Regularity of the Density of SRB Measures for Solenoidal Attractors
aa r X i v : . [ m a t h . D S ] M a y REGULARITY OF THE DENSITY OF SRB MEASURES FORSOLENOIDAL ATTRACTORS
CARLOS BOCKER AND RICARDO BORTOLOTTI
Abstract.
We show that a class of higher-dimensional hyperbolic endomorphismsadmit absolutely continuous invariant probabilities whose density are regular. Themaps we consider are given by T ( x, y ) = ( E ( x ) , C ( y ) + f ( x )), where E is a linearexpanding map of T u , C is a linear contracting map of R d , f is in C r ( T u , R d ) and r ≥
2. We prove that if | (det C )(det E ) |k C − k − s > s < r − ( u + d +1) and T satisfies a certain transversality condition, then the density of the SRB measureof T is contained in the Sobolev space H s ( T u × R d ), in particular, if s > u + d thenthe density is C k for every k < s − u + d . We also exhibit a condition involving E and C under which this tranversality condition is valid for almost every f . Introduction
The ergodic theory of hyperbolic endomorphisms was developed in the last yearsand presents similar results to the ergodic theory of invertible hyperbolic dynamicssuch as SRB measures, equilibrium states and structural stability [13, 15, 17, 18, 25].One interesting phenomena that may occur for hyperbolic endomorphisms is thatthe SRB measure needs not to be singular when the dynamic expands volume, whatdoes not happens for hyperbolic proper attractors [4, 7]. This was observed in [23, 24]and extended in [6], where was proved the absolute continuity of the SRB measureunder certain geometrical transversality condition.The absolute continuity of the SRB measure is usually associated to maps withonly positive Lyapunov exponents [2, 3]. The main feature in [5, 6, 23] is a geomet-rical condition of transversality between the images of the unstable directions thatallows to conclude properties of regularity of the SRB measure that are similar tothose that occurs for expanding maps. Due to these results, one may expect that vol-ume expanding hyperbolic attractors for endomorphisms satisfies ergodic propertiessimilar to expanding maps.Since the density of the SRB measure is smooth for expanding maps [20, 21], oneshould ask whether the property of the smoothness of the density is also valid forvolume expanding hyperbolic endomorphisms under this transversality condition.Here we prove the Sobolev regularity of the density of the SRB measure.
We study the action of the operator L on an appropriate Banach space B adaptedto the dynamic. This Banach space is defined using the method developed in [10]that was also used in [5], defining an anisotropic norm of the function correspondingto its action in the space of regular functions supported in “almost stable manifolds’(see definition 3.2). In this work, we consider maps T : T u × R d → T u × R d given by T ( x, y ) = ( E ( x ) , C ( y ) + f ( x )) , (1)where E is a linear expanding map of the torus T u , C is a linear contraction of R d , f ∈ C r ( T u , R d ) and r ≥ µ T of T . In this paper, we study the Sobolev regularity of the density φ T = dµ T /dx . The low-dimensional case u = d = 1 was previously studied in [5].Here we are focused on the higher-dimensional setting with u ≥ d ≥ E ( u ) the set of the linear expanding maps of T u , by C ( d ) the set of thelinear contractions of R d and denote T = T ( E, C, f ) for E ∈ E ( u ), C ∈ C ( d ) and f ∈ C r ( T u , R d ). Given E ∈ E ( u ), consider the following subset C s ( d ; E ) of C ( d ): C s ( d ; E ) = ( C ∈ C ( d ) , | det C || det E |k C − k − s > k C k < k E − k − | det E | u − d +1 ) . When T contracts volume ( | det E || det C | <
1) there exists no absolutely continu-ous invariant probability (ACIP). On the other hand, if T expands volume then thecondition | det C || det E |k C − k − s > s > Theorem A.
Given integers u ≥ d and ≤ s < r − ( u + d + 1) , E ∈ E ( u ) and C ∈ C s ( d ; E ) , there exists an open and dense subset U of C r ( T u , R d ) such that thecorresponding SRB measure µ T of T = T ( E, C, f ) for f ∈ U is absolutely continuouswith respect to the volume of T u × R d and its density is in H s ( T u × R d ) . The condition behind the subset U corresponds to a geometrical condition oftransversal overlaps of the images (see definition 2.1). In [6], the authors provedthat this condition is generic when C is in C ( d ; E ).Notice that if C ∈ C ( d ; E ) we obtain the absolute continuity of µ T under the con-dition | det C || det E | >
1, which is more general than the hypothesis of [6, TheoremA]. Moreover, by continuity, if C ∈ C ( d ; E ) then C ∈ C s ( d ; E ) for some s > Corollary B.
Given integers u ≥ d , E ∈ E ( u ) and C ∈ C ( d ; E ) , there exists anopen and dense subset U of C r ( T u , R d ) such that the corresponding SRB measure µ T of every map T = T ( E, C, f ) for f ∈ U is absolutely continuous with respect to thevolume of T u × R d and its density is in H s ( T u × R d ) for some s > . EGULARITY OF THE DENSITY OF SRB MEASURES 3
In the situation where s > u + d , Sobolev’s embedding theorem implies that any φ T coincides almost everywhere with a C k function for every k < s − u + d . In particular φ T is continuous almost everywhere, that implies that the attractor Λ has non-emptyinterior. Corollary C.
Under the assumptions of Theorem A, if r ≥ u + d + 2 and s > u + d ,then the corresponding attractor Λ T has non-empty interior. Consider the Ruelle-Perron-Frobenius transfer operator (or simply transfer oper-ator) L : L → L defined by L φ ( x ) = X T ( y )= x φ ( y ) | det DT ( y ) | . (2)The technical part of this paper corresponds a Lasota-Yorke inequality for thetransfer operator in a Banach space B contained in H s . This kind of approach alsoallows to conclude statistical properties as consequences of the spectral gap. Actually,for s > u/ L and, thus, exponentialdecay of correlations for T in a Banach space containing smooth observables. Theorem D.
Suppose that C ∈ C s ( d ; E ) for u/ < s < r − ( u + d + 1) and ζ ∈ (max {k E − k d u ) , ( | det E || det C |k C − k s ) } , . Then, for any f ∈ C r ( T u , R d ) in an open and dense set, there exists a Banachspace B contained in H s ( T u × R d ) and containing C r − ( D ) such that the actionof the operator L in B has spectral gap with essential spectral radius at most ζ .In particular, T has exponential decay of correlations in some linear space ˜ B withexponential rate ζ , where ˜ B is contained in B and contains C r − ( D ) . An interesting consequence of Theorem D is that the rate of exponential decay ofcorrelations can be taken uniform when the rate of contraction tends to be weaker,for instance, through the family of dynamics T t = T ( E, (1 − t ) C + tI, f ), 0 ≤ t < T has exponential decay of correlations withthe same rate ζ for an open an dense set of f ’s.The plan of the paper is as follows: Section 2 details the basic definitions (includingthe transversality condition) and statements of this work. In section 3 we introducethe norms, some properties that shall be used further and two Main Lasota-Yorkeinequalities for the transfer operator. Section 4 is dedicated to the proof of thetwo Main Lasota-Yorke inequalities. In section 5 we prove a third Lasota-YorkeInequality and we prove Theorems 1 and 2. In Section 6 we conclude Theorems A, Dand Corollaries B, C as consequence of the genericity of the transversality conditionwhen C ∈ C ( d ; E ). CARLOS BOCKER AND RICARDO BORTOLOTTI Definitions and statements
Given integers u and d , we consider the dynamic T = T ( E, C, f ) : T u × R d → T u × R d given by T ( x, y ) = (cid:0) E ( x ) , C ( y ) + f ( x ) (cid:1) , (3)where E ∈ E ( u ) is a map whose lift E : R u → R u is a linear map with k E − k − > Z u , C ∈ C ( d ) is a linear invertible map with k C k < f ∈ C r ( T u , R d ), r ≥ T is given by Λ = ∩ n ≥ T n ( D ) for some D = T u × [ − K , K ] d satisfying T ( D ) ⊂ D . Since the restriction of T to Λ is a transitive hyperbolicendomorphism, it admits a unique SRB measure µ T supported on Λ [25].We suppose in the whole text that T is volume expanding and we consider s > | det E || det C |k C − k − s > Codifying the dynamics.
Let us fix notation involving the partition of thebase space T u that codify the action of the expanding map E . This is essentially thesame notation used in [6].Fix R = {R (1) , · · · , R ( r ) } a Markov partition for E , that is, R ( i ) are disjoint opensets, the interior of each R ( i ) coincides with R ( i ), E | R ( i ) is one-to-one, S i R ( i ) = T u and E ( R ( i )) ∩R ( j ) = ∅ implies that R ( j ) ⊂ E ( R ( i )). Each R ( i ) is called a rectangleof the Markov partition. Markov partitions always exist for expanding maps witharbitrarily small diameter (see [16]).Let us suppose that diam( R ) < γ , where 0 < γ < / x ∈ T u and y ∈ E − ( x ) there exists a unique affine inverse branch g y,x : B ( x, γ ) → B ( y, γ ) such that g y,x ( x ) = y and E ( g y,x ( z )) = z (4)for every z ∈ B ( x, γ ).Consider the set I = { , · · · , r } and I n the set of words of length n with lettersin I , 1 ≤ n ≤ ∞ . Denoting by a = ( a i ) ni =1 a word in I n , define I n the subset ofadmissible words a = ( a i ) ni =1 , that is, with the property that E ( R ( a i +1 )) ∩ R ( a i ) = ∅ for every 0 ≤ i ≤ n − . (5)Consider the partition R n := ∨ n − i =0 E − i ( R ) and, for every a ∈ I n , the set R ( a ) = ∩ n − i =0 E − i ( R ( a n − i )) in R n , which is nonempty if and only if a ∈ I n . The truncationof a = ( a j ) nj =1 to length 1 ≤ p ≤ n is denoted by [ a ] p = ( a j ) pj =1 .For any x ∈ T u , fix some π ( x ) ∈ I such that x ∈ R ( π ( x )). For any c ∈ I p ,1 ≤ p < ∞ , we consider I n ( c ) the set of words a ∈ I n such that E n ( R ( a )) ∩R ( c ) = ∅ .Define I n ( x ) := I n ( π ( x )) and, for a ∈ I n ( x ), denote by a ( x ) the point y ∈ R ( a ) thatsatisfies E n ( y ) = x . EGULARITY OF THE DENSITY OF SRB MEASURES 5
For any a ∈ I n and 1 ≤ n < ∞ we consider the set D ( a ) := { x ∈ T u | a ∈ I n ( x ) } = E n ( R ( a )) = E ( R ([ a ] )), which is a union of rectangles of the Markov partition. Theimage of R ( a ) × { } by T n is the graph of the function S ( · , a ) : D ( a ) → R d given by S ( x, a ) := n X i =1 C i − f ( E n − i ( a ( x ))) = n X i =1 C i − f ([ a ] i ( x )) . (6)Consider the sets I ∞ ( x ) = { a ∈ I ∞ such that [ a ] i ∈ I i ( x ) for every i ≥ } and D ( a ) := { x ∈ T u | a ∈ I ∞ ( x ) } = ∩ + ∞ n =1 E n ( R ([ a ] n )) = E ( R ([ a ] )) for a ∈ I ∞ . If a ∈ I ∞ ( x ), we define S ( x, a ) = lim n →∞ S ( x, [ a ] n ).For any p ≥ c ∈ I p , let us denote by R ∗ ( c ) the union of atoms R (˜ c ),˜ c ∈ I p , that are adjacent to R ( c ). We suppose that the diameter of the partition R is small enough such that the diameter of R ∗ ( c ) is smaller than γ . For a ∈ I i ,let us denote by E − i c , a the inverse branch of E i satisfying E − i c , a ( R ( c )) ⊂ R ( a ) (and so E − i c , a ( R ∗ ( c )) ⊂ R ∗ ( a )). We can extend S ( x, a ) to a ball B c of radius γ containing R ∗ ( c ) by S c ( x, a ) := n X i =1 C i − f ( E − i c , a ( x )) . (7)Consider the constant α := k f k Cr −k C k . Notice also that S c ( · , a ) is of class C r and k E k j k ∂ α S c ( x, a ) k ≤ α (8)for every x ∈ R ∗ ( c ) and multi-index | α | = j , 0 ≤ j ≤ r .2.2. The transversality condition.
Given a linear map A : R u → R d , denote by m ( A ) := sup dim W = d inf k v k =1 ,v ∈ W k A ( v ) k (9)the smallest singular value of A . Denote the minimum and maximum rates of expan-sion and contraction by µ = k E − k − , µ = k E k , λ = k C − k − , λ = k C k . Consideralso N = | det E | the degree of the expanding map and θ = λµ − . Definition 2.1.
Given T = T ( E, C, f ) as above, integers ≤ p, q < ∞ , c ∈ I p and a , b ∈ I q ( c ) , we say that a and b are transversal on c if m ( DS c ( x, a ) − DS c ( y, b )) > θ q α (10) for every x, y ∈ R ∗ ( c ) . Defining the integer τ ( q ) by τ ( q ) = min p ≥ max c ∈ I p max a ∈ I q ( c ) { b ∈ I q ( c ) | a is not transversal to b on c } , (11) we say that it holds the transversality condition if lim sup q →∞ q log τ ( q ) = 0 . (12) CARLOS BOCKER AND RICARDO BORTOLOTTI
When E and C are fixed, we denote τ f ( q ) to denote its dependence on f . In [6],it was given a condition which implies that, for every β >
0, the set of f ’s satisfyinglim sup q →∞ log τ ( q ) q > β is open and dense. More precisely, considering C ( d ; E ) = ( C ∈ C ( d ) , k C k < k E − k − | det E | u − d +1 ) , it was proved that there exists a residual subset R ⊂ C r ( T u , R d ) such that if C ∈C ( d ; E ), then lim sup log τ f ( q ) q = 0 for every f ∈ R (see Proposition 6.1).Theorems A and D are obtaining putting together their more explicit formula-tions evolving the transversality condition given below with the genericity of thetransversality condition. Theorem 1.
Given ≤ s < r − ( u + d + 1) , E ∈ E ( u ) , C ∈ C ( d ) and f ∈ C r ( T u , R d ) such that | det E || det C |k C − k − s > and the transversality condition is valid, thenthere exists an open set U ⊂ C ( d ) × C r ( T u , R d ) containing f such that for every ( ˜ C, ˜ f ) ∈ U the SRB measure µ T for T = T ( E, C, ˜ f ) is absolutely continuous and itsdensity is in H s ( T u × R d ) . Notice that Theorem above is stronger than [6, Theorem 2.9] because for s = 0the condition is just | det E || det C | > L in a Banach space B ⊂ H s , such asspectral gap and exponential decay of correlations, are obtained when the transver-sality condition is valid and s > u/ Theorem 2.
Suppose that C ∈ C s ( d ; E ) for u/ < s < r − ( u + d + 1) and ζ ∈ (max {k E − k d u ) , ( | det E || det C |k C − k s ) } , . For any f ∈ C r ( T u , R d ) such that T satisfies the transversality condition, thereexists an open set U ⊂ C r ( T u , R d ) containing f such that for every ˜ f ∈ U thereexists a Banach space B contained in H s ( T u × R d ) and containing C r − ( D ) such thatthe action of the operator L ˜ f in B has spectral gap with essential spectral radius atmost ζ . In particular, T ˜ f has exponential decay of correlations in some linear space ˜ B with exponential rate ζ , where ˜ B is contained in B and contains C r − ( D ) . Description of the norms k · k † ρ and k · k H s In this Section, we define the two main norms that will be used in this work. TheMain Inequalities of this paper (Propositions 3.3, 3.9 and 5.1) are stated in terms ofthese norms.
EGULARITY OF THE DENSITY OF SRB MEASURES 7
The norm k · k † ρ . Here we define a norm k · k † ρ similar to the norms in [5, 10].Let c ∈ I , we define S ( c ) as the set of C r transformations ψ : U ψ → T u suchthat U ψ = V ψ for a bounded open set V ψ ⊂ R d , ψ ( U ψ ) ⊂ R ∗ ( c ) and k D ν ψ ( x ) k ≤ k ν for 1 ≤ ν ≤ r , for constants k , · · · , k r that will be chosen appropriately. We define S = S c ∈ I S ( c )Given ψ ∈ S ( c ), we denote by G ψ = { ( ψ ( x ) , x ) | x ∈ U ψ } the graph of ψ . For each a ∈ I n ( c ), we denote ( ˜ G ψ ) a the unique connected component of T − n ( G ψ ) which iscontained in R ∗ ( a ) × R d . Moreover, the constants k , · · · , k r will be chosen such thateach set ( ˜ G ψ ) a is the graph of a transformation ψ a : U ψ a → T u such that ψ a ∈ S .Note that T n is locally written in the form T n ( x, y ) = ( E n x, C n y + S n c , a ( E n x )) , (13)where S n c , a ( z ) = P n − j =0 C j f ( E − j − c , a z ) is a C r function with k D j S n c , a k ≤ α , 1 ≤ j ≤ r .Given ψ ∈ S ( c ) and ( ˜ G ψ ) a , a ∈ I n ( c ), the inverse branch T − n c , a is written as T − n c , a ( x, y ) = (cid:0) ( E c , a ) − n ( x ) , C − n ( y − S n c , a ( x )) (cid:1) . (14)Consider the C r diffeomorphism g a : U ψ → U ψ a such that T n ( ψ a ◦ g a ( y ) , g a ( y )) =( ψ ( y ) , y ) for all y ∈ U ψ . We have ψ a ( y ) = E − n c , a ψ ( g − a ( y )), ( ˜ G ψ ) a = G ψ a and ψ a ∈ S ,where the g a ’s are given by g a ( y ) = C − n ( y − S n c , a ( ψ ( y )) . (15)A useful estimate for the map g a is given in the following. Claim 3.1.
The map g a is a C r diffeomorphism and there exists a C r − map Q a : U ψ a → L ( R d , R d ) such that Dg − a ( z ) = Q a ( z ) C n . Moreover, k Q a k C r − ≤ K for someconstant K depending only α , k , . . . , k r . In particular, k D j g − a ( z ) k ≤ K k C k n and | D j − det Dg − a ( z ) | ≤ K | det C | n for every z ∈ U ψ a and ≤ j ≤ r .Proof. The map g a is one-to-one because g a ( y ) = g a ( z ) implies y − z = S n c , a ( ψ ( y )) − S n c , a ( ψ ( z )). But the estimates k DS n c , a k ≤ α and k Dψ k ≤ c < α − implies that y = z . The expression Dg a ( y ) = C − n (cid:0) I − DS n c , a ( ψ ( y )) Dψ ( y ) (cid:1) implies that Dg a ( y )is invertible for every y ∈ U ψ due to k DS n c , a ( ψ ( y )) Dψ ( y ) k ≤ α c <
1. This provesthat g a is a C r diffeomorphism.For every z ∈ U ψ a we have: Dg − a ( z ) = ( I − DS n c , a ( ψ ( z )) Dψ ( z )) − C n = ∞ X k =0 ( DS n c , a ( ψ ( z )) Dψ ( z )) k C n . CARLOS BOCKER AND RICARDO BORTOLOTTI
The result follows taking Q a ( z ) = P ∞ k =0 ( DS n c , a ( ψ (( g a ) − ( z ))) Dψ ( g − a ( z ))) k . (cid:3) Let us fix the cone field C = { ( u, v ) ∈ T ( x,y ) T u × R d | k u k ≤ α − k v k} , (16)which is invariant under ( DT − ) ( x,y ) for every ( x, y ) ∈ T u × R d .We suppose that k ≤ α − / k , · · · , k r > σ is a u -dimensional ball contained in a u -dimensional plane of T u × R d and Γ is a connected component of T − q ( σ ) such thatits tangent vectors are all in C , then Γ is the graphic of an element of S .For h ∈ C r ( D ) and multi-indexes α = ( α , · · · , α u ) and β = ( β , · · · , β d ), | α | + | β | ≤ r , we denote ∂ αx ∂ βy h = ∂ | α | + | β | h∂ α x · · · ∂ α u x u ∂ β y · · · ∂ β d y d . (17) Definition 3.2.
For h ∈ C r ( D ) and an integer ≤ ρ ≤ r − , we define k h k † ρ = max | α | + | β |≤ ρ sup ψ ∈S sup φ ∈C | α | + | β | ( U ψ ) Z φ ( y ) .∂ αx ∂ βy h ( ψ ( y ) , y ) dy (18) where the first supremum is taken over functions φ with supp( φ ) ⊂ Int ( U ψ ) and k φ k C | α | + | β | ≤ . Clearly, k h k † ρ is a norm that satisfies: k h k L ≤ k h k † and k h k † ρ − ≤ k h k † ρ . (19)The first main Lasota-Yorke inequality is similar to the ones in [5, 10]: Proposition 3.3 (First Main Lasota-Yorke (for k · k † )) . For any δ ∈ ( k E − k , ,there exist constants K and K ( n ) such that kL n h k † ρ ≤ Kδ ρn k h k † ρ + K ( n ) k h k † ρ − for ≤ ρ ≤ r − , (20) and kL n h k † ≤ K k h k † (21) for n ≥ and h ∈ C r ( D ) , where K ( n ) depends on n but not on h . Proposition 3.3 is proved in Section 4.1.
EGULARITY OF THE DENSITY OF SRB MEASURES 9
The Sobolev norm k · k H s . Let us remind some facts about the Fourier trans-form and the Sobolev norm that shall be used further.Given φ ∈ C r ( D ), we define ˆ φ : Z u × R d → C byˆ φ ( ξ, η ) = Z T u × R d φ ( x, y ) e − πi ( h ξ,x i + h η,y i ) dxdy (22)The Sobolev norm of is defined by k φ k H s = p h φ, φ i H s , where h φ , φ i H s := X η ∈ Z u Z R d ˆ φ ( η, ξ ) ˆ φ ( η, ξ )(1 + | ξ | + | η | ) s dη, (23)and the Sobolev space H s is the completion of C r ( D ) with respect to this norm.This norm comes from the inner productAn equivalent definition is given by the L norm of the derivatives. For multi-indexes α = ( α , · · · , α u ) and β = ( β , · · · , β d ), we denote σ = ( α, β ) and ∂ σz h = ∂ αx ∂ βy h . If s is a non-negative integer with r ≥ s and φ , φ ∈ C r ( D ), we define theinner product h φ , φ i ˜ H s = X | σ |≤ s h ∂ γz φ , ∂ γz φ i L . (24)If s is not integer, we define δ = s − ⌊ s ⌋ ∈ (0 ,
1) and h φ , φ i ˜ H s = X | σ |≤⌊ s ⌋ h ∂ γz φ , ∂ γz φ i L + X | σ | = ⌊ s ⌋ Z R u × R d Z R u × R d Φ σ ( x, y, v, w ) dvdw dxdy, (25)whereΦ σ ( x, y, v, w ) = ( ∂ σ φ ( x + v, y + w ) − ∂ σ φ ( x, y ))( ∂ σ φ ( x + v, y + w ) − ∂ σ φ ( x, y ))( | v | + | w | ) u + d + δ is defined considering the extension of φ j to R u × R d as zero if ( x, y ) / ∈ [0 , u × R d ∼ T u × R d . This inner product induces the norm k φ k H s = h φ, φ i ˜ H s . It is a standard factthat these norms are equivalent (see [12, page 241]), that is, there exists a constant
K > K k φ k H s ≤ k φ k H s ≤ K k φ k H s . (26) Remark 3.4.
Through this paper we will introduce several constants
K > depend-ing only on the objects that were fixed before, for simplicity we will keep denotingthem as K . In the cases that the constant depends on other objects that are notfixed, we will emphasize this dependence. Claim 3.5.
For ≤ t < s ≤ r and ǫ > , there is a constant K ( ǫ, t, s ) such that k φ k H t ≤ ǫ k φ k H s + K ( ǫ, t, s ) k φ k L (27) for every φ ∈ C r ( D ) .Proof. Choose 1 < p < + ∞ such that ( t − sp )( pp − ) ≤ − ( u + d ) and use the Young’sinequality to obtain (putting t = sp + t − sp and recall 1 /p +1 /q = 1 with q = p/ ( p − | ξ | + | η | ) t = (1 + | ξ | + | η | ) sp (1 + | ξ | + | η | ) t − sp ≤ ǫ (1 + | ξ | + | η | ) s + ˜ K ( ǫ, t, s )(1 + | ξ | + | η | ) ( t − sp )( pp − ) ≤ ǫ (1 + | ξ | + | η | ) s + ˜ K ( ǫ, t, s )(1 + | ξ | + | η | ) − u − d . So we have k φ k H t = X ξ ∈ Z u Z R d | ˆ φ ( ξ, η ) | (1 + | ξ | + | η | ) t dη ≤ X ξ ∈ Z u Z R d ǫ | ˆ φ ( ξ, η ) | (1 + | ξ | + | η | ) s + | ˆ φ ( ξ, η ) | ˜ K ( ǫ, t, s )(1 + | ξ | + | η | ) − u − d dη ≤ ǫ k φ k H s + K ( ǫ, t, s ) k ˆ φ k L ∞ ≤ ǫ k φ k H s + K ( ǫ, t, s ) k φ k L . (cid:3) Remark 3.6.
Given a multi-index σ , for every f : D → R and g : T u × R d → T u × R d infinitely many times differentiable, we have ∂ σ ( f ◦ g )( x ) = X ≤| σ ′ |≤| σ | ∂ σ ′ f ( g ( x )) · Q σ,σ ′ ( g ; x ) , (28) where Q σ,σ ′ ( g ; · ) is a homogeneous polynomial of degree | σ ′ | in the derivatives of g , . . . , g u + d until order | σ | − | σ ′ | + 1 .As a consequence, given F : U ⊂ T u × R d → F ( U ) ⊂ T u × R d of class C r and u : U ⊂ T u × R d → R a function in H s for some s ≤ r supported in F ( U ) , thereexists a constant K = K ( F ) depending on F and its derivatives up to order ⌊ s ⌋ suchthat k u ◦ F k H s ≤ K ( F ) k u k H s (29) Proof.
The formula for the derivative of the composition in (28) can be seen in [9] andthe estimate in (29) is an immediate consequence using the expressions for k·k ˜ H s . (cid:3) When φ and φ have disjoint supports, then we have an estimative for h φ , φ i . Claim 3.7.
For ǫ > , there exists a constant K ( ǫ, s ) such that |h φ , φ i ˜ H s | ≤ K ( ǫ, s ) k φ k L k φ k L (30) for every φ , φ ∈ C r ( D ) whose support are disjoints and the distance between themis greater than ǫ . EGULARITY OF THE DENSITY OF SRB MEASURES 11
Proof. If s is integer, by (24) the inner product is 0.If s is not integer then we use (25), the disjointness of the supports and change ofvariables to obtain h φ , φ i ˜ H s = − X | σ |≤⌊ s ⌋ Z R u × R d Z R u × R d Θ σ ( x, y, v, w ) dvdw dxdy, where Θ σ ( x, y, v, w ) = ∂ σ φ ( x + v, y + w ) ∂ σ φ ( x, y )( | v | + | w | ) u + d + δ . Integrating by parts ⌊ s ⌋ times in ( v, w ) according to each index in σ , changingvariables and integrating by parts again ⌊ s ⌋ times, we obtain: h ∂ σ φ , ∂ σ φ i ˜ H s = Z R u × R d Z R u × R d φ ( x + v, y + w ) φ ( x, y ) B ( v, w )( | v | + | w | ) u + d + δ +2 ⌊ s ⌋ dvdw dxdy, where B ( v, w ) is a polynomial of order 2 ⌊ s ⌋ . The proof follows noticing that theintegrand vanish if | v | + | w | ≤ ǫ . (cid:3) Claim 3.8.
Given ≤ s ≤ s , a linear operator L : H s → H s such that L ( H s ) ⊂ H s and constants A , A such that: k L ( u ) k H s ≤ A k u k H s and k L ( u ) k H s ≤ A k u k H s . Then L ( H s θ ) ⊂ H s θ for s θ = (1 − θ ) s + θs , θ ∈ [0 , , and k L ( u ) k H sθ ≤ A − θ A θ k u k H sθ . (31) Proof.
This corresponds to Theorem 22.3 in [22]. (cid:3)
The second main Lasota-Yorke Inequality of this work corresponds to the following.
Proposition 3.9 (Second Main Lasota-Yorke (for Sobolev norm)) . There exist aconstant B , independent of q , and K ( q ) such that for every φ ∈ C r ( D ) and everyinteger ρ with s + u + d < ρ ≤ r − , we have kL q φ k H s ≤ B τ ( q ) | det E | q | det C | q m ( C ) sq k φ k H s + K ( q ) k φ k H s k φ k † ρ . (32)Proposition 3.9 is proved in Section 4.2. Proof of the Main Inequalities
First Lasota-Yorke (for k · k † ). Proof of Proposition 3.3.
We will prove supposing that C is in the Jordan canonicalform. In particular R d = E ⊕ E ⊕· · ·⊕ E k , where the E j ’s are subspaces generated byvectors of the canonical basis, each E j is invariant by C and C | E j has all eigenvalueswith the same absolute value λ j > Claim 4.1.
It is enough to prove Lemma 3.3 supposing that C is in the Jordancanonical form.Proof. Consider P : R d → R d be an invertible linear operator and consider thetransformation ˜ T : T u × R d → T u × R d given by ˜ T ( x, y ) = ( Ex, P CP − y + P f ( x ))and the associated Perron-Frobenius operator˜ L ˜ h ( x, y ) = ( | det E || det C | ) − X ˜ T ( x ′ ,y ′ )=( x,y ) ˜ h ( x ′ , y ′ ) . Notice that the transformations T and ˜ T are linear conjugated by P : T u × R d → T u × R d , P ( x, y ) = ( x, P y ), that is P ◦ T = ˜ T ◦ P . Moreover, defining ˜ D = P ( D )and, for ˜ h ∈ C r ( ˜ D ), the norm k ˜ h k †† ρ = max ≤| α | + | β |≤ ρ sup ψ ∈S sup φ ∈ C | α | + | β | ( ψ ) Z ( φ ◦ P − )( y ) ∂ αx ∂ βy ˜ h ( ψ ◦ P − ( y ) , y ) dy, (33)then the operator U : ( C r ( D ) , k . k † ρ ) → ( C r ( ˜ D ) , k . k †† ρ ) given by U ( h ) = h ◦ P − is abounded isomorphism, that is, there is a constant B > kU ( h ) k †† ρ ≤ B k h k † ρ and kU − (˜ h ) k † ρ ≤ B k ˜ h k †† ρ . Clearly, it is valid that U ◦ L = ˜
L ◦ U .So if for some constants a ≥ b ≥ k ˜ L n ˜ h k †† ρ ≤ a k ˜ h k †† ρ + b k ˜ h k †† ρ − , (34)then kL n h k † ρ ≤ aB k h k † ρ + bB k h k † ρ − . (35) (cid:3) In the rest of this proof, we suppose that C : R d → R d is in the Jordan form.Notice that E i ⊥ E j for i = j and therefore all E j are invariants by C ∗ and C ∗| E j has all eigenvalues with the same absolute value λ j >
0. In these conditions for anycanonical vector ǫ l we havelim n →±∞ n log k C n ǫ l k = lim n →±∞ n log k ( C ∗ ) n ǫ l k = log λ l (36) EGULARITY OF THE DENSITY OF SRB MEASURES 13 and, in particular, lim n →±∞ n log( k C − n ǫ l kk ( C ∗ ) n ǫ l k ) = 0 . (37)Denote { e , . . . , e u } the canonical basis of R u and { ǫ , . . . , ǫ d } the canonical basisof R d . We have the following formula for the derivatives of L n h ( x, y ). Claim 4.2. If ≤ | α | + | β | = ρ ≤ r − , then ∂ αx ∂ βy ( L n h )( x, y ) = X ( x ′ ,y ′ ) ∈ T − n ( x,y ) X | a | + | b |≤ ρ ∂ ax ∂ b y · · · ∂ b d y d h ( x ′ , y ′ ) · Q α,β,a,b,n ( x )( | det E || det C | ) n · k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − b l , (38) where β = ( b , . . . , b l ) and the functions Q α,β,a,b,n are of class C r − −| α |−| β | + | a | + | b | andthere exists a constant K such that k Q α,β,a,b,n k C | a | + | b | ≤ K for every n ≥ , α , β , a and b with | a | + | b | ≤ | α | + | β | ≤ ρ ≤ r − .Proof. By induction in ρ , noticing that the inverse branch T − n c , a is locally written as T − n c , a ( x, y ) = ( E − n c , a x, C − n ( y − S c , a ( x ))) , (39)where S n c , a ( x ) = P n − j =0 C j f ( E − j − c , a y ) is a C r function with k D j S n c , a k ≤ α , 1 ≤ j ≤ r . (cid:3) Using this formular, for ψ ∈ Ω, φ ∈ C r ( ψ ) with k φ k C ρ ≤
1, and considering ψ , · · · , ψ N , g , · · · , g N such that T n ( ψ i ( g i ( y ) , y ) = ( ψ ( y ) , y ), we have: Z φ ( y ) · ∂ αx ∂ βy ( L n h )( ψ ( y ) , y ) dy = X ≤ i ≤ N Z X | a | + | b |≤ ρ φ ( y ) ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( g i ( y )) , g i ( y )) · Q α,β,a,b,n ( ψ ( y ))( | det E | · | det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − b l dy = X ≤ i ≤ N X | a | + | b |≤ ρ Z Ψ α,β,a,b,n ; i ( y ′ ) · ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ )( | det E || det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − β l dy ′ where Ψ α,β,a,b,n ; i ( y ′ ) = φ ( g − i ( y ′ )) · Q α,β,a,b,n (( ψ i ◦ g − i )( y ′ )) · | det Dg − i ( y ′ ) | . Note that Ψ α,β,a,b,n ; i has C | a | + | b | -norm uniformly bounded by some constant K ,depending on the constants k , k , · · · , k r on the definition of Ω but not on h . Inparticular, we have X ≤ i ≤ N X | a | + | b | <ρ Z Ψ α,β,a,b,n ; i ( y ′ ) · ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ )( | det E || det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − β l dy ′ ≤ K ( n ) k h k † ρ − . (40) We will estimate the sum X ≤ i ≤ N X | a | + | b | = ρ Z Ψ α,β,a,b,n ; i ( y ′ ) · ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ )( | det E || det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − β l dy ′ . (41)To integrate by parts, fix i ∈ { , . . . , N } and multi-index ( a, b ) such that | a | + | b | = ρ and note that for b ≥ ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) = ∂ y [ ∂ ax ∂ b − y ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ )] − u X j =1 ∂ x j (cid:0) ∂ ax ∂ b − y ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) (cid:1) h Dψ i ( y ′ ) ǫ k , e j i , If b = 1 then the partial derivative with respect to y disappear, otherwise werepeat the process until the partial derivative with respect to y disappear. So: ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) = ∂ y [ ∂ ax ∂ b − y ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ )]+ b − X m =1 ( − m X | a ′ | = m ∂ y (cid:0) ∂ a + a ′ x ∂ b − m − y ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) (cid:1) u Y j =1 ( h Dψ i ( y ′ ) ǫ , e j i ) a ′ j + ( − b X | a ′ | = b (cid:0) ∂ a + a ′ x ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) (cid:1) u Y j =1 ( h Dψ i ( y ′ ) ǫ , e j i ) a ′ j , which may rewritten as ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) = b − X | a ′ | =0 ( − | a ′ | ∂ y (cid:0) ∂ a + a ′ x ∂ b −| a ′ |− y ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) (cid:1) u Y j =1 ( h Dψ i ( y ′ ) ǫ , e j i ) a ′ j + X | a ′ | = b ( − | a ′ | (cid:0) ∂ a + a ′ x ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) (cid:1) u Y j =1 ( h Dψ i ( y ′ ) ǫ , e j i ) a ′ j , Applying repeatedly the same process to the last sum, but considering derivativeswith respect to y , . . . , y d successively, we obtain that: ∂ ax ∂ b y · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) = d X k =1 F k ( y ′ ) + F ( y ′ ) , EGULARITY OF THE DENSITY OF SRB MEASURES 15 where F k ( y ′ ) = X | a (1) | = b · · · X | a ( k − | = b k − b k − X | a ( k ) | =0 ( − P kj =1 | a ( j ) | ( ∂ y k H a (1) ,...,a ( k ) ( y ′ )) G a (1) ,...,a ( k ) ( y ′ ) ,H a (1) ,...,a ( k ) ( y ′ ) = ∂ a + a (1) + ··· + a ( k ) x ∂ b k −| a ( k ) |− y k ∂ b k +1 y k +1 · · · ∂ b d y d h ( ψ i ( y ′ ) , y ′ ) .G a (1) ,...,a ( k ) ( y ′ ) = k Y l =1 u Y j =1 ( h Dψ i ( y ′ ) ǫ l , e j i ) a ( l ) j and F ( y ′ ) = X | a (1) | = b · · · X | a ( d ) | = b d ( − | b | ˜ H a (1) ,...,a ( d ) ( y ′ )) ˜ G a (1) ,...,a ( d ) ( y ′ ) , ˜ H a (1) ,...,a ( d ) ( y ′ ) = ∂ a + a (1) + ··· + a ( d ) x h ( ψ i ( y ′ ) , y ′ ) , ˜ G a (1) ,...,a ( d ) ( y ′ ) = d Y l =1 u Y j =1 ( h Dψ i ( y ′ ) ǫ l , e j i ) a ( l ) j . Integrating by parts it is easy to note that Z Ψ α,β,a,b,n ; i ( y ′ ) · P dk =1 F k ( y ′ )( | det E || det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − β l dy ′ ≤ K ( n ) k h k † ρ − . (42)By Claim 3.1, the derivatives D j g − a ( z ) = D j − Q a ( z ) C n , D j − det Dg − a ( z ) = D j − (det Q a ( z )) det C n and Dψ a ( z ) = E − n Dψ ( g − a ( z )) Q a ( z ) C n are uniformly boundedby some constant K , since Q a is C r − uniformly bounded.So we have k Ψ α,β,a,b,n ; i ( y ′ ) k C ρ ≤ K | det C | n (43)and k ˜ G a (1) ,...,a ( d ) ( y ′ ) k C ρ ≤ K k E − n k | b | d Y =1 k C n ǫ l k b l , (44)hence Z Ψ α,β,a,b,n ; i ( y ′ ) · F ( y ′ )( | det E || det C | ) n k E − n k −| a | Q dl =1 k ( C − n ) ∗ ǫ l k − β l dy ′ ≤ K ( n ) k h k † ρ , (45)where K ( n ) = K ( | det E | ) n k E − n k − ρ Q dl =1 ( k ( C − n ) ∗ ǫ l kk C n ǫ l k ) − β l . (46) Therefore, by (42) and (45), we conclude that (41) is bounded by K ( n ) k h k † ρ − + K k E − n k ρ d Y l =1 ( k ( C − n ) ∗ ǫ l kk C n ǫ l k ) β l k h k † ρ . From (37) we have that log( k ( C − n ) ∗ ǫ l kk C n ǫ l k ) n converges to zero, which implies(20). The estimate in (21) is analogous and easier. (cid:3) Second Lasota-Yorke (for Sobolev norm).
Through this Section we fix aninteger q and fix p such that τ ( q, ˜ p ) = τ ( q ) for every ˜ p ≥ p .Since L φ ( x ) = | det DT | − P φ ◦ T − c , a ( x ), Remark 3.6 and (39) imply that L is abounded operator in H s , that is kL ( φ ) k H s ≤ K k φ k H s . Let us consider the dual cone fields C ∗ = { ( u, v ) ∈ R u × R d | k v k ≤ α − k u k} (47)and C ∗ = { ( u, v ) ∈ R u × R d | k v k ≤ α − k u k} . (48)Notice that for all ( ξ , η ) = 0 in C ∗ there is a u -dimensional subspace W containedin C ∗ such that ( ξ , η ) ∈ W . Indeed, it is enough to take W = [ { ( ξ , η ) , ( ξ , , . . . , ( ξ u − , } ] , where { ξ k ξ k , ξ , . . . , ξ u − } is an orthonormal base of R u .By continuity of ( x, y ) ( DT q ( x,y ) ) ∗ and noticing that this map does not depend on y , it follows that if ( DT q ( x ,y ) ) ∗ ( ξ, η ) ∈ C ∗ then there exists a u -dimensional subspace W such that ( ξ, η ) ∈ W and a constant R = R ( q ) > DT q ( x,y ) ) ∗ W ⊂ C ∗ for every x ∈ B ( x , R ) and y ∈ R d . More precisely, we conclude that( DT q ( x,y ) ) ∗ (( DT q ( x ,y ) ) ∗ ) − C ∗ ⊂ C ∗ . (49)Consider p sufficiently large such that R ∗ ( ca ) ⊂ B ( x, R ) for all x ∈ R ( ca ), where R = R ( q ) is given as above.The following lemma gives a comparision between k φ k † ρ and the Fourier transformof iterates of L q h ( ξ, η ) when ( DT q ) ∗ ( ξ, η ) is in C ∗ . The main point behind thiscomparison between is that the condition ( DT q ) ∗ ( ξ, η ) ∈ C ∗ allows to consider σ with ( ξ, η ) ∈ σ ⊥ such that σ ⊥ = T q (˜ σ ) with ˜ σ ∈ Ω. EGULARITY OF THE DENSITY OF SRB MEASURES 17
Lemma 4.3.
Let ρ be an integer with s +1 < ρ ≤ r − . Let a ∈ I q and c ∈ I p , and χ : T u × R d → R a C ∞ function supported on R ( ca ) × R d . If = ( ξ, η ) ∈ Z u × R d satisfies ( DT qx ) ∗ ( ξ, η ) ∈ C ∗ for some x ∈ R ( ca ) × R d . Then, for any φ ∈ C r ( D ) , (1 + k ξ k + k η k ) ρ |F ( L q ( χ.φ ))( ξ, η ) | ≤ K ( χ, q ) k φ k † ρ (50) where K ( χ, q ) depends only on χ and q .Proof. We will consider a u -dimensional subspace W as described above satisfying( DT qx ) ∗ W ⊂ C ∗ , for all x ∈ B ( x , R ) ⊃ R ( ca ) and ( ξ, η ) ∈ W .Let 0 = ( ξ, η ) ∈ W ∩ Z u × R d , then the standard property of Fourier transform F ( ∂ x k u ) = iξ k F u gives: |F ( L q ( χφ ))( ξ, η ) | ( k ξ k ρ + k η k ρ ) ≤ K |F ( ∂ ρ x j L q ( χφ ))( ξ, η ) | , (51)where the ρ derivatives are taken with respect to the variable x j ( ξ j or η j ) that hasgreatest absolute value ( | x j | = max {| ξ j | , | η j |} ).Define the partition Γ of D ∩ ( R ( c ) × R d ) formed by the intersections σ of D ∩ ( R ( c ) × R d ) with the d -dimensional affine manifolds orthogonal to W .Since the support of L q ( χφ ) is contained in D ∩ ( R ( c ) × R d ), Rokhlin’s disinte-gration theorem gives: |F ( ∂ ρ x j L q ( χφ ))( ξ, η ) | ≤ Z R ( c ) × R d | ∂ ρ x j L q ( χφ ))( x, y ) | dm ≤ Z Γ Z σ | ∂ ρ x j L q ( χφ ))( x, y ) | dm σ ( x, y ) d ˆ m ( σ ) ≤ ˆ m (Γ) sup σ ∈ Γ Z σ | ∂ ρ x j L q ( χφ )( x, y ) | dm σ . Each m σ above is the d -dimensional Lebesgue measure on σ and ˆ m is identifiedwith the u -dimensional Lebesgue measure on the set of the points w ∈ W such that( w + W ⊥ ) ∩ σ = ∅ for some σ ∈ Γ. In particular, ˆ m (Γ) is finite, because the set ofpoints w ∈ W such ( w + W ⊥ ) ∩ σ = ∅ for some σ ∈ Γ is bounded.For each σ ∈ Γ, there is a unique e σ contained in R ( ca ) × R d such that T q ( e σ ) = σ .For x ∈ e σ and ( u, v ) tangent to σ at T q ( x ), we have0 = h ( u, v ) , ( w , w ) i = h ( DT qx ) − ( u, v ) , ( DT qx ) ∗ ( w , w ) i for all ( w , w ) ∈ W .Since ( DT qx ) ∗ W is a u -dimensional subspace contained in C ∗ , we have ( DT qx ) − ( u, v ) ∈C . So, we conclude that ˜ σ = T − q σ ∩ ( R ( ca ) × R d ) is the graph of some ˜ ψ in S . Since χ is supported in R ( ca ) × R d , we have that L q ( χφ ) = ( χφ ) ◦ b | det DT q | for the inversebranch g : R ( c ) × R d → R ( ca ) × R d of the restriction of T q to R ( ca ) × R d . Then | det DT q || ∂ ρ x j L q ( χφ )( x, y ) | = | ∂ ρ x j (( χφ ) ◦ g )( x, y ) | = | X k α,β,γ ∂ α χ ( g ( x, y )) ∂ β φ ( g ( x, y )) ∂ γ g ( x, y ) |≤ K ( χ ) K ( b ) X β | ∂ β φ ( g ( x, y )) | Integrating and changing variables, we obtain: Z σ | ∂ ρ L q ( χφ )( x, y ) | dm σ ≤ | det DT q | − K ( χ, q ) X β Z σ | ∂ β φ ( b ( x, y )) | dm σ ≤ K ( χ, q ) Z ˜ σ | ∂ β φ (˜ x, ˜ y ) | dm ˜ σ ≤ K ( χ, q ) k φ k † ρ Putting it together, we have that |F ( L q ( χφ ))( ξ, η ) | ( k ξ k ρ + k η k ρ ) ≤ K ˆ m (Γ) K ( χ, q ) k φ k † ρ . Finally, the result follows noticing that |F ( L q ( χφ ))( ξ, η ) | ≤ k φ k L ≤ K k φ k † ρ and(1 + | ξ | + | η | ) ρ ≤ K (1 + | ξ | ρ + | η | ρ ). (cid:3) One Lemma concerning the transversality that shall be used in the proof of theLasota-Yorke inequality is the following:
Lemma 4.4.
Let ( ξ, η ) ∈ Z u × R d \ { } . If a is transversal to b on R ∗ ( c ) theneither ( DT qx ) ∗ ( ξ, η ) ∈ C ∗ for all x ∈ E − q c , a ( R ∗ ( c )) or ( DT qx ) ∗ ( ξ, η ) ∈ C ∗ for all x ∈ E − q c , b ( R ∗ ( c )) .Proof. Note that if E q ( x a ) = x for some x a ∈ E − q c , a ( R ∗ ( c ))then( DT q ( x a , y )) ∗ = (cid:18) ( E q ) ∗ ( E q ) ∗ ( DS c ( x, a )) ∗ C q ) ∗ (cid:19) . Supposing that ( DT q ( x a , y )) ∗ ( ξ, η )
6∈ C ∗ for some x ∈ E − q c , a ( R ∗ ( c )), then we claimthat ( DT q ( x b , y )) ∗ ( ξ, η ) ∈ C ∗ for all x b ∈ E − q c , b ( R ∗ ( c )).In fact, if both vectors are not in C ∗ , then k ( C q ) ∗ η k > / α − k ( E q ) ∗ ξ +( E q ) ∗ ( DS c ( x, a )) ∗ η k and k ( C q ) ∗ η k > / α − k ( E q ) ∗ ξ + ( E q ) ∗ ( DS c (˜ x, b )) ∗ η k . Then, summing and usingtriangular inequality, we have that2 k ( C q ) ∗ η k > / α − k ( E q ) ∗ ( DS c ( x, a ) − DS c (˜ x, b )) ∗ η k . EGULARITY OF THE DENSITY OF SRB MEASURES 19
On the other hand, the transversality implies that k ( DS c ( x, a ) − DS c (˜ x, b )) ∗ η k ≥ α k C k q k E − k q k η k . So, by the last inequality,2 k C q kk η k > k E − q k − k C k q k E − k q k η k . Since k E − q k ≤ k E − k q , it follows k E − q kk E − k q ≥
1, and therefore 2 k C q kk η k > k C k q k η k , which is a contradiction. (cid:3) To make the local argument we will consider a fixed partition of unity. For thispurpose, consider { χ c : T u → R } c ∈A p a family of C ∞ functions that form a partitionof unity subordinated to the covering {R ∗ ( c ) } .We define { χ c , a : T u → R } c ∈A p by χ c , a ( E − q c , a ( x )) = χ c ( x ) (52)if x ∈ R ∗ ( c ) and 0 elsewhere. Notice that { χ c , a } is another partition of unitysubordinated to {R ∗ ( ca ) } .The following lemma compares the H s norm of φ with the sums of H s norm of χ c φ , defined by χ c φ ( x, y ) := χ c ( x ) φ ( x, y ). Lemma 4.5.
There exists a constant K such that, for any φ ∈ C r ( D ) , it holds X ( a , c ) ∈ A q × A p k χ c , a φ k H s ≤ k φ k H s + K k φ k L (53) and k φ k H s ≤ K X c ∈ A p k χ c φ k H s + K k φ k L . (54) Proof.
First consider the case s ∈ N . Then X ( a , c ) ∈ A q × A p k χ c , a φ k H s = X ( a , c ) ∈ A q × A p X | σ |≤ s k ∂ σ ( χ c , a φ ) k L ≤ K ( s )( I + I ) , where I = X ( a , c ) ∈ A q × A p X | σ |≤ s Z T u × R d | χ c , a ( x, y ) | | ∂ σ φ ( x, y ) | dxdy and I = X ( a , c ) ∈ A q × A p X | σ |≤ s X σ ′ <σ (cid:18) σσ ′ (cid:19) Z T u × R d | ∂ σ − σ ′ χ c , a ( x, y ) | | ∂ σ ′ φ ( x, y ) | dxdy. Since χ c , a is a partition of unity, I is bounded by X | σ |≤ s Z T u × R d | ∂ σ φ ( x, y ) | dxdy = k φ k H s and I is bounded by K ( s, p, q ) k φ k H s − . Using Young’s inequality, it follows (53).For the case s N , let t be the largest integer that is less than s and δ = s − t ∈ (0 , X a , c k χ c , a φ k H s ≤ X a , c X | σ |≤ t k ∂ σ ( χ c , a φ ) k L + X ( a , c ) ∈ A q × A p X | σ | = t R ( a , c , σ ) = S + S , where, considering X = ( x, u ) and V = ( v, w ), R ( a , c , σ ) = Z T u × R d Z R u × R d | ∂ σ ( χ c , a φ )( X + V ) − ∂ σ ( χ c , a φ )( X ) | ( | V | ) u + d +2 δ dXdV. As in the previous case, S is bounded by k φ k H t . To estimate S , let us write R ( a , c , σ ) = R ( a , c , σ ) + R ( a , c , σ ) + R ( a , c , σ ) + R ( a , c , σ )with R = Z T u × R d Z R u × R d | χ c , a ( X ) | | ∂ σ φ ( X + W ) − ∂ σ φ ( X ) | ( | W | ) u + d +2 δ dXdWR = Z T u × R d Z R u × R d | χ c , a ( X + W ) − χ c , a ( X ) | ( | W | ) u + d +2 δ | ∂ σ φ ( X + W ) | dXdWR = X σ ′ <σ Z T u × R d Z R u × R d | ∂ σ − σ ′ χ c , a ( X ) | | ∂ σ ′ φ ( X + U ) − ∂ σ ′ φ ( X ) | ( | W | ) u + d +2 δ dXdWR = X σ ′ <σ Z T u × R d Z R u × R d | ∂ σ − σ ′ χ c , a ( X + U ) − ∂ σ − σ ′ χ c , a ( X ) | ( | W | ) u + d +2 δ | ∂ σ ′ φ ( X + W ) | dXdW So, S + P ( a , c ) ∈ A q × A p P | σ | = t R ( a , c , σ ) is bounded by k φ k H s and R + R + R isbounded by K ( s ) k φ k H t . Using Young’s inequality again, we have (53).For inequality (54), note that the closure of each R ∗ ( c ) intersects at most r closuresof the sets R ∗ (˜ c ), since the Markov partition is formed by r sets. Then: k φ k H s = X c,c ′ h χ c φ, χ c ′ φ i H s = X R ∗ ( c ) ∩R ∗ ( c ′ )= ∅ h χ c φ, χ c ′ φ i H s + X R ∗ ( c ) ∩R ∗ ( c ′ ) = ∅ h χ c φ, χ c ′ φ i H s . If R ∗ ( c ) ∩R ∗ ( c ′ ) = ∅ , then Remark 3.7 implies that h χ c φ, χ c ′ φ i H s ≤ K k χ c φ k L k χ c ′ φ k L ,which gives: X R ∗ ( c ) ∩R ∗ ( c ′ )= ∅ h χ c φ, χ c ′ φ i H s ≤ K k φ k L . EGULARITY OF THE DENSITY OF SRB MEASURES 21 If R ∗ ( c ) ∩ R ∗ ( c ′ ) = ∅ , then h χ c φ, χ c ′ φ i H s ≤ k χ c φ k Hs + k χ c ′ φ k Hs . So, we have X R ∗ ( c ) ∩R ∗ ( c ′ ) = ∅ h χ c φ, χ c ′ φ i H s ≤ r X c ∈ A p k χ c φ k H s . (cid:3) Lemma 4.6.
Given ≤ s ≤ r , a ∈ A q and c ∈ A p , there exists a constant K > such that kL q ( χ ca φ ) k H s ≤ K ( | det E || det C | m ( C ) s )) q k χ c , a φ k H s (55) for every φ ∈ C r ( D ) .Proof. Let us first consider s integer. Recalling that T − q c , a is an inverse branch definedover R ( c ) × R d by T − q c , a ( x, y ) = ( E − q c , a x, C − q ( y − S c ( x, a ))). If we call by g , g , . . . , g u + d the components of T − q c , a then we may observe that | ∂ σ g j k ≤ α k C − q k for all σ multi-index with | σ | ≤ r .Noticing that L q ( χ c , a φ ) = ( χ c , a φ ) ◦ T − q c , a | det DT | q and recalling the formula for differentiationof the composition (Remark 3.6), we have | det DT | q kL q ( χ c , a φ ) k H s = | det DT | q X | σ |≤ s Z | ∂ σ L q ( χ c , a φ )( z ) | dz ≤ X | σ |≤ s Z | ∂ σ [( χ c , a φ ) ◦ T − q c , a ]( z ) | dz ≤ X | σ |≤ s Z | X | σ ′ |≤| σ | ( ∂ σ ′ χ c , a φ ) ◦ T − q c , a ( z ) ψ σ ′ ,σ ( z ) | dz Above we used Remark 3.6 to write ∂ σ [( χ c , a φ ) ◦ T − q c , a ]( z ) as P | σ ′ |≤| σ | ψ σ ′ ,σ ( z )( ∂ σ ′ χ c , a φ ) ◦ T − q c , a ( z ) , where ψ σ ′ ,σ is a polynomial function of degree at most s in the variables ∂ γ g j ,where γ goes through the multi-indexes with | γ | ≤ | σ ′ | and j = 1 , , . . . , u + d .Noticing that | ψ σ ′ ,σ ( z ) | ≤ K k C − q k s for some constant K , we have | det DT | q kL q ( χ c , a φ ) k H s ≤ K k C − q k s X | σ |≤ s Z (cid:0) X | σ ′ |≤| σ | | ( ∂ σ ′ χ c , a φ ) ◦ T − q c , a ( z ) | (cid:1) dz ≤ K k C − q k s | det DT q | X | σ |≤ s Z (cid:0) X | σ ′ |≤| σ | | ( ∂ σ ′ χ c , a φ )( z ) | (cid:1) dz ≤ K k C − q k s | det DT q | X | σ |≤ s Z | ( ∂ σ χ c , a φ )( z ) | dz = K k C − q k s | det DT q |k χ c , a φ k H s . This implies (55) in this case.For non-integers values of s , we consider integers s and s with 0 ≤ s ≤ s ≤ s ≤ r . Since φ is C r ( D ), then it is in H s and H s . Applying Claim 3.8 for K = (cid:0) K ( | det E |·| det C | m ( C ) s ) q (cid:1) and K = (cid:0) K ( | det E |·| det C | m ( C ) s ) q (cid:1) , it follows (55). (cid:3) Now we can proceed to the Proof of Lemma 3.9.
Proof of Lemma 3.9.
ByLemma 4.5, we have kL q φ k H s ≤ K X c ∈ A p k χ c L q φ k H s + K k φ k L ≤ K X c ∈ A p k X a ∈ A q L q ( χ ca φ ) k H s + K k φ k L = K X c ∈ A p , a , b ∈ A q hL q ( χ ca φ ) , L q ( χ cb φ ) i H s + K k φ k L . In the following, we estimate hL q ( χ ca φ ) , L q ( χ cb φ ) i H s dividing it into 2 cases: when a ⋔ c b and when a ⋔ c b If a ⋔ c b , by Lemma 4.4, for every ( ξ, η ) = (0 ,
0) we have either ( DT qx ) ∗ ( ξ, η ) ∈ C ∗ for all x ∈ R ( ca ) or ( DT qx ) ∗ ( ξ, η ) ∈ C ∗ for all x ∈ R ( cb ). Denote by U the set of( ξ, η ) such that the first occurs and V the set such that the second occurs.Then, if ( ξ, η ) ∈ U , by Lemma 4.3 we have |F ( L q ( χ c , a φ ))( ξ, η ) | ≤ K (1 + | ξ | + | η | ) − ρ k φ k † ρ . (56)Remind that kL q φ k H s ≤ K q k φ k H s since the operator is bounded in H s (see Re-mark 3.6). So, by Cauchy-Schwarz EGULARITY OF THE DENSITY OF SRB MEASURES 23 (cid:12)(cid:12)(cid:12) X ξ Z U (1 + | ξ | + | η | ) s F L q ( χ c , a φ )( ξ, η ) F L q ( χ c , b φ )( ξ, η ) dη (cid:12)(cid:12)(cid:12) ≤ X ξ Z U (1 + | ξ | + | η | ) s |F L q ( χ c , a φ )( ξ, η ) | dη ! / kL q ( χ c , b φ ) k H s ≤ K X ξ Z (1 + | ξ | + | η | ) s − ρ k φ k † ρ d η ≤ K ( q ) k φ k † ρ k φ k H s , where we used that the integral is finite since s − ρ < − ( u + d ) / V instead of U , we obtain that hL q ( χ ca φ ) , L q ( χ cb φ ) i H s ≤ K ( q ) k φ k † ρ k φ k H s (57)If a ⋔ c b , we use hL q ( χ ca φ ) , L q ( χ cb φ ) i H s ≤ kL q ( χ ca φ ) k H s + kL q ( χ cb φ ) k H s τ ( q ) to obtain X a ⋔ c b hL q ( χ ca φ ) , L q ( χ cb φ ) i H s ≤ τ ( q ) X a kL q ( χ ca φ ) k H s (59)Using it, Lemma 4.5 and Lemma 4.6, we have X c ∈ A p , a ⋔ c b hL q ( χ c , a φ ) , L q ( χ c , b φ ) i H s ≤ Kτ ( q )( | det E || det C | m ( C ) s ) q X a , c k χ c , a φ k H s ≤ Kτ ( q )( | det E || det C | m ( C ) s ) q k φ k H s + K ( q ) k φ k L . (60)Since k · k L ≤ k · k † ρ and k · k L ≤ K k · k L ≤ K k · k H s , we may use (57) and (60)to get the estimate below kL q φ k H s ≤ X c ∈ A p , a ⋔ c b hL q ( χ ca φ ) , L q ( χ cb φ ) i H s + X c ∈ A p , a ⋔ c b hL q ( χ ca φ ) , L q ( χ cb φ ) i H s + K k φ k L ≤ Kτ ( q )( | det E || det C | m ( C ) s ) q k φ k H s + K k φ k L + K ( q ) k φ k † ρ k φ k H s ≤ Kτ ( q )( | det E || det C | m ( C ) s ) q k φ k H s + K ( q ) k φ k † ρ k φ k H s (cid:3) Proof of Theorems 1 and 2
Let us proceed to put together the two main Lasota-Yorke inequalities to obtainthe third Lasota-Yorke inequality of this work, from which will follow Theorems 1and 2.5.1.
Third Lasota-Yorke (for k φ k = k φ k H s + k φ k † ρ ). Putting the two Main In-equalities together, we obtain a third Lasota-Yorke.
Proposition 5.1 (Third Lasota-Yorke) . Given q ∈ N satisfying B τ ( q )( | det DT | m ( C ) s ) q < and integers ≤ ρ < ρ ≤ r − with s < ρ − u − d , consider ν = ν ( ρ , ρ ) := P ρ j = ρ +1 1 j and some ζ ∈ (cid:16) max (cid:8) k E − k ν , (cid:0) { B q τ ( q ) /q | det DT | m ( C ) s (cid:1) (cid:9) , (cid:17) . (61) Consider also the norm k φ k := k φ k H s + k φ k † ρ , then there exists a constant K suchthat for all n ∈ N , kL n φ k ≤ Kζ n k φ k + K k φ k † ρ . (62) Proof of Proposition 5.1.
Let us begin this proof stating two consequences of the firstMain Inequality (Lemma 3.3).
Claim 5.2.
Let δ ∈ ( k E − k , . There exists K > such that, for ≤ ρ ≤ r − ,for n ∈ N , kL n h k † ρ ≤ Kδ ρn k h k † ρ + K k h k † ρ − . (63) Proof.
Take N ∈ N such that K k E − k ρN ≤ δ ρN . Then, by Lemma 3.3, we have kL N h k † ρ ≤ δ ρN k h k † ρ + K ( N ) k h k † ρ − (64)Moreover, using k h k † ρ − ≤ k h k † ρ , there exists K = K ( N ) > kL j h k † ρ ≤ δ rN K k h k † ρ ≤ δ ρj K k h k † ρ (65)for all 1 ≤ j ≤ N − ≤ ρ ≤ r − ρ that there exists a constant K ρ > n ∈ N kL n h k † ρ ≤ K ρ δ ρn k h k † ρ + K ρ k h k † ρ − and kL n h k † ρ ≤ K ρ k h k † ρ . (66) EGULARITY OF THE DENSITY OF SRB MEASURES 25
Write n = kN + j , with 1 ≤ j ≤ N −
1. For ρ = 1 we have kL n h k † = kL kN ( L j h ) k † ≤ δ kN kL j h k † + K ( N ) k − X i =0 δ iN kL ( k − − i ) N ( L j h ) k † ≤ Kδ n k h k † + K ( N ) K − δ N k h k † ≤ K δ n k h k † + K k h k † and kL n h k † ≤ K k h k † , where K = 2 max { K, K ( N ) A − δ N } .Now, suppose the result is true for ρ −
1, we prove for ρ . kL n h k † ρ = kL kN ( L j h ) k † ρ ≤ δ kN kL j h k † ρ + K ( N ) N − X i =0 δ iN kL ( k − − i ) N ( L j h ) k † ρ − ≤ Kδ n k h k † ρ + K ( N ) K ρ − − δ N k h k † ρ − ≤ K ρ δ n k h k † ρ + K ρ k h k † ρ − and kL n h k † ρ ≤ K ρ k h k † ρ , where K ρ = 2 max { K, K ( N ) K ρ − − δ N } . The result follows taking K = max ≤ i ≤ r − { K i } . (cid:3) Claim 5.3.
Given δ ∈ ( k E − k , and integers ≤ ρ < ρ ≤ r − , let ν ( ρ , ρ ) beas before. Then there exists K > such that, for every n ∈ N , kL n h k † ρ ≤ Kδ n/ν ( ρ ,ρ ) k h k † ρ + K k h k † ρ . (67) Proof.
Let n be a multiple of ( r − ρ ∈ [ ρ + 1 , ρ ]that kL n ( ν ( ρ,ρ )) h k † ρ ≤ K ρ δ n k h k † ρ + K ρ k h k † ρ . (68)Actually, the case ρ = ρ + 1 is immediately because ν ( ρ + 1 , ρ ) = 1 / ( ρ + 1).Also, using Claim 5.2, the relation nν ( ρ + 1 , ρ ) = nν ( ρ, ρ ) + nρ +1 and the induction hypothesis, we have: kL n ( ν ( ρ +1 ,ρ )) h k † ρ +1 = kL nρ +1 ( L n ( ν ( ρ,ρ )) h ) k † ρ +1 ≤ Kδ n kL n ( ν ( ρ,ρ )) h k † ρ +1 + K kL n ( ν ( ρ,ρ )) h k † ρ ≤ Kδ n (cid:0) Kδ nν ( ρ,ρ )( ρ +1) k h k ρ +1 + K k h k † ρ (cid:1) + (cid:0) K ρ δ n k h k † ρ + K ρ k h k † ρ (cid:1) ≤ K ρ +1 δ n k h k † ρ +1 + K ρ +1 k h k † ρ . So we have the lemma for multiples of ( r − ν ( ρ , ρ ). For the general case, justnotice that Claim 5.2 also implies that L is a bounded operator with respect to thenorm k · k † ρ . (cid:3) Now we proceed to prove Lemma 5.1, noticing first that for a, b >
0, we have that √ a + b ≤ √ a + √ b and √ ab ≤ ǫa + ǫ − b . So Lemma 3.9 implies that for every ǫ > kL q φ k H s ≤ (cid:16) K τ ( q ) /q | det DT | m ( C ) s (cid:17) q/ k φ k H s + ǫ k φ k H s + K ( ǫ ) k φ k † ρ . Since ( K /q τ ( q ) /q | det DT | m ( C ) s ) q/ < ζ q , for ǫ = ǫ ( q ) small we have kL q φ k H s ≤ ζ q k φ k H s + K ( q ) k φ k † ρ . Iterating it l times: kL lq φ k H s ≤ ζ lq k φ k H s + K ( l ) k φ k † ρ . (69)Now, taking δ slightly smaller than ζ ν and l large enough such that K ( δ ν ) l q < ζ l q , Claim 5.3 implies for l kL l q φ k ρ ≤ ζ l q k φ k ρ + K ( l ) k φ k † ρ . (70)Let us consider the auxiliary norm k φ k ∗ := k φ k H s + 2 K ( l ) ζ − l q k φ k † ρ , which isequivalent to k · k . Adding (69) and (70), it follows that: kL l q φ k ∗ ≤ ζ l q k φ k ∗ + ˜ K ( l ) k φ k † ρ . (71)Iterating this inequality, it follows what we want for every n but for the norm k · k ∗ .Since they are equivalent norms, it follows the result for the norm k · k . (cid:3) Proof of Theorem 1.
Proof of Theorem 1.
Since | det DT | m ( C ) s >
1, the transversality condition impliesthat we can consider q such that ω = B τ ( q )( | det DT | m ( C ) s ) q < ρ = r − ρ = 0. Since s < r − u/ − d/ −
1, we have that s + u/ d/ < ρ , so we can apply Lemma 5.1 for some ζ between ω and 1. EGULARITY OF THE DENSITY OF SRB MEASURES 27
Let us fix some non-negative function ψ ∈ C r ( D ) with k ψ k L = 1, ν = ψ m , ψ n = n ( ψ + L ψ + · · · + L n − ψ ) and ν n = ψ n m . Then ν n = n P n − j =0 T j ∗ ν .Since µ is the SRV measure for T , for every φ ∈ C ( D ) we have that n P n − j =0 φ ◦ T j ( x ) converges to R φ dµ for Lebesgue almost every x , therefore Z φdν n = Z n n − X j =0 φ ◦ T j dν → Z φ dµ. (72)On the other hand, Lemma 5.1 implies that there exists a constant K >
0, suchthat kL n ψ k ≤ K k ψ k , for all n . In particular, k ψ n k H s ≤ k ψ n k ≤ K k ψ k for every n . So, Banach-Alaoglu theorem implies that there is a subsequence { ψ n k } k whichconverges weakly to some function ψ ∞ ∈ H s , then Z φdν n k = Z φ ψ n k dm → Z φ ψ ∞ dm (73)for every φ ∈ C r ( D ) with compact support. Hence µ = ψ ∞ m is an absolutelycontinuous invariant probability.The openness in ( C, f ) follows from the fact that τ ( q ) is upper semi-continuous on( C, f ) ∈ C ( d ) × C r ( T u , R d ) and from the openness of the condition B τ ( q ) | det E || det C | m ( C ) s <
1, what concludes the proof of the theorem. (cid:3)
Remark 5.4.
It is important to mention that the transversality condition defined inDefinition 2.1 is not an open condition. What is open in ( C, f ) is the condition B τ f ( q ) | det DT q | m ( C ) sq < for fixed q . Spectral Gap.
When s > u/
2, we can apply a theorem of Hennion to obtainspectral properties of the action of the operator L in a Banach space B contained in H s and containing C r − ( D ).Let us denote the spectral radius of L : B → B by ρ ( L ) = lim n →∞ n p kL n k . Wesay that L has spectral gap if there exist bounded operators P and N such that L = λ P + N , with P = P , dim(im( P )) = 1, ρ ( N ) < | λ | and PN = N P = 0.The spectral gap can be obtained as a standard consequence of a Theorem due toHennion, Ionescu Tulcea-Marinescu, et al [11, 19]:
Theorem (Hennion) . Let L : ( B , k · k ) → ( B , k · k ) be a bounded operator and k · k ′ be a norm in B such that (1) k · k ′ is continuous in k · k . (2) For every bounded sequence { φ n } ∈ B , there exists a subsequence { φ n k } and ψ ∈ B such that k φ n k − ψ k ′ → . (3) k Lφ k ′ ≤ M k φ k ′ for some M > and every φ ∈ B . (4) There exists r ∈ (0 , ρ ( L )) and K > such that for all n ∈ N k L n φ k ≤ r n k φ k + K k φ k ′ . (74)(5) There exists a unique eigenvalue λ with | λ | = ρ ( L ) and dim ker( L − λI ) = 1 .Then L has spectral gap. Inequality (74) is sometimes known as Lasota-Yorke [14] or Doeblin-Fortet [8]inequality for L with respect to the spaces B and B ′ . It is exactly the same kind ofinequality that we proved in Lemmas 3.3, 3.9 and 5.1. Definition 5.5.
We say that ( T, µ ) has exponential decay of correlations in avector space B ⊂ L ( µ ) with exponential rate at most ζ < if for every φ ∈ B and ψ ∈ L ∞ ( µ ) , there exists a constant K ( φ, ψ ) > such that (cid:12)(cid:12)(cid:12)(cid:12)Z φ ( ψ ◦ T n ) dµ − Z φdµ Z ψdµ (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ( φ, ψ ) ζ n . (75)When the transfer operator L has spectral gap, it follows that the dynamics hasexponential decay of correlations with exponential rate at most ρ ( N ) <
1, as givenin the following Proposition.
Proposition 5.6.
Supposing that L has spectral gap in some Banach space B embed-ded continuously in L ( m ) with ρ ( L|B ) = 1 and ρ ( N ) = ζ < , if we consider φ ∈ B a nonnegative fixed point of L satisfying R φ dm = 1 and µ = φ m , then ( T, µ ) hasexponential decay of correlations in ˜ B := { φ ∈ B , φφ ∈ B} with exponential rate atmost ζ . In particular, if B is a Banach algebra then ( T, µ ) has exponential decay ofcorrelations in B .Proof. Since L has spectral gap in B , for each φ ∈ B we write φ = a ( φ ) φ + φ with kL n φ k B ≤ ζ n k φ k B . Then the property R L u dm = R u dm and L φ = φ impliesthat R φ dm = 0 and a ( φ ) = R φdm . We also have L n ( φ · ψ ◦ T n ) = ψ · L n φ and k φ k B ≤ K k φ k B .Given ψ ∈ L ∞ ( µ ) and φ ∈ ˜ B , it follows that: EGULARITY OF THE DENSITY OF SRB MEASURES 29 (cid:12)(cid:12)(cid:12) Z φ ( ψ ◦ T n ) dµ − Z φdµ Z ψdµ (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z φ ( ψ ◦ T n ) φ dm − Z φφ dm Z ψφ dm (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z (cid:2) L n ( φφ ) − (cid:16) Z φφ dm (cid:17) φ (cid:3) ψdm (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Z L n h ( φφ ) i ψdm (cid:12)(cid:12)(cid:12) ≤ k ψ k L ( m ) kL n ( φφ ) k L ( m ) ≤ K k ψ k L ( m ) kL n ( φφ ) k B ≤ K k ψ k L ( m ) k ( φφ ) k B ζ n = K ( φ, ψ ) ζ n . So (
T, µ ) has exponential decay of correlations in ˜ B . (cid:3) These facts will be used to prove Theorem 2.
Proof of Theorem 2.
Consider the smallest integer ρ and the greatest integer ρ suchthat ρ + u/ < s < ρ − u/ − d/ . Consider t ∈ ( ρ + u/ , s ) and an integer q such that B τ ( q ) < ( | det DT | m ( C ) s ) q .Since ρ − ρ ≤ u + d +2 and P nj =1 /j ≤ n − ν ≤ u + d +1) := a . So, if ζ ∈ (cid:16) max {k E − k a , ( ( B τ ( q )) /q | det DT | m ( C ) s ) } , (cid:17) then ζ is in the interval in (61).We will verify that the conditions of Theorem 5.3 are satisfied considering B thecompletion of C r ( D ) with respect to the norm k · k = k · k H s + k · k † ρ and B ′ thecompletion of C r ( D ) with respect to the norm k · k † ρ .Obviously k · k ρ ≤ k · k ρ ≤ k · k , which implies condition (1) in the theorem ofHennion. Condition (3) is and immediate consequence of Lemma 3.3 and condition(4) follows from Lemma 5.1 with r = ζ . Condition (5) is immediate since T is mixingin Λ.It remains to verify the compactness (condition (2)). The embedding of H s ( D )in H t ( D ) is compact, by Sobolev’s embedding theorem ( s > t ). So, it is sufficientto prove that the embedding of H t ( D ) in B ′ is continuous, which will be proved inLemma 5.7.Finally, we notice that C r − ( D ) ⊂ B . The definition of k · k † ρ gives that (cid:12)(cid:12)(cid:12) Z U ψ φ ( y ) ∂ αx ∂ βy h ( ψ ( y ) , y ) dy (cid:12)(cid:12)(cid:12) ≤ Z U ψ (cid:12)(cid:12)(cid:12) ∂ αx ∂ βy h ( ψ ( y ) , y ) (cid:12)(cid:12)(cid:12) dy ≤ vol( D ) k h k C r − whenever | α | + | β | ≤ ρ , ψ ∈ S , φ ∈ C | α | + | β | ( U ψ ) and k φ k C | α | + | β | ≤
1. This impliesimmediately that k h k † ρ ≤ K k h k C r − and that C r − ( D ) ⊂ B . (cid:3) Lemma 5.7.
Consider ≤ ρ < ρ ≤ r − such that ρ + u/ < t < s < ρ − u/ − d/ . Then the embedding of H t ( D ) in B ′ is continuous, that is, there exists a constant K > such that k u k † ρ ≤ K k u k H t . Proof of Lemma 5.7.
Consider ψ = ( ψ , . . . , ψ u ) ∈ S a C r transformation as in theSubsection 3.1. Recall that k D ν ψ k ≤ k ν for k , k , . . . , k r previously fixed. Con-sidering an extension of ψ , we may suppose that the domain U ψ of ψ contains π ( D ) = [ − K , K ] d . In these conditions, we establish Claim 5.8.
Let u : T u × R d → R be a C r function with compact support in D .Define v ( x, y ) = u ( x + ψ ( y ) , y ) for y ∈ U ψ and v ( x, y ) = 0 if y U ψ . Then, for everymulti-index γ with | γ | ≤ r and for every y ∈ U ψ , we have ( ∂ γ u )( x + ψ ( y ) , y ) = X | ˜ γ |≤| γ | a ˜ γ,γ ( y )( ∂ ˜ γ v )( x, y ) and (76)( ∂ γ v )( x, y ) = X | ˜ γ |≤| γ | b ˜ γ,γ ( y )( ∂ ˜ γ u )( x + ψ ( y ) , y ) , (77) where a ˜ γ,γ and b ˜ γ,γ are polynomials of degree at most | γ | in the variables ∂ β ψ k ,with ≤ | β | ≤ | γ | . Consequently ∂ α a ˜ γ,γ and ∂ α b ˜ γ,γ are bounded by some constant K which depends on only k , k , . . . , k r , for all multi-index α = ( α , . . . , α d ) , with ≤ | α | ≤ r − | γ | .Proof. Follows by induction on ρ = | γ | . (cid:3) Claim 5.9.
Let u : T u × R d → R be a C r function with compact support in D .Define v ( x, y ) = u ( x + ψ ( y ) , y ) for y ∈ U ψ and v ( x, y ) = 0 if y U ψ . Then, for any < t < r , we have k v k H t ( R u + d ) ≤ K k u k H t ( R u + d ) , where K depends on only k , k , . . . , k r .Proof. Follows using inequality (77). (cid:3)
From the definition of k . k † ρ and Cauchy-Schwarz inequality, we have that k u k † ρ ≤ vol( D ) max | γ |≤ ρ sup ψ ∈S k ∂ γ u ( ψ ( . ) , . ) k L ( R d ) . (78)By Claim 5.8, the right-hand side of (78) is bounded byvol( D ) K X | γ |≤ ρ k ∂ γ v (0 , . ) k L ( R d ) = K k v (0 , . ) k H ρ . (79) EGULARITY OF THE DENSITY OF SRB MEASURES 31
Due to [1, Theorem 7.58(iii)] applied with p = q = 2, ˜ s = ρ + u/ χ = ρ , k = d , n = u + d , we have: k v (0 , . ) k H ρ ≤ K k v ( ., . ) k H ˜ s ( R u + d ) . By t > ρ + u = ˜ s and Claim 5.9, we conclude that k v ( ., . ) k H ˜ s ( R u + d ) ≤ K k v ( ., . ) k H t ( R u + d ) ≤ K k u k H t ( R u + d ) . (80)Therefore we have that k u k † ρ ≤ K k u k H t . (cid:3) Genericity
In [6, Theorem 2.12] the authors proved that if C ∈ C ( d ) satisfies k C k < k E − k − | det E | u − d +1 ,then there exists a family f t , t ∈ R m , with f = f such that the set (cid:8) t ∈ R m , lim sup q →∞ q log τ f t ( q ) > log J (cid:9) has zero Lebesgue measure (where J = | det E || det C | − k C − k − d > β > J , thatis, if we define C ( d ; E ) = ( C ∈ C ( d ) , k C k < k E − k − | det E | u − d +1 ) , (81)then the set T β := { t ∈ R m , lim sup q →∞ q log τ f t ( q ) > β } has zero Lebesgue measure. Proposition 6.1 ([6], Theorem 2.12) . Given β > , integers u ≥ d ≥ , E ∈ E ( u ) and C ∈ C ( d, E ) , there exist C ∞ -functions φ k : T u → R d , k = 1 , , . . . , m such thatfor f ∈ C ( T u , R d ) and its corresponding family f t = f + P sk =1 t k φ k , the set ofparameters t = ( t , t , . . . , t m ) such that t / ∈ T β has full Lebesgue measure. As a consequence, for every n ≥ R n ∈ C r ( T u , R d ) suchthat lim sup q → + ∞ q log τ f ( q ) < n . Then R = ∩ n ≥ R n is also a residual subset of C r ( T u , R d )such that lim sup q → + ∞ q log τ f ( q ) = 0 for every f ∈ R .6.1. Proof of the Main Theorems.
Putting Proposition 6.1 together with The-orems 1 and 2, it follows Theorems A and D, and the immediate Corollaries B andC.
Proof of Theorem A.
Consider β = log | det C | det E |k C − k − s > R as given bythe consequence of Proposition 6.1 and V ⊂ C r ( T u , R d ) the set of f ’s such that thecorresponding SRB measure µ T of T ( C, E, f ) is absolutely continuous with repectto the Lebesgue measure and k dµ T /d vol T u × R d k H s < + ∞ . As Theorem 1 is valid for every f ∈ R , we have a corresponding open set U f suchthat the conclusion of Theorem 1 is valid for every g ∈ U f . Taking U = ∪ f ∈R U f , itfollows that R ⊂ U and that U is dense. So U is open and dense and Theorem A isvalid for every f ∈ U . (cid:3) Proof of Corollary B.
Notice that the inequality | det E || det C |k C − k − s > s ∈ R . So if it is valid for s = 0, then is also valid for some s > (cid:3) Proof of Corollary C.
This corollary is immediate from Theorem A and Sobolev’sembedding Theorem (the elements of H s ( T u × R d ) are continuous up to a nullLebesgue set when s > u + d ). (cid:3) Proof of Theorem D.
Consider the same residual set
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Department of Mathematics, UFPB, Jo˜ao Pessoa-PB, Brazil
E-mail address : [email protected] (Ricardo Bortolotti) Department of Mathematics, UFPE, Recife-PE, Brazil
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