ¯ u ≥ Z u,ψ ⊂ { x ∈ Λ : ¯ u ( x ) = 0 } . Periodic Approximation.
For α ∈ (0 , Z ⊂ M and a segment S of Φ, wedefine the α -deviation of S with respect to Z by d α,Z ( S ) = Z |S| d α (cid:0) φ t (cid:0) S L (cid:1) , Z (cid:1) dt. For P ≥
0, using O P Λ denote the collection of periodic segments in Λ with length notlarger than P . Now we have the following version of Quas and Bressaud’s periodicapproximation Lemma. WEN HUANG, ZENG LIAN, XIAO MA, LEIYE XU, AND YIWEI ZHANG
Lemma 3.11.
Let Z ⊂ Λ be a Φ -invariant compact subset of Λ . Then, for all α ∈ (0 , , k ≥ , lim P → + ∞ P k min S∈O P Λ d α,Z ( S ) = 0 . Proof of Theorem 2.2.
This section contains two main subsections 4.1 and 4.2 corresponding to the proofsof Part I and Part II of Theorem 2.2 respectively. Indeed, Proposition 4.7 in Subsection4.1.3 plays the key role in the proof of Theorem 2.2, based on which the Part I resultfollows immediately and the Part II result follows in a straightforward way with thehelp of an approximation lemma. We also note that throughout the whole section δ ′ , ǫ ′ , δ, ǫ, λ, β, C are the fixed constants as in Remark 3.2.4.1. Proof of Part I) of Theorem 2.2.
This section mainly contains three parts:4.1.1, 4.1.2 and 4.1.3. As mentioned above, Proposition 4.7 in Subsection 4.1.3 is thekey to prove the main theorem of this paper, proving which is the main aim of Section4.1. While Section 4.1.1 and 4.1.2 are devoted to building notions, tools and resultsas preparations for the proof of Proposition 4.7, of which Section 4.1.1 investigate thebasic properties of periodic orbits and Section 4.1.2 constructs periodic orbits with”good shapes”.4.1.1.
Locking Property of Periodic Segments.
In this subsection, we show that periodicsegments have locking property in some sense.For 0 < η ≤ δ , the η -disk of x is defined by D ( x, η ) = { y ∈ Λ : d ( x, y ) ≤ η, W sǫ ( x ) ∩ W uǫ ( y ) = ∅} . (4.1) D ( x, η ) has the following properties:(a) W sη ( x ) ⊂ D ( x, η ) and φ t ( W sη ( x )) ⊂ D ( φ t ( x ) , Ce − λt η ) for t ≥ W uη ( x ) ⊂ D ( x, η ) and φ t ( W uη ( x )) ⊂ D ( φ t ( x ) , Ce λt η ) for t ≤ φ t ( D ( x, η )) ⊂ D (cid:0) φ t ( x ) , Ce β | t | η (cid:1) for t ∈ R satisfying Ce β | t | η < δ .(d) for η ≤ δC and x, y ∈ Λ with d ( x, y ) ≤ η , there exists a unique time v = v ( x, y )with | v | ≤ Cd ( x, y ) such that y ∈ D ( φ v ( x ) , δ ). In fact, v is the one given by the basic canonical setting .Now we define D : Λ × Λ → [0 , + ∞ ) by D ( x, y ) = (cid:26) δ ′ , if y / ∈ D ( x, δ ′ ) ,d ( x, y ) , if y ∈ D ( x, δ ′ ) . (4.2) By a periodic segment O of Φ | Λ , we define the gap of O by D ( O ) = min x ∈O ,
Let O be a periodic segment of Φ | Λ . If x, y ∈ O satisfy d ( x, y ) < D ( O ) C ,then φ v ( x ) = y where v = v ( x, y ) .Proof. Let δ ′ , ǫ ′ , δ, ǫ, λ, β, C be the constants as in Remark 3.2. Then, d ( x, y ) ≤ D ( O ) C < δ ′ ≪ δ. Hence, by the basic canonical setting , there is a constant v = v ( x, y ) such that y ∈ D ( φ v ( x ) , Cd ( x, y )) ⊂ D ( φ v ( x ) , δ ) . If φ v ( x ) = y , then D ( O ) ≤ d ( φ v ( x ) , y ) ≤ Cd ( x, y ) < D ( O ) , which is impossible. Thus, φ v ( x ) = y . This ends the proof. (cid:3) By a periodic segment O of Φ, the periodic measure µ O is defined by µ O = 1 |O| Z |O| δ φ t ( O L ) dt. By an ergodic measure µ ∈ M e (Λ , Φ | Λ ), a point x ∈ M is called a generic point of µ ifthe following holds lim T → + ∞ T Z T f ( φ t ( x )) dt = Z f dµ for all f ∈ C ( M ) . The following Lemma shows that periodic segments have locking property in somesense.
Lemma 4.2.
Let O be a periodic segment of Φ | Λ and x ∈ M . If d ( φ t ( x ) , O ) ≤ D ( O )4 C e β for all ≤ t ≤ T, (4.4) then there is a y ∈ O such that d ( φ t ( x ) , φ t ( y )) ≤ Cd ( φ t ( x ) , O ) for all ≤ t ≤ T. Especially, if d ( φ t ( x ) , O ) ≤ D ( O )4 C e β for all t ≥ , then x is a generic point of µ O . Proof.
Let δ ′ , ǫ ′ , δ, ǫ, λ, β, C be the constants as in Remark 3.2. Take a positive constant θ ≪ δ such that d ( φ t ( z ) , z ) ≤ D ( O )4 C for all | t | ≤ θ and z ∈ M . By assumption (4.4),there are y ′ t ∈ O such that d ( φ t ( x ) , y ′ t ) = d ( φ t ( x ) , O ) ≤ D ( O )4 C e β ≪ δ for all 0 ≤ t ≤ T. Let y t = φ v ( y ′ t ,φ t ( x )) ( y ′ t ) for 0 ≤ t ≤ T . Then φ t ( x ) ∈ D ( y t , Cd ( φ t ( x ) , O )) ⊂ D (cid:18) y t , D ( O )4 Ce β (cid:19) for all 0 ≤ t ≤ T. (4.5)Fix t , t ∈ [0 , T ] with | t − t | ≤ θ . Then, by (4.5), d ( y t , y t ) ≤ d ( y t , φ t ( x )) + d ( φ t ( x ) , φ t ( x )) + d ( φ t ( x ) , y t ) ≤ D ( O )4 Ce β + D ( O )4 C + D ( O )4 Ce β < D ( O ) C .
Thus, by Lemma 4.1, there is a constant τ satisfying | τ | ≤ Cd ( y t , y t ) ≪ δ such that φ τ ( y t ) = y t . By the uniqueness of v given in the basic canonical setting and the smallness of both θ and | τ | , one has τ = t − t . Hence, y t = φ t − t ( y t ) for all t , t ∈ [0 , T ] with | t − t | ≤ θ. Therefore, y t = φ t ( y ) for all t ∈ [0 , T ] by induction, which implies that d ( φ t ( y ) , φ t ( x )) = d ( y t , φ t ( x )) ≤ Cd ( y ′ t , φ t ( x )) = Cd ( φ t ( x ) , O ) for all 0 ≤ t ≤ T. Thus, y = y is the point as required.Now we assume that d ( φ t ( x ) , O ) ≤ D ( O )4 C e β for all t ≥
0. Then by the arguments above,there is y ∈ O such that d ( φ t ( x ) , φ t ( y )) ≤ D ( O )4 Ce β ≤ δ ′ for all t ≥ . Then by Lemma 3.4, we have d ( φ t φ v ( x ) , φ t ( y )) → t → + ∞ , where v = v ( x, y ). Then x must be a generic point of µ O . This ends the proof. (cid:3) Lemma 4.3.
There is positive constant τ such that for any x ∈ Λ φ t ( x ) / ∈ D ( x, δ ) for all < | t | ≤ τ . Proof.
Let δ ′ , ǫ ′ , δ, ǫ, λ, β, C be the constants as in Remark 3.2. Take τ > d ( φ t ( z ) , z ) ≤ δ ′ C for all | t | ≤ τ and z ∈ M . Suppose that there is an x ∈ Λ and 0 < | τ | ≤ τ such that φ τ ( x ) ∈ D ( x, δ ). Note w = W sǫ ( x ) ∩ W uǫ ( φ τ ( x )) . Then, by (1)(b) of the basic canonical setting , one has thatmax { d ( w, x ) , d ( w, φ τ ( x )) } ≤ Cd ( x, φ τ ( x )) ≤ δ ′ C .
Then for t ≥ d ( φ t ( w ) , φ t ( x )) ≤ Ce − λt d ( w, x ) ≤ Cd ( w, x ) ≤ δ ′ C , and for t < d ( φ t ( w ) , φ t ( x )) ≤ d ( φ t ( w ) , φ t φ τ ( x )) + d ( φ τ φ t ( x ) , φ t ( x )) ≤ δ ′ C + δ ′ C ≤ δ ′ , where we used w = W sǫ ( x ) ∩ W uǫ ( φ τ ( x )) and the selection of τ . Then by Lemma 3.4,there is a constant l with | l | ≪ δ such that w = φ l ( x ) = φ l − τ φ τ ( x ). It is clear thatat least one of l and l − τ is not zero since τ = 0. Without loss of any generality, weassume that l = 0, then { x, φ l ( x ) } ⊂ W sǫ ( x ) (otherwise { φ l ( x ) , φ τ ( x ) } ⊂ W uǫ ( φ τ ( x ))).Thus W sǫ ( x ) ∩ W uǫ ( φ l ( x )) = ∅ and W sǫ ( x ) ∩ W uǫ ( x ) = ∅ , which is impossible by the uniqueness of function v given in the basic canonicalsetting and A4) of Definition 2.1. This ends the proof. (cid:3) Remark 4.4.
Lemma 4.3 provides a lower bound τ of the periods of periodic segments. Remark 4.5.
We say a periodic segment O is pure if φ t ( y ) = y for all y ∈ O and0 < t < |O| . By Lemma 4.3, a periodic segment O is pure if and only if |O| = |O| min .4.1.2. Good periodic orbits.
In this subsection, we mainly demonstrate that for a givencompact invariant set, there exist periodic segments being closed enough as well as withreasonable large gap, which are the candidates to support certain minimizing measures.
Proposition 4.6.
For any α ∈ (0 , , a given ˜ L > and Φ -forward-invariant non-empty subset Z ⊂ Λ (i.e. φ t ( Z ) ⊂ Z, ∀ t ≥ ), there exists a periodic segment O of Φ Λ such that D α ( O ) d α,Z ( O ) > ˜ L. (4.6) Proof.
Fix 0 < α ≤
1, and recall that δ ′ , ǫ ′ , δ, ǫ, λ, β, C are as in Remark 3.2, and K , L are as in Lemma 3.5. For the sake of convenience, we assume that K ≥ L additionally.By Lemma 3.11, for any k ∈ N , there exists a periodic segment O of Φ | Λ with period P large enough such that d α,Z ( O ) < P − k << δ ′ . (4.7)We remark here that the period of a periodic segment is always assumed to be the MINIMUM period, which will avoid unnecessary complexity without harming theargument.If D α ( O ) > ˜ Ld α,Z ( O ), the proof is done. Otherwise, one has that D α ( O ) ≤ ˜ Ld α,Z ( O ) < ˜ LP − k . (4.8)Since P , k can be chosen as large as needed, one can request that D ( O ) < δ ′ . There-fore, by definition of D ( O ) (see (4.3)), there exist y ∈ O and t ∈ (0 , P ) such that φ t ( y ) ∈ D ( y, D ( O )) . Split the periodic segment O into two segments which are noted by Q : [0 , t ] → Λ : t → φ t ( y ); Q : [ t , P ] → Λ : t → φ t ( y ) . We choose the segment with smaller length and note it by Q . Then either Q L ∈D ( Q R , δ ′ ) or Q R ∈ D ( Q L , δ ′ ). It is clear that in either case d ( Q L , Q R ) ≤ δ ′ and d α,Z ( Q ) ≤ d α,Z ( O ) . Next, we will estimate the increment of orbit deviation after orbit splitting. We willemploy different discussions for two different situations according to the length of thesegment for which we set 3 K as a landmark. Case 1.
If the following condition holds |Q | > K, (4.9)also note that d ( Q L , Q R ) ≤ δ ′ , then Lemma 3.5 is applicable here, by which one hasthat there exists a periodic segment O such that ||Q | − |O || ≤ Ld ( Q L , Q R ) ≤ LD ( O ) < L ˜ LP − k , (4.10) d (cid:0) φ t (cid:0) Q L (cid:1) , φ t (cid:0) O L (cid:1)(cid:1) ≤ Ld ( Q L , Q R ) ∀ < t < max {|Q | , |O |} . (4.11)Since K ≥ L and δ ′ <<
1, (4.10) together with the assumption |Q | > K implies that |O | ≤ |Q | ≤ |O | . (4.12) d α,Z ( O ) = Z |O | d α (cid:0) φ t ( O L ) , Z (cid:1) dt = Z |O | d α (cid:16) φ t φ v ( O L , Q L ) ( O L ) , Z (cid:17) dt ≤ Z |Q | d α (cid:16) φ t φ v ( O L , Q L ) ( O L ) , Z (cid:17) dt + Z |Q | + ||Q |−|O ||||Q |−|O || d α (cid:16) φ t φ v ( O L , Q L ) ( O L ) , Z (cid:17) dt ≤ Z |Q | d α (cid:16) φ t φ v ( O L , Q L ) ( O L ) , φ t ( Q L ) (cid:17) dt ( Int ( a ))+ Z |Q | d α (cid:0) φ t ( Q L ) , Z (cid:1) dt ( Int ( b ))+ Z |Q | d α (cid:16) φ ||Q |−|O || φ t φ v ( O L , Q L ) ( O L ) , Z (cid:17) dt ( Int ( c )) . By applying Ash2) of Lemma 3.4, one has that
Int ( a ) ≤ Z |Q | C e − λ min( t, |Q |− t ) Ld α (cid:0) Q L , Q R (cid:1) dt ≤ C L ) α λα d α ( Q L , Q R ) . (4.13)By definition, one has that Int ( b ) = d α,Z ( Q ) . (4.14)By (3) of the basic canonical setting , one has that Int ( c ) = Z |Q | d α (cid:16) φ ||Q |−|O || φ t φ v ( O L , Q L ) ( O L ) , φ ||Q |−|O || Z (cid:17) dt ≤ (cid:0) Ce β ||Q |−|O || (cid:1) α Z |Q | d α (cid:16) φ t φ v ( O L , Q L ) ( O L ) , Z (cid:17) dt ≤ (cid:0) Ce β ||Q |−|O || (cid:1) α ( Int ( a ) + Int ( b )) . (4.15)By taking P and k large, one is able to make ||Q | − |O || <
1. Therefore, one has thefollowing simplified estimate d α,Z ( O ) ≤ L d α ( Q L , Q R ) + L d α,Z ( Q ) ≤ ˆ Ld α,Z ( O ) , (4.16)where L = (cid:0) C α e αβ + 1 (cid:1) C L ) α λα ; L = C α e αβ + 1;ˆ L = L ˜ L + L = (cid:0) C α e αβ + 1 (cid:1) (cid:18) C L ) α λα ˜ L + 1 (cid:19) . Once d α,Z ( O ) > ˜ Ld α,Z ( O ), O is the periodic segment as required, the splittingprocess stops. Otherwise, repeat the operation above as long as it is doable. Notethat such a process will stop at a finite time, since the operation above will reduce theperiod of periodic segment at least by (4.12). Therefore, under the assumption thatthe operation above is always doable, the process will end on an periodic segment O m for some m ∈ N ∪ { } , which either satisfies the requirement of this Proposition or |O m | ≥ K while |Q m | < K . In either cases, one has that m ≤ log P − log(3 K )log 1 . , and d α,Z ( O i ) ≤ ˆ L i d α,Z ( O ) , ∀ ≤ i ≤ m. In order to make each operation above doable, one needs that D ( O i ) < δ ′ , ∀ ≤ i ≤ m − , which can be done by assuming the largeness of P and k in advance. To be precise,one can take k > log ˆ L log 1 . P log ˆ L log 1 . − k < ( δ ′ ) α ˜ L ˆ L , (4.17)where the second inequality above implies that for all 0 ≤ i ≤ m − Ld α,Z ( O i ) ≤ P − k ˜ L ˆ L m < ( δ ′ ) α , which ensures the existence of O i +1 and ˜ Ld α,Z ( O m ) < ( δ ′ ) α . Case 2.
We will deal with the case that |Q m | ≤ K , which is the counterpart of thecase when (4.9) holds. We will show that by rearranging extra largeness of P and k ,one can make O m satisfy the requirement of Proposition 4.6. We will prove this bycontradiction.Before going to further discussion, we should note first that the union of all periodicorbits of Φ | Λ with period ≤ K is a nonempty compact subset of Λ, which is denotedby P er K . Once Z ∩ P er K = ∅ Proposition 4.6 holds automatically; otherwise, thereexists σ > d ( x, Z ) > σ ∀ x ∈ P er K . (4.18)Suppose that D α ( O m ) ≤ ˜ Ld α,Z ( O m ) < ( δ ′ ) α . (4.19)When K ≤ |Q m | ≤ K, (4.20)by the exactly same argument as on Q , one has that there exists a periodic segment O m +1 such that |O m +1 | ≤ K and d α,Z ( O m +1 ) ≤ ˆ Ld α,Z ( O m ) ≤ ˆ L m +1 d α,Z ( O ) < ˆ L m +1 P − k . By choosing P and k large enough one can make d α,Z ( O m +1 ) < σ which implies ancontradiction with (4.18). Therefore (4.19) and (4.20) can not hold simultaneously for P and k large enough.When |Q m | < K, (4.21)Lemma 3.5 is not applicable directly. For sake of convenience, note l = |Q m | . By (4.19), Q Lm ∈ D ( Q Rm , δ ′ ) or Q Rm ∈ D ( Q Lm , δ ′ ). Then by Lemma 4.3, l > τ . Let q be the integersuch that K ≤ ql ≤ K and thus 2 ≤ q ≤ (cid:20) Kτ (cid:21) . Note S i : [0 , l ] → M : t → φ il + t ( Q Lm ) for i = 0 , , , · · · , q − , and S : [0 , ql ] → M : t → φ t ( Q Lm ) . Then, d ( S L , S R ) = d ( Q Lm , Q Rm ) d ( S L , S R ) = d ( φ l ( S L ) , φ l ( S R )) ≤ Ce βl d ( S L , S R ) ≤ Ce βK d ( Q Lm , Q Rm ) ,. . .d ( S Lq − , S Rq − ) ≤ ( Ce βK ) q − d ( Q Lm , Q Rm ) . Therefore, d ( S L , S R ) ≤ q − X i =0 ( Ce βK ) i d ( Q Lm , Q Rm ) ≤ ( Ce βK ) h Kτ i − Ce βK − D ( O m ) , which together with (4.19) implies that d ( S L , S R ) ≤ ( Ce βK ) h Kτ i − Ce βK − (cid:16) ˜ Ld α,Z ( O m ) (cid:17) α ≤ ( Ce βK ) h Kτ i − Ce βK − (cid:16) ˜ L ˆ L m P − k (cid:17) α . (4.22)By taking P and k large enough, one can make d ( S L , S R ) < δ ′ . Also note that |S| ≥ K ,then Lemma 3.5 is applicable to S . Therefore, there exists a periodic segment O ∗ suchthat |O ∗ | ≤ |S| + Ld ( S L , S R ) ≤ K and d (cid:0) φ t φ v ( O L ∗ , S L ) ( O L ∗ ) , φ t ( S L ) (cid:1) ≤ L ( Ce βK ) h Kτ i − Ce βK − (cid:16) ˜ L ˆ L m P − k (cid:17) α ∀ ≤ t ≤ |S| , (4.23)where the right hand side of the above inequality can be make smaller than σ bytaking P and k large enough. On the other hand, d α,Z ( S ) = d α,Z ( Q m ) d α,Z ( S ) = Z l d α ( φ l + t ( S L ) , Z ) ≤ ( Ce βl ) α d α,Z ( S ) ≤ ( Ce βK ) α d α,Z ( Q m ) ,. . . d α,Z ( S q − ) ≤ ( Ce βK ) ( q − α d α,Z ( Q m ) . Thus, d α,Z ( S ) = q − X i =0 d α,Z ( S i ) ≤ q − X i =0 ( Ce βK ) iα d α,Z ( Q m ) ≤ [ Kτ ] − X i =0 C iα e iβKα d α,Z ( O m ) ≤ [ Kτ ] − X i =0 C iα e iβKα ˆ L m d α,Z u ( O ) ≤ ( Ce βK ) h Kτ i α − Ce βK ) α − L m P − k , (4.24)which can be make smaller than σ by taking P and k large enough. Since |S| ≥ K > t ∗ ∈ [0 , |S| ] such that d ( φ t ∗ ( S L ) , Z ) ≤ σ . Therefore, by (4.23), by taking P and k large enough, we have d ( φ t ∗ φ v ∗ ( O L ∗ ) , Z ) ≤ d (cid:0) φ t ∗ φ v ( O L ∗ , S L ) ( O L ∗ ) , φ t ∗ ( S L )) + d ( φ t ∗ ( S L ) , Z (cid:1) ≤ σ < σ which contradicts with (4.18) as K ≤ |S| ≤ K .Hence (4.19) can not hold for large enough P and k . This ends the proof. (cid:3) Here, we remark that there is no fixed point under the setting of this paper by A4)of Definition 2.1, thus the orbit splitting process cannot stop at a fixed point, whichis the main difference comparing to the discrete time case in [HLMXZ]. The Case 2.above is mainly taking care this issue.4.1.3.
Main Proposition.
In this subsection, we state and prove our main proposition.For a continuous function u and a segment S of Φ, define the integration of u along S with time interval [ a, b ] and starting point x by the following hS , u i := Z ba u (cid:0) φ t (cid:0) S L (cid:1)(cid:1) dt, (4.25)also recall that the definition of u γ for γ > u γ ( x ) = 1 γ Z γ u ( φ t ( x )) dt. Now we have the following proposition. Proposition 4.7.
Given < ε ≤ , < α ≤ , a strictly positive function ψ ∈ C ,α ( M ) and u ∈ C ,α ( M ) , if a periodic segment O of Φ | Λ satisfies the following comparisoncondition D α ( O ) d α,Z u,ψ ( O ) > C α ( k ¯ u k α + 10 ε + k ¯ u k +1 ψ min k ψ γ k α ) λαετ + 1 + 1 τ ! k ¯ u k α k ψ k ψ min · C e β ) α ε , (4.26) where ¯ u is defined in Remark 3.9 and τ is the constant in Lemma 4.3, then the periodicmeasure µ O ∈ M min ( u + εd α ( · , O ) + h ; ψ, Λ , Φ) , where h ∈ C ,α ( M ) satisfying k h k α < ε and k h k < min ε (cid:16) D ( O )4 C e β (cid:17) α (cid:18) C α ( k ¯ u k α +10 ε + k ¯ u k ψmin k ψ γ k α ) λαε + |O| + 1 (cid:19) k ψ k + ψ min ψ min , . Proof.
Fix ε, α, O , ψ, u, h as in the Proposition, δ ′ , ǫ ′ , δ, ǫ, λ, β, C as in Remark 3.2,¯ u, Z u,ψ , γ, ψ γ as in Remark 3.10, τ as in Lemma 4.3. Note G = ¯ u + εd α ( · , O ) + h − a O ψ γ where a O := hO , ¯ u + εd α ( · , O ) + h ihO , ψ γ i = hO , ¯ u + h ihO , ψ γ i . By straightforward computation, one has that | a O | ≤ hO , k ¯ u k α d α ( · , Z u,ψ ) + h ihO , ψ min i ≤ k ¯ u k α d α,Z u,ψ ( O ) |O| ψ min + k h k ψ min , (4.27)where we used ¯ u | Z u,ψ = 0. Notice that for all µ ∈ M (Λ , Φ) R ( u + εd α ( · , O ) + hd ) µ R ψdµ = R ( u γ + εd α ( · , O ) + h ) dµ R ψ γ dµ = R (¯ u + εd α ( · , O ) + h ) dµ R ψ γ dµ + β ( u ; ψ, Λ , Φ)= R Gdµ R ψdµ + a O + β ( u ; ψ, Λ , Φ) . Then, in order to show that µ O ∈ M min ( u + εd α ( · , O ) + h ; ψ, Λ , Φ), it is enough to showthat µ O ∈ M min ( G ; ψ γ , Λ , Φ). Since ψ is strictly positive and R Gdµ O = 0, it is enoughto show that Z Gdµ ≥ µ ∈ M e (Λ , Φ) . (4.28) Define a compact set
R ⊂ M by R = ( y ∈ M : d ( y, O ) ≤ (cid:18) | a O |k ψ k + k h k ε (cid:19) α ) . We have the following Claim.
Claim 1. R contains all x ∈ M with G ( x ) ≤ .Proof of Claim 1. Given x ∈ M \ R , we are to show that G ( x ) >
0. Note that¯ u + h − a O ψ γ ≥ −| a O |k ψ γ k − k h k . (4.29)where we used ¯ u ≥ k ψ γ k ≤ k ψ k . Then G ( x ) = ¯ u ( x ) + εd α ( x, O ) + h ( x ) − a O ψ γ ≥ εd α ( x, O ) − | a O |k ψ k − k h k > ε · (cid:18) | a O |k ψ k + k h k ε (cid:19) α ! α − | a O |k ψ k − k h k = 0 . This ends the proof of Claim 1. (cid:3)
Define a compact set R ′ ⊂ M by R ′ = ( y ∈ M : d ( y, O ) ≤ (cid:18) | a O |k ψ k + k h k ) ε (cid:19) α ) . It is easy to see that R is in the interior of R ′ and the following holds because of (4.26),(4.27) and the range of k h k d ( y, O ) ≤ (cid:18) a O k ψ k + k h k ) ε (cid:19) α ≤ D ( O )4 C e β , ∀ y ∈ R ′ . (4.30)By Claim 1, there is a constant τ with 0 < τ < G ( φ t ( x )) > x ∈ M \ R ′ and | t | ≤ τ .Now we claim the following assertion: Claim 2. If z ∈ M is not a generic point of µ O , then there is m ≥ τ such that Z m G ( φ t ( z )) dt > . Next we prove the Proposition by assuming the validity of Claim 2, while the proofof Claim 2. is left to the end of this section. For a given ergodic measure µ ∈ M e (Λ , Φ), if µ = µ O , (4.28) obviously holds. Otherwise, let z be a generic point of µ , thus z isnot a generic point of µ O . Therefore, by Claim 2, there is a t ≥ τ such that Z t G ( φ t ( z )) dt > . Since φ t ( z ) is still not a generic point of µ O , by using claim 2 agian, one has a t ≥ t + τ such that Z t t G ( φ t ( z )) dt > . By repeating the above process, one has 0 ≤ t < t < t < · · · with all gaps not lessthan τ such that Z t i +1 t i G ( φ t ( z )) dt > i = 0 , , , , · · · , where we assign t = 0. Therefore Z Gdµ = lim l → + ∞ l Z l G ( φ t ( z )) dt = lim i → + ∞ t i (cid:18)Z t t G ( φ t ( z )) dt + Z t t G ( φ t ( z )) dt + · · · + Z t i t i − G ( φ t ( z )) dt (cid:19) ≥ . Thus, µ O ∈ M min ( u + εd α ( · , O ) + h ; ψ, Λ , Φ). This ends the proof. (cid:3)
Remark 4.8.
It is not difficult to see that for any ε ′ > ε , µ O is the unique measure in M min ( u + ε ′ d α ( · , O ) + h ; ψ, Λ , Φ) whenever k h k α < ε and k h k is sufficiently small.The Proposition shows that there is an open set of C ,α ( M ) near u such that these α -H¨older functions in the open set has the same unique minimizing measure with respectto ψ being supported on a periodic orbit. Proof of Claim 2. If z / ∈ R ′ , just take m = τ , we have nothing to prove since G ( φ t ( z )) > | t | ≤ τ . Therefore, one needs only to consider the case that z ∈ R ′ . Also notethat, since z is not a generic point of µ O , Lemma 4.2 implies that the following inequality d ( φ t ( z ) , O ) ≤ D ( O )4 C e β CANNOT hold for all t ≥
0. Thus there is an m > d ( φ m ( z ) , O ) > D ( O )4 C e β . Let m > d ( φ m ( z ) , O ) = D ( O )4 C e β , (4.31) where the existence of such m is ensured by (4.30) and the continuity of the flow.Then, by (3) of the basic canonical setting , one has the following d ( φ m − t ( z ) , O ) > D ( O )4 C e β , ∀ < t ≤ , which together with (4.30) implies that φ m − t ( z ) / ∈ R ′ for all 0 < t ≤ . (4.32)Thus Z m m − G ( φ t ( z )) dt = Z m m − ¯ u ( φ t ( z )) + εd α ( φ t ( z ) , O ) + h ( φ t ( z )) − a O ψ γ ( φ t ( z )) dt ≥ Z m m − εd α ( φ t ( z ) , O ) − | a O |k ψ k − k h k dt ≥ ε · (cid:18) D ( O )4 C e β (cid:19) α − | a O |k ψ k − k h k , (4.33)where we used (4.29).Now since R ′ is compact, there is an m which is the largest time such that 0 ≤ m ≤ m and φ t ( z ) ∈ R ′ . By (4.32), it is clear that m ≤ m −
1. Then by Claim 1,for all m < t < m − G ( φ t ( z )) > , (4.34)where we used the fact R ⊂ R ′ . On the other hand, since m < m , by the choice of m (see (4.31)), one has that d ( φ t ( z ) , O ) ≤ (cid:18) a O k ψ k + k h k ) ε (cid:19) α < D ( O )4 C e β for all 0 ≤ t ≤ m . Therefore, by Lemma 4.2, there is y ∈ O such that d ( φ t ( z ) , φ t ( y )) ≤ C (cid:18) a O k ψ k + k h k ) ε (cid:19) α ≤ δ ′ for all t ∈ [0 , m ] . By using Lemma 3.4, we have for all 0 ≤ t ≤ m , d ( φ t φ v ( y ,z ) ( y ) , φ t ( z )) ≤ C e − λ min( t,m − t ) C (cid:18) a O k ψ k + k h k ) ε (cid:19) α . Hence, Z m d α ( φ t ( z ) , φ t φ v ( y ,z ) ( y )) dt ≤ Z m C (cid:18) | a O |k ψ k + k h k ) ε (cid:19) α (cid:0) e − λt + e − λ ( m − t ) (cid:1)! α dt ≤ C α λα · | a O |k ψ k + k h k ε . Therefore, Z m G ( φ t ( z )) − G ( φ t φ v ( y )) dt = Z m ¯ u ( φ t ( z )) + εd α ( φ t ( z ) , O ) + h ( φ t ( z )) − ¯ u ( φ t + v ( y )) − h ( φ t + v ( y )) − a O (cid:0) ψ γ ( φ t ( z )) − ψ γ ( φ t + v ( y )) (cid:1) dt ≥ Z m ¯ u ( φ t ( z )) − ¯ u ( φ t + v ( y )) + h ( φ t ( z )) − h ( φ t + v ( y )) − a O (cid:0) ψ γ ( φ t ( z )) − ψ γ ( φ t + v ( y )) (cid:1) dt ≥ − ( k ¯ u k α + k h k α + | a O |k ψ γ k α ) Z m d α ( φ t ( z ) , φ t + v ( y )) dt ≥ − ( k ¯ u k α + k h k α + | a O |k ψ γ k α ) · C α λα · | a O |k ψ k + k h k ε , where we write v short for v ( y , z ). Also note that | a O | = (cid:12)(cid:12)(cid:12)(cid:12) hO , ¯ u + h ihO , ψ γ i (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ¯ u k + k h k ψ min ≤ k ¯ u k + 1 ψ min . Thus, one has that Z m G ( φ t ( z )) − G ( φ t φ v ( y )) dt ≥ − ( k ¯ u k α + k h k α + k ¯ u k + 1 ψ min k ψ γ k α ) · C α λα · | a O |k ψ k + k h k ε , (4.35)Rewrite m = p |O| + r for some nonnegative integer p and real number 0 ≤ r ≤ |O| .By applying (4.29) and R Gdµ O = 0, one has that Z m G ( φ t φ v ( y )) dt = Z m m − r G ( φ t φ v ( y )) dt ≥ −|O| · ( | a O |k ψ k + k h k ) . (4.36)Combining (4.27), (4.33), (4.34), (4.35) and (4.36), we have Z m G ( φ t ( z )) dt ≥ Z m G ( φ t ( z )) dt + Z m m − G ( φ t ( z )) dt by (4 . = Z m ( G ( φ t ( z )) − G ( φ t + v ( y ))) dt + Z m G ( φ t + v ( y )) dt + Z m m − G ( φ t ( z )) dt ≥ − ( k ¯ u k α + k h k α + k ¯ u k + 1 ψ min k ψ γ k α ) · C α λα · | a O |k ψ k + k h k ε by (4 . − |O| · ( | a O |k ψ k + k h k ) by (4 . ε (cid:18) D ( O )4 C e β (cid:19) α − | a O |k ψ k − k h k by (4 . ≥ ε (cid:18) D ( O )4 C e β (cid:19) α − C α ( k ¯ u k α + k h k α + k ¯ u k +1 ψ min k ψ γ k α ) λαε |O| + 1 + 1 |O| ! k ¯ u k α k ψ k ψ min d α,Z u,ψ ( O ) − C α ( k ¯ u k α + k h k α + k ¯ u k +1 ψ min k ψ γ k α ) λαε + |O| + 1 ! k ψ k + ψ min ψ min k h k by (4 . ≥ ε (cid:18) D ( O )4 C e β (cid:19) α − C α ( k ¯ u k α + 10 ε + k ¯ u k +1 ψ min k ψ γ k α ) λαετ + 1 + 1 τ ! k ¯ u k α k ψ k ψ min d α,Z u,ψ ( O ) − C α ( k ¯ u k α + 10 ε + k ¯ u k +1 ψ min k ψ γ k α ) λαε + |O| + 1 ! k ψ k + ψ min ψ min k h k > , where we used Remark 4.4 and condition (4.26). Therefore, m = m is the time asrequired since m ≥ ≥ τ . This ends the proof of Claim 2. (cid:3) Proof of Part II) of Theorem 2.2.
Firstly, we state a technical result on func-tion approximation, which plays a key role in proving Proposition 4.11. Proposition4.11 can be viewed as a C -version of Proposition 4.7 which implies the part II) ofTheorem 2.2. Theorem 4.9 ([GW]) . Let M be a smooth compact manifold. Then C ∞ ( M ) ∩ C , ( M ) is Lip-dense in C , ( M ) . Remark 4.10.
In this theorem, C ∞ ( M ) ∩ C , ( M ) is Lip-dense in C , ( M ) meansthat for any g ∈ C , ( M ) and ε > g ∈ C ∞ ( M ) such that k g − g k < ε and k g k < ε + k g k . Especially, k D M g k < ε + k g k , where D M g isthe derivative of function g with respect to space variables. Proposition 4.11.
Given < ε ≤ , a strictly positive ψ ∈ C , ( M ) and u ∈ C , ( M ) ,if a periodic segment O of Φ | Λ satisfies the following comparison condition D ( O ) > (cid:16) C ( k ¯ u k + 10 ε + k ¯ u k +1 ψ min k ψ γ k ) λετ + 1 + 1 τ (cid:17) k ¯ u k k ψ k ψ min · C e β ε · d ,Z u,ψ ( O ) , where ¯ u is defined in Remark 3.9 and τ is the constant in Lemma 4.3, then there isa function w ∈ C ∞ ( M ) with k w k < ε · diam ( M ) and k D M w k < ε such that theprobability measure µ O ∈ M min ( u + w + h ; ψ, Λ , Φ) , where h is any C function with k D M h k < ε and k h k < · min ε (cid:16) D ( O )4 C e β (cid:17)(cid:18) C ( k ¯ u k +10 ε + k ¯ u k ψmin k ψ γ k ) λε + |O| + 1 (cid:19) k ψ k + ψ min ψ min , . Proof.
By Theorem 4.9, there exists a function w ∈ C ∞ such that k D M w k < k εd ( · , O ) k + ε ≤ ε and k w − εd ( · , O ) k < min (cid:18) H , ε · diam ( M ) (cid:19) , where H = min ε (cid:16) D ( O )4 C e β (cid:17)(cid:18) C ( k ¯ u k +10 ε + k ¯ u k ψmin k ψ γ k ) λε + |O| + 1 (cid:19) k ψ k + ψ min ψ min , . Next we show that w is the function as required. Note that u + w + h = u + εd ( · , O ) + ( w − εd ( · , O ) + h ) . Notice that, k w − εd ( · , O ) + h k ≤ k D M w k + k εd ( · , O ) k + k h k ≤ ε + ε + 5 ε < ε, and k w − εd ( · , O ) + h k ≤ k w − εd ( · , O ) k + k h k < H H H. Then by Proposition 4.7, we have that µ O ∈ M min ( u + w + h ; ψ, Λ , Φ) . Additionally, k w k < k εd ( · , O ) k + ε · diam ( M ) ≤ ε · diam ( M ) . This ends the proof. (cid:3)
Remark 4.12.
Let e w ∈ C , ( M ) be such that k e w k , < ε , e w | O = 0 and e w | M \O > µ O is the unique measure in M min ( u + e w + w + h ; ψ, Λ , Φ) whenever k h k < ε and k h k is sufficiently small. The Proposition shows that there is an open set of C , ( M )near u such that functions in the open set have the same unique minimizing measurewith respect to ψ and the measure supports on a periodic orbit.5. Proofs of Technical Lemmas
Note that throughout this section, δ, ǫ, λ, β, C, ǫ ′ , δ ′ are same as the ones in Remark3.2.5.1. Proof of Lemma 3.4.
Proof.
We put a small positive constant τ with τ ≪ | s ( t ) − s ( t ) | ≤ η forall | t − t | ≤ τ and t , t ∈ [0 , T ] . Since η ≤ C e β +10 λ e λ − δ ′ , for all 0 ≤ t ≤ T, there exists r ( t ) with | r ( t ) | < Cη such that w ( φ t + s ( t )+ r ( t ) ( y ) , φ t ( x )) = W sǫ ′ ( φ t + s ( t )+ r ( t ) ( y )) ∩ W uǫ ′ ( φ t ( x )) . (5.1)Then for t ′ ∈ [ − τ, τ ] and t ∈ [ τ, T − τ ], φ t ′ ( w ( φ t + s ( t )+ r ( t ) ( y ) , φ t ( x ))) ∈ W sǫ ′ ( φ t + s ( t )+ r ( t )+ t ′ ( y )) ∩ W uǫ ′ ( φ t + t ′ ( x )) . On the other hand, one has that w ( φ t + t ′ + s ( t + t ′ )+ r ( t + t ′ ) ( y ) , φ t + t ′ ( x )) = W sǫ ′ ( φ t + t ′ + s ( t + t ′ )+ r ( t + t ′ ) ( y )) ∩ W uǫ ′ ( φ t + t ′ ( x )) . Since | ( t + t ′ + s ( t + t ′ ) + r ( t + t ′ )) − ( t + s ( t ) + r ( t ) + t ′ ) | ≤ (2 C + 1) η ≪ δ, by the uniqueness of v ( φ t + s ( t )+ r ( t )+ t ′ y, φ t + t ′ x ) given by (1) of the basic canonical set-ting , one has that t + t ′ + s ( t + t ′ ) + r ( t + t ′ ) = t + s ( t ) + r ( t ) + t ′ , and w ( φ t + t ′ + s ( t + t ′ )+ r ( t + t ′ ) ( y ) , φ t + t ′ ( x )) = φ t ′ ( w ( φ t + s ( t )+ r ( t ) ( y ) , φ t ( x ))) , for all t ′ ∈ [ − τ, τ ] and t ∈ [ τ, T − τ ]. Since τ can be taken arbitrarily small, one hasthe following by induction s ( t ) + r ( t ) = s ( τ ) + r ( τ ) = s (0) + r (0) = r (0) = v ( y, x ) , ∀ t ∈ [0 , T ] . (5.2)Thus | s ( t ) | ≤ | r ( t ) | + | r (0) | ≤ Cη, and for all t ∈ [ τ, T − τ ] and t ′ ∈ [ − τ, τ ] w ( φ t + t ′ + v ( y,x ) ( y ) , φ t + t ′ ( x )) = φ t ′ ( w ( φ t + v ( y,x ) ( y ) , φ t ( x ))) , which implies that w ( φ t + v ( y,x ) ( y ) , φ t ( x )) = φ t ( w ( φ v ( y,x ) ( y ) , x )) ∀ t ∈ [0 , T ] . (5.3)Now, we prove Ash 2). Note w = w ( φ v ( y,x ) ( y ) , x ) = W sǫ ′ ( φ v ( y,x ) ( y )) ∩ W uǫ ′ ( x ). Thenby (5.1), (5.2) and (5.3), one has φ t ( w ) = W sǫ ′ ( φ t φ v ( y,x ) ( y )) ∩ W uǫ ′ ( φ t ( x )) for all t ∈ [0 , T ] . Thus, for all t ∈ [0 , T ], by (b) of the basic canonical setting , d ( φ t ( w ) , φ t φ v ( y,x ) ( y )) < Cd ( φ t ( x ) , φ t ( y )) and d ( φ t ( w ) , φ t ( x )) < Cd ( φ t ( x ) , φ t ( y )) . Therefore d ( φ t ( w ) , φ t ( x )) ≤ Ce − λ ( T − t ) d ( φ T ( w ) , φ T ( x )) ≤ C e − λ ( T − t ) d ( φ T ( x ) , φ T ( y )) , where we used w ∈ W uǫ ′ ( x ) and d ( φ t ( w ) , φ t φ v ( y,x ) ( y )) ≤ Ce − λ ( t ) d ( w, φ v ( y,x ) ( y )) ≤ C e − λ ( t ) d ( x, y ) . where we used w ∈ W sǫ ′ ( φ ( y,x ) v ( y )). By summing up, we have d ( φ t φ v ( y,x ) ( y ) , φ t ( x )) ≤ C e − λ min( t,T − t ) ( d ( x, y ) + d ( φ T ( x ) , φ T ( y ))) for 0 ≤ t ≤ T. Now we assume that d ( φ t + s ( t ) ( y ) , φ t ( x )) ≤ η for all t ≥ , then by the argumentsabove. We have for all t ≥ d ( φ t φ v ( y,x ) ( y ) , φ t ( x )) ≤ C e − λ min( t, t − t ) ( d ( x, y ) + d ( φ t ( x ) , φ t ( y ))) ≤ C ηe − λ min( t, t − t ) → t → + ∞ . This ends the proof. (cid:3)
Proof of Lemma 3.5.
Proof.
We partially follow Bowen’s arguments in [Bowen]. Firstly we fix a constant K ≫ C with 2 C e − λK ≪ S as in Lemma 3.5. We let τ = |S| and η = d ( S L , S R ). Then η < δ ′ and 2 Ce − λτ ≪
1. Therefore, we have the following claim.
Claim A.
For the segment S in Lemma 3.5, there is a y ∈ Λ and a continuous function ˆ s : R → R with ˆ s (0) = 0 and Lip (ˆ s ) ≤ Cητ such that d ( φ iτ + t + s ( iτ + t ) ( y ) , φ t ( S L )) ≤ L η for all t ∈ [0 , τ ] and i ∈ Z where L = 2 C ( e λ − + e β + 2) . Since the proof of Claim A is long, we postpone the proof of Claim A to the nextsubsection. Let y ∈ M and ˆ s : R → R be as in Claim A. We divide the following proofinto two steps. Step 1.
At first, we show that y is a periodic point. By Claim A, d ( φ t +ˆ s ( t ) ( y ) , φ t + τ +ˆ s ( t + τ ) ( y )) ≤ L η for t ∈ R . Since
Lip (ˆ s ) ≪ s (0) = 0, g ( t ) = t + ˆ s ( t ) is a homomorphism of R onto itself, theabove inequality can be rewritten as the following d ( φ t ( y ) , φ g − ( t )+ τ +ˆ s ( g − ( t )+ τ ) ( y )) ≤ L η for t ∈ R . We note y ′ = φ g − (0)+ τ +ˆ s ( g − (0)+ τ ) ( y )and s ( t ) = g − ( t ) + ˆ s ( g − ( t ) + τ ) − g − (0) − ˆ s ( g − (0) + τ ) − t. Then d ( φ t ( y ) , φ t + s ( t ) ( y ′ )) ≤ L η for t ∈ R and s (0) = 0 . Therefore, by Lemma 3.4, one has φ v ( y ′ ) = y and | v | ≤ CL η, where v = v ( y ′ , y ). Thus φ g − (0)+ τ +ˆ s ( g − (0)+ τ )+ v ( y ) = y. Notice that g − (0) = 0 since g (0) = 0. Thus, | g − (0) + ˆ s ( g − (0) + τ ) + v | ≤ | ˆ s ( τ ) | + | v | ≤ (2 C + 2 CL ) η ≪ τ. Therefore, y is a periodic point. Step 2.
There is a periodic segment O such that ||S| − |O|| ≤ Ld ( S L , S R )and d ( φ t ( O L ) , φ t ( S L )) ≤ Ld ( S L , S R ) for all 0 ≤ t ≤ max( |S| , |O| ) , where L = 2 CL + L and L = 2 C L + C L . By Claim A, d ( φ t +ˆ s ( t ) ( y ) , φ t ( S L )) ≤ L η for t ∈ [0 , τ ] . By Lemma 3.4, for t ∈ [0 , τ ] , | ˆ s ( t ) | ≤ CL η and d ( φ t φ v ( y ) , φ t ( S L )) ≤ C e − λ min( t,τ − t ) ( d ( y, S L ) + d ( φ τ ( y ) , φ τ ( S L ))) ≤ C ( d ( y, S L ) + d ( φ τ +ˆ s ( τ ) ( y ) , φ τ ( S L )) + d ( φ τ +ˆ s ( τ ) ( y ) , φ τ ( y ))) ≤ L η, (5.4)where v = v ( y, S L ). Now we put y ∗ = φ v y and we have a periodic segment, O : [0 , τ + g − (0) + ˆ s ( g − (0) + τ ) + v ] → M : t → φ t ( y ∗ ) , where v is as in Step 1. It is clear that ||S| − |O|| ≤ | g − (0) + ˆ s ( g − (0) + τ ) + v | ≤ L η. If |O| ≤ |S| , by (5.4), d ( φ t ( y ∗ ) , φ t ( S L )) ≤ L η, for t ∈ [0 , max( |S| , |O| )] , where we used τ = max( |S| , |O| ) . If |O| > |S| , by (5.4), d ( φ t ( y ∗ ) , φ t ( S L )) ≤ L η, for t ∈ [0 , |S| ] , and for t ∈ ( |S| , |O| ], d ( φ t ( y ∗ ) , φ t ( S L )) ≤ d ( φ t ( y ∗ ) , φ τ ( y ∗ )) + d ( φ τ ( y ∗ ) , φ τ ( S L )) + d ( φ τ ( S L ) , φ t ( S L )) ≤ Lη, where L = 2 CL + L . This ends the proof since L ≤ L . (cid:3) Proof of Claim A.Proof.
Recall that S is a segment of Φ | Λ with |S| = τ ≥ K and d ( S L , S R ) = η < δ ′ where K satisfies 2 C e − λK ≪
1. We define x − k , ζ − k recursively for k ≥ x = S R , ζ = 0and ζ − k − = v ( φ − τ ( x − k ) , S R ) , x − k − = W sǫ ′ ( φ − τ + ζ − k − ( x − k )) ∩ W uǫ ′ ( S R ) for k = 1 , , · · · . We have the following two assertions.
Assertion 1. x − k and ζ − k are well defined and d ( x − k , S R ) ≤ Cη for k ≥ . Proof.
In the case k = 0, it is obviously true. Now assume that we have ζ − k , x − k and d ( x − k , S R ) ≤ Cη . Then d ( φ − τ ( x − k ) , S R ) ≤ d ( φ − τ ( x − k ) , φ − τ ( S R )) + d ( S R , φ − τ ( S R )) ≤ Ce − λτ · Cη + η ≤ η, (5.5)where we used x − k ∈ W uǫ ′ ( S R ). Since 2 η ≤ δ ′ , x − k − is well defined as well as ζ − k − ,and moreover one has that d ( x − k − , S R ) ≤ Cη.
This ends the proof. (cid:3)
By (5.5), we have that | ζ k | ≤ Cη ≪ . (5.6)Next we denote x ( − k ) = φ kτ − P ki =0 ζ − i ( x − k ) for k ≥
0. For k ∈ N , we define s ∗− k : R → R by s ∗− k ( t ) = ζ , if t > , P l − i =0 ζ − i , if − lτ < t ≤ − ( l − τ, l ∈ { , , · · · , k } , P ki =0 ζ − i , if t ≤ − kτ. Assertion 2.
There exists a constant L such that for t = − jτ − t satisfying t ∈ [0 , τ ) and j ∈ { , , · · · , k − } , the following holds d (cid:16) φ t + s ∗− k ( t ) (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:17) ≤ L η. Proof.
We fix t = − jτ − t for some j ∈ { , , · · · , k − } and t ∈ [0 , τ ). Since x − j ∈ W uǫ ′ ( S R ), we have d (cid:0) φ − t ( x − j ) , φ − t (cid:0) S R (cid:1)(cid:1) ≤ Ce − λt d (cid:0) x − j , S R (cid:1) ≤ C η. (5.7)Note that τ − ζ − j − − t ≥ − x − j − ∈ W sǫ ′ ( φ − τ + ζ − j − ( x − j )) ∩ W uǫ ′ ( S R ), we have d (cid:0) φ τ − ζ − j − − t ( x − j − ) , φ − t ( x − j ) (cid:1) = d (cid:0) φ τ − ζ − j − − t ( x − j − ) , φ τ − ζ − j − − t φ − τ + ζ − j − ( x − j ) (cid:1) ≤ e β d (cid:0) x − j − , φ − τ + ζ − j − ( x − j ) (cid:1) ≤ C e β η, (5.8)where we used (2) and (3) of the basic canonical setting for the case τ − ζ − j − − t > τ − ζ − j − − t ≤
0, respectively. Note that | ζ − l | ≪ , τ ≫ t ∈ [0 , τ ), i.e., τ − ζ − j − − t ≥ − τ − ζ − i >τ − >
1. Since x − ( k − l ) ∈ W sǫ ′ ( φ − τ + ζ − ( k − l ) ( x − ( k − j − )) ∩ W uǫ ′ ( S R ), we have k − j − X l =0 d (cid:16) φ ( k − j − l − τ − P k − li = j +1 ζ − i − t (cid:0) x − ( k − l ) (cid:1) , φ ( k − j − l − τ − P k − l − i = j +1 ζ − i − t (cid:0) x − ( k − l − (cid:1)(cid:17) ≤ k − j − X l =0 Ce − λ ( ( k − j − l ) τ − P k − li = j +1 ζ − i − t ) d (cid:16) x − ( k − l ) , φ − τ + ζ − ( k − l ) (cid:0) x − ( k − j − (cid:1)(cid:17) ≤ k − j − X l =0 Ce − λ ( ( k − j − l ) τ − P k − li = j +1 ζ − i − t )4 Cη ≤ C ηe − λ (2 τ − ζ − j − − ζ − j − − t ) k − l − X l =0 e − λ · ( τ − · l ≤ C ηe − λ ( τ − − e − λ = 4 C η e λ − , (5.9)where we used 1b) of the basic canonical setting . Combining (5.7), (5.8) and (5.9),we have that for t = − jτ − t d (cid:16) φ t + s ∗− k ( t ) (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:17) = d (cid:16) φ − jτ + P ji =0 ζ − i − t (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:17) = d (cid:16) φ ( k − j ) τ − P ki = j +1 ζ − i − t ( x − k ) , φ − t (cid:0) S R (cid:1)(cid:17) ≤ k − j − X l =0 d (cid:16) φ ( k − j − l − τ − P k − li = j +1 ζ − i − t (cid:0) x − ( k − l ) (cid:1) , φ ( k − j − l − τ − P k − l − i = j +1 ζ − i − t (cid:0) x − ( k − l − (cid:1)(cid:17) + d ( φ τ − ζ − j − − t ( x − j − ) , φ − t ( x − j )) + d (cid:0) φ − t ( x − j ) , φ − t ( S R ) (cid:1) ≤ L η, where L = 2 C ( e λ − + e β + 1). This ends the proof of Assertion 2. (cid:3) Now for k ∈ N , we define ¯ s − k : R → R by¯ s − k ( t ) = ζ , if t > , P l − i =0 ζ − i − t +( l − ττ ζ − l , if − lτ < t ≤ − ( l − τ, l ∈ { , , · · · , k } , P ki =0 ζ − i , if t ≤ − kτ. It is clear that ¯ s − k is Lipschitz continuous with Lip (¯ s − k ) ≤ max i ∈{ , , , ··· ,k } | ζ i | τ ≤ Cητ , and | ¯ s − k ( t ) − s ∗− k ( t ) | ≤ max i ∈{ , , , ··· ,k } | ζ i | ≤ Cη.
Therefore, by Assertion 2, when t = − jτ − t for some j ∈ { , , , · · · , k − } and t ∈ [0 , τ ), one has that d (cid:0) φ t +¯ s − k ( t ) (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:1) ≤ d (cid:16) φ t +¯ s − k ( t ) (cid:0) x ( − k ) (cid:1) , φ t + s ∗− k ( t ) (cid:0) x ( − k ) (cid:1)(cid:17) + d (cid:16) φ t + s ∗− k ( t ) (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:17) ≤ L η, (5.10)where L = L + 2 C . Now for k ∈ N , we define s − k : R → R by s − k ( t ) = ¯ s − k ( t − kτ ) − k X i =0 ζ − i . It is clear that s − k (0) = 0. On the other hand, we note y k = φ − τk + P ki =0 ζ − i ( x ( − k ) ).Thus, when t = − jτ − t for some j ∈ {− k, − k + 1 , · · · , k − } and t ∈ [0 , τ ), (5.10)implies that d (cid:0) φ t + s − k ( t ) ( y k ) , φ − t (cid:0) S R (cid:1)(cid:1) = d (cid:0) φ t − τk +¯ s − k ( t − τk ) (cid:0) x ( − k ) (cid:1) , φ − t (cid:0) S R (cid:1)(cid:1) ≤ L η. Notice that s − k are Lipschitz with Lip ( s − k ) ≤ Cητ ≪ η for all k ∈ N . Applying theAscoli-Azel´a theorem, there exists a subsequence ( s − k i ) + ∞ i =1 that converges to a Lipschitzcontinuous function ˆ s : R → R with Lip (ˆ s ) ≤ Cητ ≪ η and ˆ s (0) = 0. Without losingany generality, we assume that y k i → y as i → + ∞ . By the continuity, if t = − jτ − t for some j ∈ Z and t ∈ [0 , τ ), then d ( φ t +ˆ s ( t ) ( y ) , φ − t ( S R )) ≤ L η. That is to say, if t = − ( j + 1) τ + ( τ − t ) for some j ∈ Z and t ∈ [0 , τ ), then d (cid:0) φ t +ˆ s ( t ) ( y ) , φ τ − t ( S L ) (cid:1) ≤ L η. Note that y k i ∈ Λ for each i ∈ N , thus y ∈ Λ. Let t = τ − t , then the proof of ClaimA is completed. (cid:3) Proof of Lemma 3.7.
In this section, we mainly prove a version of the so calledMa˜n`e-Conze-Guivarc’h-Bousch’s Lemma. The proof partially follows Bousch’s argu-ments in [Bo3].5.3.1.
Integration along segment.
Recall that, for a continuous function u and a segment S of Φ, the integration of u along S is defined by hS , u i := Z ba u ( φ t ( x )) dt. Lemma 5.1.
Let u : M → R be an α -H¨older function with β ( u ; 1 , Λ , Φ) ≥ . Then fora segment S of Φ | Λ satisfying |S| ≥ K and d ( S L , S R ) ≤ δ ′ , the following holds hS , u i ≥ − K d α ( S L , S R ) , where K = ( CL ) α λα k u k α + L k u k .Proof. Since d ( S L , S R ) ≤ δ ′ and |S| ≥ K , by Anosov Closing Lemma, there exists aperiodic segment O of Φ | Λ such that ||S| − |O|| ≤ Ld ( S L , S R )and d ( φ t ( O L ) , φ t ( S L )) ≤ Ld ( S L , S R ) for all 0 ≤ t ≤ max( |S| , |O| ) . Therefore, Letting v = v ( O L , S L ) as in Lemma 3.4, one has that hS , u i − hO , u i = Z |O| u ( φ t ( S L )) − u ( φ t φ v ( O L )) dt + Z |S||O| u ( φ t ( S L )) dt ≥ −k u k α Z |O| d α ( u ( φ t ( S L )) , u ( φ t φ v ( O L ))) dt − k u k ||S| − |O||≥ −k u k α Z |O| (cid:0) Ce − λ min( t,T − t ) Ld ( S L , S R ) (cid:1) α dt − k u k Ld ( S L , S R ) ≥ − (cid:18) ( CL ) α λα k u k α + L k u k (cid:19) d α ( S L , S R ) , where we used the assumption 0 < α ≤ < d ( S L , S R ) < δ ′ ≪
1. Then theLemma is immediately from the fact hO , u i ≥ β ( u ; 1 , Λ , Φ) ≥
0. This ends theproof. (cid:3)
Lemma 5.2.
Let P be a finite partition of M with diameter smaller than δ ′ and u : M → R be an α -H¨older function with β ( u ; 1 , Λ , Φ) ≥ . Then for a given segment S of Φ | Λ , the following holds hS , u i ≥ − K δ ′ α , where K = ♯ P · (cid:16) K k u k δ ′ α + K (cid:17) and K , K are as in Lemma 3.5 and 5.1 respectively.Proof. For x ∈ M , denote P ( x ) the element in P which contains x . Assume |S| =( n − K + r for some n ≥ ≤ r < K . Let t i = iK for 0 ≤ i ≤ n − t n = |S| . We define the function w : N → [0 , n ] ∩ N inductively by letting w (0) = 0 w ( k ) = min { η ( w ( k − , n } . where η : [0 , n − ∩ N → [0 , n − ∩ N is the function that maps each i to the largest j ∈ [0 , n − ∩ N such that P ( φ t i ( S L )) = P ( φ t j ( S L )). Let s ≥ for which η ( w ( s )) = n −
1. Then P ( φ t w ( i ) ( S L )) = P ( φ t w ( j ) ( S L )) for 0 ≤ i < j ≤ s whichimplies s ≤ ♯ P . For 0 ≤ j ≤ s , we have two cases: If η ( w ( j )) = w ( j ) Z t η ( w ( j )) t w ( j ) u ( φ t ( x )) dt = 0 and Z t η ( w ( j ))+1 t η ( w ( j )) u ( φ t ( x )) dt ≥ − K k u k . (5.11)If η ( w ( j )) > w ( j ), by using Lemma 5.1, Z t η ( w ( j )) t w ( j ) u ( φ t ( x )) dt ≥ − (cid:18) ( CL ) α λα k u k α + L k u k (cid:19) δ ′ α , (5.12)where we use the fact d ( φ t i ( S L ) , φ t j ( S L )) < δ ′ since P ( φ t i ( S L )) = P ( φ t j ( S L )). On theother hand, as in (5.11), Z t η ( w ( j ))+1 t η ( w ( j )) u ( φ t ( S L )) dt ≥ − K k u k . (5.13)Combining (5.11), (5.12) and (5.13), one has hS , u i = s − X j =0 Z t η ( w ( j )) t w ( j ) + Z t η ( w ( j ))+1 t η ( w ( j )) u ( φ t ( S L )) dt ≥ − s (cid:18) K k u k + (cid:18) ( CL ) α λα k u k α + L k u k (cid:19) δ ′ α (cid:19) ≥ − ♯ P · (cid:18) K k u k + (cid:18) ( CL ) α λα k u k α + L k u k (cid:19) δ ′ α (cid:19) , which completes the proof. (cid:3) In the following, we deal with the so called shadowing property for two finite timesegments, which will allow one to use one segment to shadow two segments of whichthe ending point of one segment is close to the beginning point of the other. Let S and S be two segments of Φ | Λ , suppose that d ( S R , S L ) < δ ′ . Then there exist v ( S L , S R ) and w ( S L , S R ) = W sǫ ′ ( φ v ( S L , S R ) ( S L )) ∩ W uǫ ′ ( S R ). Define anew segment S ∗ S : (cid:2) −|S | , |S | − v ( S L , S R ) (cid:3) by letting S ∗ S ( t ) = φ t (cid:0) w ( S L , S R ) (cid:1) ∀ t ∈ (cid:2) −|S | , |S | − v ( S L , S R ) (cid:3) . (5.14)We remark here that the definition of S ∗ S above is not the unique way for describingthe shadowing property. Nevertheless, it is the most convenient way for the rest of theproof. Lemma 5.3.
Given < α ≤ and a large constant γ = γ ( α ) ≫ satisfying that C α e − γαλ ≪ , when two segments S and S of Φ | Λ satisfy the following d ( S R , S L ) ≤ δ ′ and min {|S | , |S |} ≥ γ, then for all u ∈ C ,α ( M ) , |hS ∗ S , u i − hS , u i − hS , u i| d α ( S R , S L ) − d α (( S ∗ S ) R , S R ) − d α (( S ∗ S ) L , S L ) ≤ K , where K = C k u k + C α k u k αλα − C α e − ( γ − αλ and the denominator of the left side of the above inequalityis always positive by the choice of γ .Proof. Fix α, γ, u, S , S as in this Lemma. Note v = v ( S L , S R ), w = w (cid:0) S L , S R (cid:1) and e S : [0 , |S | − v ] : t → φ t + v ( S L ) . Thus, we have (cid:12)(cid:12)(cid:12) hS ∗ S , u i − D e S , u E − hS , u i (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z |S |− v u ( φ t ( w )) − u ( φ t φ v ( S L )) dt + Z |S | u ( φ − t ( w )) − u ( φ − t ( S R )) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z |S |− v k u k α d α ( φ t ( w ) , φ t φ v ( S L )) dt + Z |S | k u k α d α ( φ − t ( w ) , φ − t ( S R )) dt ≤ Z |S |− v k u k α ( Ce − λt ) α d α ( w, S L ) dt + Z |S | k u k α ( Ce − λt ) α d α ( w, S R ) dt ≤ k u k α C α λα d α ( S R , S L ) , and (cid:12)(cid:12)(cid:12)D e S , u E − hS , u i (cid:12)(cid:12)(cid:12) ≤ k u k | v | ≤ k u k Cd ( S R , S L ) ≤ k u k Cd α ( S R , S L ) . Therefore |hS ∗ S , u i − hS , u i − hS , u i| ≤ (cid:18) C k u k + 2 C α k u k α λα (cid:19) d α ( S R , S L ) (5.15)On the other hand, one has that d α (( S ∗ S ) L , S L ) ≤ C α e − αλ ( γ − v ) d α ( w, S L ) ≤ C α e − αλ ( γ − d α ( S R , S L ) , and d α (( S ∗ S ) R , S R ) ≤ C α e − αλγ d α ( w, S R ) ≤ C α e − αλγ d α ( S R , S L ) , which combining with (5.15) and the choice of γ implies what needed, thus accomplishthe proof. (cid:3) Proof of Lemma 3.7.
Before the main proof, we first state a technical Lemmawhich can be deduced from the Lemma 1.1 of [Bo3].
Lemma 5.4.
Given < α ≤ , A > , γ ∈ R and a continuous function u : M → R ,the following are equivalent (1). For all n ≥ and x i ∈ M, i ∈ Z /n Z , X i ∈ Z /n Z u ( x i ) + A X i ∈ Z /n Z d α ( φ γ x i , x i +1 ) ≥ . (5.16)(2). There exists an α -H¨older function v : M → R with k v k α ≤ A such that u ≥ v ◦ φ γ − v . Now we prove Lemma 3.7.
Proof.
Let K , K , K be the constants as in Lemmas 5.1, 5.2 and 5.3. We fix a γ > N satisfying the condition in Lemma 5.3 and a large number Q such that Q > max { K , K , K } . For n ≥
1, we note i ( n ) = i + n Z ∈ Z /n Z for i ∈ [0 , n − ∩ Z . Now we fix an integer n ≥ x i ( n ) ∈ Λ , i ( n ) ∈ Z /n Z . Note S i ( n ) : [0 , γ ] → Λ : t → φ t ( x i ( n ) ) for i ( n ) ∈ Z /n Z , L ( n ) = {S i ( n ) , i ( n ) ∈ Z /n Z } and Σ ( n ) = X i ( n ) ∈ Z /n Z hS i ( n ) , u i + Q X i ( n ) ∈ Z /n Z d α ( S Ri ( n ) , S Li ( n ) +1 ( n ) ) . If there is some j ( n ) ∈ Z /n Z such that d ( S Rj ( n ) , S Lj ( n ) +1 ( n ) ) < δ ′ , just take S ( n − = S j ( n ) ∗ S j ( n ) +1 ( n ) and S i ( n − = S j ( n ) + i ( n ) for i = 2 , , · · · n − L ( n − = {S i ( n − , i ( n − ∈ Z / ( n − Z } and Σ ( n − = X i ( n − ∈ Z / ( n − Z hS i ( n − , u i + Q X i ( n − ∈ Z /n Z d α ( S Ri ( n − , S Li ( n − +1 ( n − ) . Note that by Lemma 5.3Σ ( n ) − Σ ( n − ≥ − (cid:12)(cid:12)(cid:10) S j ( n ) ∗ S j ( n ) +1 ( n ) , u (cid:11) − (cid:10) S j ( n ) , u (cid:11) − (cid:10) S j ( n ) +1 ( n ) , u (cid:11)(cid:12)(cid:12) + Qd α (cid:16) S Rj ( n ) , S L ( j +1) ( n ) (cid:17) − Q (cid:16) d α (cid:16) S Lj ( n ) , ( S j ( n ) ∗ S ( j +1) ( n ) ) L (cid:17) − d α (cid:16) S R ( j +1) ( n ) , ( S j ( n ) ∗ S ( j +1) ( n ) ) R (cid:17)(cid:17) ≥ . That is Σ ( n ) ≥ Σ ( n − . (5.17)Repeat the above process until L (1) with d ( S R (1) , S L (1) ) < δ ′ OR some m ∈ [1 , n ] ∩ N with d (cid:16) S Rj ( m ) , S L ( j +1) ( m ) (cid:17) ≥ δ ′ for all j ∈ Z /m Z . In the case that the process ends at L (1) with d ( S R (1) , S L (1) ) < δ ′ . We have by Lemma5.1 that Σ (1) = hS (1) , u i + Qd α ( S R (1) , S L (1) ) ≥ − K d α ( S R (1) , S L (1) ) + Qd α ( S R (1) , S L (1) ) ≥ . (5.18)In the case that the process ends at some m ∈ [1 , n ] ∩ N with d (cid:16) S Rj ( m ) , S L ( j +1) ( m ) (cid:17) ≥ δ ′ for all j ( m ) ∈ Z /m Z . We have by Lemma 5.2 thatΣ ( m ) = X i ( m ) ∈ Z /m Z hS i ( m ) , u i + Q X i ( m ) ∈ Z /m Z d α ( S Ri ( m ) , S Li ( m ) +1 ( m ) ) ≥ − mK δ ′ α + mQδ ′ α ≥ . (5.19)Combining the inequality (5.18), (5.19) and the fact Σ ( n ) ≥ Σ ( n − ≥ Σ ( n − ≥ · · · by(5.17), one has Σ ( n ) ≥ . Then X i ( n ) ∈ Z /n Z u γ ( x i ( n ) ) + Qγ X i ( n ) ∈ Z /n Z d α ( S Ri ( n ) , S Li ( n ) +1 ( n ) ) = Σ ( n ) γ ≥ . By Lemma 5.4, there is an α -H¨older function v on Λ with k v k α ≤ Qγ such that u γ | Λ ≥ v ◦ φ γ | Λ − v. This ends the proof. (cid:3)
Finally , we give the proof of Lemma 3.9.
Proof of Lemma 3.9. (1). By Lemma 3.7, we only need to show that Z u − β ( u ; ψ, Λ , Φ) ψdµ ≥ µ ∈ M (Φ | Λ ) . It is immediately from the fact R udµ R ψdµ ≥ β ( u ; ψ, Λ , Φ) for all µ ∈ M (Φ | Λ )since ψ is strictly positive.(2). Given a probability measure µ ∈ M min ( u ; ψ, Λ , Φ), one has Z u γ + v ◦ φ γ − v − β ( u ; ψ, Λ , Φ) ψ γ dµ = Z u − β ( u ; ψ, Λ , Φ) ψdµ = 0 . Combining (1) and the fact u γ | Λ + v ◦ φ γ | Λ − v − β ( u ; ψ, Λ , Φ) ψ γ | Λ is continuous on Λ,one has supp ( µ ) ⊂ { x ∈ Λ : ( u γ + v ◦ φ γ − v − β ( u ; ψ, Λ , Φ) ψ γ ) | Λ ( x ) = 0 } . Therefore, Z u,ψ ⊂ { x ∈ Λ : ( u γ | Λ + v ◦ φ γ | Λ − v − β ( u ; ψ, Λ , Φ) ψ γ | Λ )( x ) = 0 } . This ends the proof. (cid:3)
Proof of Lemma 3.11.
In this section, we mainly prove the periodic approxima-tion. The proof partially follows the arguments in [BQ].5.4.1.
Joining of segments.
Recall that the definition of the joining of two segments S ⋆ S are given by (5.14), we give some properties of jointed segments. Lemma 5.5.
If two segments S and S satisfy |S | ≥ and d ( S R , S L ) ≤ δ ′ , then (1). max x ∈S ∗S d ( x, S ∪ S ) ≤ C d ( S R , S L );(2). |S | + |S | − ≤ |S ∗ S | ≤ |S | + |S | + 1 . Proof. (1). Note v = v ( S L , S R ), w = w ( S L , S R ) and e S : [ v, |S | ] : t → φ t ( S L ) . Then for t ∈ [ −|S | , d ( φ t ( w ) , S ) ≤ d ( φ t ( w ) , φ t ( S R )) ≤ Ce λt d ( w, S R ) ≤ C e λt d ( S R , S L ) ≤ C d ( S R , S L ) , where we used w ∈ W uǫ ( S R ) . For t ∈ [0 , |S | − v ], d ( φ t ( w ) , e S ) ≤ d ( φ t ( w ) , φ t ( w ) φ v ( S L )) ≤ Ce − λt d ( w, φ v ( S L )) ≤ C e − λt d ( S R , S L ) . where we used w ∈ W sǫ ( φ v ( S L )) . Thus, for t ∈ [0 , |S | − v ], d ( φ t ( w ) , S ) ≤ d ( φ t ( w ) , e S ) + max x ∈ e S d ( S , e S ) ≤ C e − λt d ( S R , S L ) + d ( S R , φ v ( S R )) ≤ C d ( S R , S L ) , where we used C ≫ basic cononical setting . Thus, by summing up,max x ∈S ∗S d ( x, S ∪ S ) = max t ∈ [ −|S | , |S |− v ] d ( φ t ( w ) , S ∪ S ) ≤ C d ( S R , S L ) . (2). One has |S ∗ S | = |S | + |S | − v ≥ |S | + |S | − Cd ( S R , S L ) ≥ |S | + |S | − Cδ ′ > |S | + |S | − , where we used the assumption δ ′ ≪ C . On the other hand, one also has that |S ∗ S | = |S | + |S | − v ≤ |S | + |S | + Cδ ′ ≤ |S | + |S | + 1 . This ends the proof. (cid:3)
Lemma 5.6.
There exists a large constant P > such that if two segments S and S satisfy |S | ≥ P , |S | ≥ P and d ( S R , S L ) ≤ δ ′ , then d ( S R , S L ) ≥ d ( S L , ( S ∗ S ) L ) + 2 d (( S ∗ S ) R , S R ) . Proof.
First we fix a large constant P ≫ C e − λ ( P − + C e − λP < . Fixtwo segments S and S as in Lemma. Note v = v ( S L , S R ) and w = w ( S L , S R ). Then d ( S L , ( S ∗ S ) L ) = d ( φ −|S | ( S R ) , φ −|S | ( w )) ≤ Ce − λ |S | d ( S R , w ) ≤ C e − λP d ( S R , S L ) , where we used w ∈ W uǫ ( S R ) . On the other hand, d ( S R , ( S ∗ S ) R ) = d ( φ |S |− v φ v ( S L ) , φ |S |− v ( w )) ≤ Ce − λ ( |S |− v ) d ( φ v ( S L ) , w ) ≤ C e − λ ( P − d ( S R , S L ) , where we used w ∈ W uǫ ( φ v ( S L )) . By assumption, we have d ( S L , ( S ∗ S ) L ) + d ( S R , ( S ∗ S ) R ) ≤ C e − λ ( P − d ( S R , S L ) + C e − λP d ( S R , S L ) ≤ d ( S R , S L ) . This ends the proof. (cid:3)
Periodic approximation.
For integer n ≥
1, let Σ n = { , , , · , n − } N and σ bea shift on Σ n . Assume F is a subset of S i ≥ { , , , · , n − } i , then the subshift withforbidden F is denoted by ( Y F , σ ) where Y F = (cid:8) x ∈ { , , , · , n − } N , w does not appear in x for all w ∈ F (cid:9) . The following lemma is Lemma 5 of [BQ], which will be used later.
Lemma 5.7 ([BQ]) . Suppose that ( Y, σ ) is a shift of finite type ( with forbidden wordsof length 2 ) with M symbols and entropy h . Then ( Y, σ ) contains a periodic point ofperiod at most M e (1 − h ) . Now we are ready to prove Lemma 3.11, which partially follow the argument in [BQ].
Proof of Lemma 3.11.
Fix a positive constant δ ′′ ≪ δ ′ C e β . Let P = { B , B , · · · , B m } be a finite partition of Λ with diameter smaller than δ ′′ . For x ∈ Λ, b x ∈ { , , , · · · , m } N is defined by b x ( n ) = j if φ n ( x ) ∈ B j , n = 0 , , , · · · . Denote b Z = { b x : x ∈ Z } and W n the collection of length n string that appears in b Z .One has ♯W n = K n e nh where h = h top ( b Z, σ ) and K n grows at a subexponential rate. Let Y n = { y y y · · · ∈ W N n : y i y i +1 ∈ W n for all i ∈ N } and ( Y n , σ n ) is the 1-step shift of finite type on W n . From Lemma 5.7, the shortestperiodic orbit in Y n is at most 1 + K n e nh e − nh = 1 + eK n . Denote one of the shortestperiodic orbit in Y n by z z z · · · z p n − z z z · · · for some p n ≤ eK n and z i ∈ W n , i =0 , , , · · · , p n −
1. For i = 0 , , , · · · , p n −
1, there is x i ∈ Z such that the leading 2 n string of b x i is z i z i +1 (we note z p n = z , x p n = x , S p n = S , · · · ). Choose segments S i by S i : [ n , n v i ] → M : t → φ t ( x i ) for i = 0 , , , · · · , p n − , where v i = v ( φ n ( x i ) , x i +1 ). We have the following Claim. Claim Q1. d ( S Ri , S Li +1 ) ≤ C e − nλ δ ′′ for i = 0 , , , · · · , p n − .Proof of Claim Q1. Note that the leading 2 n string of b x i is z i z i +1 and leading n stringof d x i +1 is z i +1 , which means P ( φ n + j ( x i )) = P ( φ j ( x i +1 )) for j = 0 , , · · · , n − . Therefore, d ( φ n + j ( x i ) , φ j ( x i +1 )) < δ ′′ for j = 0 , , · · · , n − . Thus d ( φ n + t ( x i ) , φ t ( x i +1 )) < Ce β δ ′′ < δ ′ for t ∈ [0 , n ] . Then by Lemma 3.4, we have d (cid:16) φ n + v i ( x i ) , φ n ( x i +1 ) (cid:17) ≤ C e − nλ ( d ( φ n ( x i ) , x i +1 ) + d ( φ n ( x i ) , φ n ( x i +1 ))) ≤ C e − nλ δ ′′ . This ends the proof of Claim Q1. (cid:3)
Now we define segments S i recursively for 0 ≤ i ≤ p n − S := S and S i := S i − ∗ S i for 1 ≤ i ≤ p n − . Based on Claim Q1, we have the following claim.
Claim Q2.
There is a positive integer N such that for any n ≥ N , one has (1). S i is well defined for ≤ i ≤ p n − d ( S Ri , S Li +1 ) ≤ C p n e − nλ δ ′′ < δ ′ for ≤ i ≤ p n − d ( S Rp n − , S Lp n − ) ≤ C p n e − nλ δ ′′ < min (cid:8) δ ′ , L (cid:9) , where L is as in Lemma 3.5; (4). ( n − p n ≤ |S p n − | ≤ ( n + 1) p n ;(5). max x ∈S pn − d ( x, Z ) ≤ C p n e − nλ δ ′′ . Proof of Claim Q2.
Since p n grows at a subexponential rate, we can take N largeenough such that N > P and 4 p n C e − nλ δ ′′ < min (cid:26) δ ′ , L (cid:27) for all n ≥ N, (5.20)where P is the constant as in Lemma 5.6. For 0 ≤ i ≤ p n −
2, we define χ ( i ) = d ( S Ri , S Li +1 ) + d ( S Ri +1 , S Li +2 ) + · · · + d ( S Rp n − , S Lp n − ) + d ( S Rp n − , S Li ) . By Claim Q1, χ (0) ≤ C p n e − nλ δ ′′ . Now we are to show that χ ( i ) and S i are well defined, which satisfy that χ ( i ) ≤ δ ′ and |S i | > P for i = 0 , , , · · · , p n − . These are clearly true for i = 0. Now we assume that these are true for some i ∈{ , , · · · , p n − } . Then for i + 1, since χ ( i ) ≤ δ ′ , one has d ( S Ri , S Li +1 ) ≤ δ ′ . Thus, wecan join S i and S i +1 by letting S i +1 = S i ∗ S i +1 . It is clear that |S i +1 | > P by Lemma 5.5 (2). On the other hand, by triangle inequality and Lemma 5.6, one has that χ ( i ) − χ ( i + 1)= d ( S Ri , S Li +1 ) + d ( S Ri +1 , S Li +2 ) − d ( S Ri +1 , S Li +2 ) + d ( S Rp n − , S Li ) − d ( S Rp n − , S Li +1 ) ≥ d ( S Ri , S Li +1 ) − d ( S Li , S Li +1 ) − d ( S Ri +1 , S Ri +1 ) ≥ d ( S Ri , S Li +1 ) . (5.21)Therefore, χ ( i + 1) ≤ χ ( i ) ≤ δ ′ . By induction, we can finish the proof of (1).By (5.21), one has d ( S Ri , S Li +1 ) ≤ χ ( i ) ≤ · · · ≤ χ (0) ≤ C p n e − nλ δ ′′ for i = 0 , , · · · , p n − , and we can deduce the following by triangle inequality: d ( S Rp n − , S Lp n − ) ≤ χ ( p n − ≤ · · · χ (0) ≤ C p n e − nλ δ ′′ . This ends the proof of (2) as well as (1), (3), and (4) follows from Lemma 5.5 (2).Now we denote D i := S i S ∪ p n − j = i +1 S j for i = 0 , , , · · · , p n − . Then by Lemma 5.5(1), max x ∈ D i +1 d ( x, D i ) ≤ max x ∈S i ∗S i +1 d ( x, S i ∪ S i +1 ) ≤ C d ( S Ri , S Li +1 ) ≤ C p n e − nλ δ ′′ . Therefore, by triangle inequality and the fact that D ⊂ Z ,max x ∈S pn − d ( x, Z ) ≤ max x ∈ D d ( x, Z ) + p n − X i =0 max x ∈ D i +1 d ( x, D i ) ≤ C p n e − nλ δ ′′ . This ends the proof of Claim Q2. (cid:3)
Recall that P is the constant as in Lemma 5.6, K, L are the constants as in Lemma3.5 and N is the constant as in Claim Q2. We fix an integer n > max( P , K, N ) + 1and let S p n − be the segment as in Claim Q2. Then by Claim Q2 (3), |S p n − | > K and d ( S Rp n − , S Lp n − ) ≤ δ ′ . Applying the Anosov Closing Lemma, we have a periodic segment O n such that (cid:12)(cid:12) |S p n − | − |O n | (cid:12)(cid:12) ≤ Ld (cid:16) S Lp n − , S Rp n − (cid:17) (5.22)and d (cid:16) φ t ( O Ln ) , φ t ( S Lp n − ) (cid:17) ≤ Ld ( S Lp n − , S Rp n − ) ∀ t ∈ (cid:2) , max (cid:0) |S p n − | , |O n | (cid:1)(cid:3) . (5.23)We claim the following: Claim Q3. max x ∈O n d ( x, Z ) ≤ (2 C Lp n + C p n ) e − nλ δ ′′ . Proof of Claim Q3. If |O| ≤ |S p n − | , by (5.23),max x ∈O d ( x, S p n − ) ≤ Ld (cid:16) S Lp n − , S Rp n − (cid:17) . If |O| > |S p n − | , note t ∗ = min( t, |S p n − | ) for t ∈ [0 , |O| ]. Then, by (5.22) and (5.23),one has that for t ∈ [0 , |O| ] d (cid:0) φ t ( O L ) , S p n − (cid:1) ≤ d (cid:0) φ t ( O L ) , φ t ∗ ( O L ) (cid:1) + d (cid:16) φ t ∗ ( O L ) , φ t ∗ ( S Lp n − ) (cid:17) ≤ CLd (cid:16) S Lp n − , S Rp n − (cid:17) + Ld (cid:16) S Lp n − , S Rp n − (cid:17) ≤ C Ld (cid:16) S Lp n − , S Rp n − (cid:17) , where C ≫ x ∈O d (cid:0) x, S p n − (cid:1) = max t ∈ [0 , |O| ] d (cid:0) φ t ( O L ) , S p n − (cid:1) ≤ C Ld (cid:16) S Lp n − , S Rp n − (cid:17) . Combining with (3), (5) of Claim Q2, we havemax x ∈O d ( x, Z ) ≤ max x ∈O d (cid:0) x, S p n − (cid:1) + max x ∈S pn − d ( x, Z ) ≤ C L · C p n e − nλ δ ′′ + C p n e − nλ δ ′′ = (2 C Lp n + C p n ) e − nλ δ ′′ . This ends the proof of Claim Q3. (cid:3)
By Claim Q3 and (5.22), we have d α,Z ( O n ) ≤ |O n | (cid:18) max x ∈O d ( x, Z ) (cid:19) α ≤ (cid:16) |S p n − | + Ld (cid:16) S Lp n − , S Rp n − (cid:17)(cid:17) · (cid:16) (2 C Lp n + C p n ) e − nλ δ ′′ (cid:17) α ≤ H n e − nαλ , where H n = (cid:0) (2 C Lp n + C p n ) δ ′′ (cid:1) α · (( n + 1) p n + 1). Note that H n grows at a subex-ponential rate as n increases as p n does. Hencelim sup P → + ∞ P k min O∈O P Λ d α,Z ( O ) ≤ lim sup n → + ∞ (( n + 1) p n + 1) k · H n e − nαλ = 0 , where we used the fact that p n , H n grow at a subexponential rate as n increases and |O n | ≤ np n + 1. The proof of Lemma 3.11 is completed. (cid:3) Further discussions on the case of C s,α -observables For s ∈ N , 0 ≤ α ≤ ψ on M , P er ∗ s,α ( M, ψ ) is definedas the collection of C s,α -continuous functions on M , such that for each u ∈ P er ∗ s,α ( M, ψ ), M min ( u ; ψ, Λ , Φ) contains at least one periodic measure. And
Loc s,α ( M, ψ ) is definedby
Loc s,α ( M, ψ ) := { u ∈ P er ∗ s,α ( M, ψ ) : there is ε > M min ( u + h ; ψ, Λ , Φ) = M min ( u ; ψ, Λ , Φ) for all k h k r,α < ε } . In the case s ≥ α > s ≥
2, we do not have analogous result like Proposition4.7. However, we have the following weak version.
Proposition 6.1.
Let O be a periodic segment of Φ | Λ with D ( O ) > and u ∈ C ( M ) with u | O = 0 and u | M/ O > . Then there exists a constant ̺ > such that the probabilitymeasure µ O ∈ M min ( u + h ; ψ, Λ , Φ) , where h is any C , ( M ) function with k h k < ̺ . Remark 6.2.
As in Remark 4.12, for s ∈ N and 0 ≤ α ≤
1, we let e w ∈ C s,α ( M )such that k e w k s,α < ε , e w | O = 0 and e w | M \O >
0. Then µ O is the unique measure in M min ( u + e w + h ; ψ, Λ , Φ) whenever k h k s,α < ̺ . The Proposition shows that there isan open subset of C s,α ( M ) near u such that functions in the open set have the sameunique minimizing measure with respect to ψ and the probability measure supports ona periodic orbit.By using Remark 6.2, we have the following result. Theorem 6.3.
Loc s,α ( M, ψ ) is an open dense subset of P er ∗ s,α ( M, ψ ) w.r.t. k · k s,α forinteger s ≥ and real number ≤ α ≤ .Proof. Given s ≥ ≤ α ≤
1. The openness is clearly true. We prove
Loc s,α ( M, ψ )is dense in
P er ∗ s,α ( M, ψ ) w.r.t. k · k s,α . Since Z udµ = Z ¯ udµ for all µ ∈ M (Φ | Λ ) , we have M min ( u ; ψ, Λ , Φ) = M min (¯ u ; ψ, Λ , Φ). Then the theorem follows from Remark6.2 immediately. (cid:3)
Proof of Proposition 6.1.
Now we finish the proof of Proposition 6.1.
Proof of Proposition 6.1.
Let O be a periodic segment of Φ | Λ and u ∈ C ( M ) with u | O =0 and u | M/ O >
0. For 0 ≤ ρ ≤ D ( O ), we note θ ( ρ ) = min { u ( x ) : d ( x, O ) ≥ ρ, x ∈ M } . It is clear that θ (0) = 0, θ ( ρ ) > ρ = 0 and θ is non-decreasing. Since D ( O ) > ρ , ρ satisfy0 < ρ < ρ < D ( O )4 C e β . (6.1)Next we will show that µ O ∈ M min ( u + h ; ψ, Λ , Φ) for all h ∈ C , ( M ) with k h k < ̺ ,where the constant ̺ is positive and ̺ <
12 min ( ψ min θ ( ρ )1 + ψ min , θ ( D ( O )4 C e β )(1 + k ψ k ψ min ) · C ρ λ + |O| · ψ min ψ min + ψ min ψ min ) . (6.2)Now we fix a function h as above. Note G = u + h − a O ψ where a O = hO , u + h ihO , ψ i ≤ k h k ψ min . (6.3)Then R Gdµ R ψdµ = R u + hdµ R ψdµ − a O . Therefore, to show that µ O ∈ M min ( u + h ; ψ, Λ , Φ), it isenough to show that Z Gdµ ≥ µ ∈ M e (Φ | Λ ) , where we used the assumption ψ is strictly positive and the fact R Gdµ O = 0. Now welet Area := { y ∈ M : d ( y, O ) ≤ ρ } . We have the following claim.
Claim F1.
Area contains all x ∈ M with G ( x ) ≤ .Proof of Claim F1. For x / ∈ Area , we have G ( x ) = u ( x ) + h ( x ) − a O ψ ≥ θ ( ρ ) − k h k − a O k ψ k ≥ θ ( ρ ) − ψ min ψ min k h k > . where we used (6.2) and (6.3). This ends the proof of Claim F1. (cid:3) Note
Area = { y ∈ M : d ( y, O ) ≤ ρ } . It is clear that
Area is in the interior of Area . Thus, d ( Area , M \ Area ) >
0. Therefore, by Claim F1, we can fix a constant0 < τ < G ( φ t ( x )) > x ∈ M \ Area and | t | ≤ τ . Claim F2. If z ∈ Λ is not a generic point of µ O , then there is m ≥ τ such that R m G ( φ t ( z )) dt > . Next we prove Proposition 6.1 by assuming the validity of Claim F2, proof of whichis left to the next subsection. Same as the argument at the beginning of the proof, itis enough to show that for all µ ∈ M e (Φ | Λ ) Z Gdµ ≥ . Given µ ∈ M e (Φ | Λ ), in the case µ = µ O , it is obviously true. In the case µ = µ O , justlet z be a generic point of µ . Note that z is not a generic point of µ O . By Claim F2,we have t ≥ τ such that Z t G ( φ t ( z )) dt > . Note that φ t ( z ) is still not a generic point of µ O . Apply Claim F2 again, we have t ≥ t + τ such that Z t t G ( φ t ( z )) dt > . By repeating the above process, we have 0 ≤ t < t < t < · · · with the gap not lessthan τ such that Z t i +1 t i G ( φ t ( z )) dt > i = 0 , , , , · · · , where t = 0. Therefore, Z Gdµ = lim m → + ∞ m Z m G ( φ t ( z )) dt = lim i → + ∞ t i (cid:18)Z t t G ( φ t ( z )) dt + Z t t G ( φ t ( z )) dt + · · · + Z t i t i − G ( φ t ( z )) dt (cid:19) ≥ . That is, µ O ∈ M min ( u + h ; ψ, Λ , Φ). This ends the proof of the Proposition. (cid:3)
Proof of Claim F2.
Assume that z is not a generic point of µ O , if z / ∈ Area ,let m = τ , we have nothing to prove since G ( φ t ( z )) > | t | ≤ τ . Now we assumethat z ∈ Area . In the case that d ( φ t ( z ) , O ) < D ( O )4 C e β for all t ≥
0, by Lemma 4.2, z is ageneric point of µ O , which contradicts to our assumption. Hence, there must be some m > d ( φ m ( z ) , O ) ≥ D ( O )4 C e β . We can assume m > d ( φ m ( z ) , O ) ≥ D ( O )4 C e β . (6.4)The existence of m is ensured by (6.1). Then for 0 ≤ t ≤ d ( φ m − t ( z ) , O ) ≥ D ( O )4 C e β . Then Z m m − G ( φ t ( z )) dt = Z m m − u ( φ t ( z )) + h ( φ t ( z )) − a O ψ ( φ t ( z )) dt ≥ Z m m − θ (cid:18) D ( O )4 C e β (cid:19) − k h k − a O k ψ k dt ≥ θ (cid:18) D ( O )4 C e β (cid:19) − ψ min ψ min k h k . (6.5)where we used (6.3) and the definition of θ ( · ). On the other hand, one has that D ( O )4 C e β >ρ , which implies that φ m − t ( z ) / ∈ Area for all 0 ≤ t ≤ . (6.6)Since Area is compact, we can take m the largest time with 0 ≤ m ≤ m such that φ m ( z ) ∈ Area , where we use the assumption z ∈ Area . By (6.6), it is clear that m < m −
1. Thenby Claim F1 and the fact that
Area ⊂ Area , G ( φ t ( z )) > m < t < m − . (6.7)Since m < m , one has by (6.4) that d ( φ t ( z ) , O ) < D ( O )4 C e β < δ ′ for all 0 ≤ t ≤ m . Therefore, by Lemma 4.2, there is y ∈ O such that d ( φ t ( z ) , φ t ( y )) ≤ Cd ( φ t ( z ) , O ) ≤ D ( O )4 Ce β for all t ∈ [0 , m ] . Also notice that d ( z, y ) ≤ Cρ and d ( φ m ( z ) , φ m ( y )) ≤ Cρ , where we used z, φ m ( z ) ∈ Area . By using Lemma 3.4, we have for all 0 ≤ t ≤ m , d ( φ t ( z ) , φ t φ v ( y )) ≤ C e − λ min( t,m − t ) ( d ( z, y ) + d ( φ m ( z ) , φ m ( y ))) ≤ C e − λ min( t,m − t ) ρ , where v = v ( y , z ). Hence, Z m d ( φ t ( z ) , φ t φ v ( y )) dt ≤ Z m C ρ ( e − λt + e − λ ( m − t ) ) dt ≤ C ρ λ . Since u ( φ t ( y )) = 0 for all t ∈ R and u ≥
0, one has Z m G ( φ t ( z )) − G ( φ t + v ( y )) dt = Z m u ( φ t ( z )) + h ( φ t ( z )) − u ( φ t + v ( y )) − h ( φ t + v ( y )) − a O ( ψ ( φ t ( z )) − ψ ( φ t + v ( y ))) dt ≥ Z m h ( φ t ( z )) − h ( φ t + v ( y )) − a O ( ψ ( φ t ( z )) − ψ ( φ t + v ( y ))) dt ≥ − ( k h k + | a O |k ψ k ) Z m d ( φ t ( z ) , φ t + v ( y )) dt ≥ − ( k h k + k h k k ψ k ψ min ) · C ρ λ ≥ − k h k (1 + k ψ k ψ min ) · C ρ λ . (6.8)By assuming that m = p |O| + q for some nonnegative integer p and real number0 ≤ q ≤ |O| , one has by (6.4) that Z m G ( φ t + v ( y )) dt = Z m m − q G ( φ t + v ( y )) dt ≥ −|O| · ψ min ψ min k h k , (6.9)where we used R Gdµ O = 0. Combining (6.3), (6.5), (6.7), (6.8) and (6.9), we have Z m G ( φ t ( z )) dt ≥ Z m G ( φ t ( z )) dt + Z m m − G ( φ t ( z )) dt = Z m G ( φ t ( z )) − G ( φ t + v ( y )) dt + Z m G ( φ t + v ( y )) dt + Z m m − G ( φ t ( z )) dt ≥ −k h k (1 + k ψ k ψ min ) · C ρ λ − |O| · ψ min ψ min k h k + θ (cid:18) D ( O )4 C e β (cid:19) − ψ min ψ min k h k = θ (cid:18) D ( O )4 C e β (cid:19) − (cid:18) (1 + k ψ k ψ min ) · C ρ λ + |O| · ψ min ψ min + 1 + ψ min ψ min (cid:19) k h k > , where we used assumption (6.2). Therefore, m = m is the time as required since m ≥ > τ by (6.7). This completes the proof of Claim F2. Acknowledgement
At the end, we would like to express our gratitude to Tianyuan Mathematical Centerin Southwest China, Sichuan University and Southwest Jiaotong University for theirsupport and hospitality.
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College of Mathematical Sciences, Sichuan University, Chengdu, Sichuan,610016, China
E-mail address , Z. Lian: [email protected], [email protected] (Xiao Ma)
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academyof Sciences and Department of Mathematics, University of Science and Technologyof China, Hefei, Anhui, China
E-mail address , X. Ma: [email protected] (Leiye Xu)
Wu Wen-Tsun Key Laboratory of Mathematics, USTC, Chinese Academyof Sciences and Department of Mathematics, University of Science and Technologyof China, Hefei, Anhui, China
E-mail address , L. Xu: [email protected] (Yiwei Zhang)
School of Mathematics and Statistics, Center for Mathematical Sci-ences, Hubei Key Laboratory of Engineering Modeling and Scientific Computing, Hua-Zhong University of Sciences and Technology, Wuhan 430074, China
E-mail address , Y. Zhang:, Y. Zhang: