Backward volume contraction for endomorphisms with eventual volume expansion
aa r X i v : . [ m a t h . D S ] O c t BACKWARD VOLUME CONTRACTION FORENDOMORPHISMS WITH EVENTUAL VOLUMEEXPANSION
J. F. ALVES, ARMANDO CASTRO, AND VILTON PINHEIRO
Abstract.
We consider smooth maps on compact Riemannianmanifolds. We prove that under some mild condition of eventualvolume expansion Lebesgue almost everywhere we have uniformbackward volume contraction on every pre-orbit of Lebesgue al-most every point.
R´esum´e
Nous consid´erons des transformations diff´erentiables sur des va-riet´es Riemannienes compactes. Nous montrons que dans une cer-taine condition mod´er´ee d’expansion de volume nous pouvons d´eduireque pour Lebesgue presque chaque point nous avons contractionuniforme de volume en arri`ere de chaque pr´e-orbite. Statement of results
Let M be a compact Riemannian manifold and let Leb be a volumeform on M that we call Lebesgue measure. We take f : M → M anysmooth map. Let 0 < a ≤ a ≤ a ≤ . . . be a sequence converging toinfinity. We define h ( x ) = min { n > | det Df n ( x ) | ≥ a n } , (1)if this minimum exists, and h ( x ) = ∞ , otherwise. For n ≥
1, we takeΓ n = { x ∈ M : h ( x ) ≥ n } . (2) Theorem 1.1.
Assume that h ∈ L p (Leb) , for some p > , and take γ < ( p − / ( p − . Choose any sequence < b ≤ b ≤ b ≤ . . . suchthat b k b n ≥ b k + n for every k, n ∈ N , and assume that there is n ∈ N such that b n ≤ min { a n , Leb(Γ n ) − γ } for every n ≥ n . Then, for Leb almost every x ∈ M , there exists C x > such that | det Df n ( y ) | > C x b n for every y ∈ f − n ( x ) . Work carried out at the Federal University of Bahia. Partially supported byFCT through CMUP and UFBA.
We say that f : M → M is eventually volume expanding if thereexists λ > x ∈ M sup n ≥ n log | det Df n ( x ) | > λ. (3)Let h and Γ n be defined as in (1) and (2), associated to the sequence a n = e λn . Corollary 1.2. If f is eventually volume expanding, then for Lebesguealmost every x ∈ M there are C x > and σ n → ∞ such that | det Df n ( y ) | >C x σ n for every y ∈ f − n ( x ) . Moreover, given α > there is β > suchthat (1) if Leb(Γ n ) ≤ O ( e − αn ) , then we may take σ n ≥ e βn ; (2) if Leb(Γ n ) ≤ O ( e − αn τ ) for some τ > , then we may take σ n ≥ e βn τ ; (3) if Leb(Γ n ) ≤ O ( n − α ) and α > , then we may take σ n ≥ n β . Specific rates will be obtained in Section 4 for some eventually vol-ume expanding endomorphisms. In particular, non-uniformly expand-ing maps such as quadratic maps and Viana maps will be considered.For the proof of our results we give abstract versions of the tech-niques developed by Armando Castro in his PhD. thesis [3] and articles([4], [5]). More precisely, we adapt his chain concatenation ideas andRedundance Elimination Algorithm to noninvertible contexts. This ismain target in the next section.2.
Concatenated collections
Let ( U n ) n be a collection of measurable subsets of M whose unioncovers a full Lebesgue measure subset of M . We say that ( U n ) n is a concatenated collection if: x ∈ U n and f n ( x ) ∈ U m ⇒ x ∈ U n + m . Given x ∈ S n ≥ U n , we define u ( x ) as the minimum n ∈ N for which x ∈ U n . Note that by definition we have x ∈ U u ( x ) . We define the chaingenerated by x ∈ S n ≥ U n as C ( x ) = { x, f ( x ) , . . . , f u ( x ) − ( x ) } . Just as Corollary 2.9 in [5], the next Lemma is a consequence ofBorel-Cantelli Lemma. It says that, if we see the collection of chainsgiven by a concatenated collection ( U n ) n as a tower, and this towerhas finite measure, than a.e. point in M is contained in just a finitenumber of chains. ACKWARD VOLUME CONTRACTION FOR ENDOMORPHISMS WITH EVENTUAL VOLUME EXPANSION3
Lemma 2.1.
Let ( U n ) n be a concatenated collection. If X n ≥ n − X j =0 Leb( f j ( u − ( n ))) < ∞ , then we have sup (cid:8) u ( y ) : y ∈ S n ≥ U n and x ∈ C ( y ) (cid:9) < ∞ for Lebesguealmost every x ∈ M . Assume that for a given x ∈ M there exists an infinite numberof chains C j = { y j , f ( y j ) , . . . , f s j − ( y j ) } , j ≥
1, containing x with s j → ∞ . For each j ≥ ≤ r j < s j be such that x = f r j ( y j ).First we verify that lim r j = ∞ . If not, then replacing by a subse-quence, we may assume that there is N > r j < N forevery j ≥
1. This implies that y j ∈ S Ni =1 f − i ( x ) for every j ≥
1. Since S Ni =1 f − i ( x )) < ∞ and the number of chains is infinite, we have acontradiction. Since r j → ∞ and x = f r j ( y j ) ∈ f r j ( u − ( s j )), we have x ∈ S n ≥ k S n − j =0 f j ( u − ( n )) for every k ≥
1. Since we are assuming P n ≥ P n − j =0 Leb( f j ( u − ( n ))) < ∞ , we have Leb (cid:0) S n ≥ k S n − j =0 f j ( u − ( n )) (cid:1) → , when k → ∞ . This completes the proof of Lemma 2.1.If a point x is contained just in a finite number of chains, as in theLemma above, it is obvious that the greatest length of such chains isbounded by N = N ( x ). Such points have a interesting property. Lemma 2.2.
Let ( U n ) n be a concatenated collection.If sup { u ( y ) : y ∈ ∪ n ≥ U n and x ∈ C ( y ) } ≤ N , then f − n ( x ) ⊂ U n ∪· · · ∪ U n + N for all n ≥ . The idea of the proof is inspired in Prop. 2.15-2.19 in [4] and Prop.2.12-2.14 in [5]. Due to the concatenation property, one can glue chainsfrom z = f − n ( x ) up to some moment less then n , when for the firsttime we glue a chain containing x . Such chain can not have lengthgreater then N , and therefore z ∈ U n ∪ . . . U n + N . Let us write downthese arguments in detail.Assume that sup { u ( y ) : y ∈ ∪ n ≥ U n and x ∈ C ( y ) } ≤ N and take z ∈ f − n ( x ). Let z j = f j ( z ) for each j ≥
0. We distinguish the cases x ∈ C ( z ) and x / ∈ C ( z ). If x ∈ C ( z ), then n ≤ u ( z ) ≤ n + N . Hence z ∈ U u ( z ) ⊂ U n ∪ · · · ∪ U n + N . If x / ∈ C ( z ), then letting u = u ( z )we must have u < n . Let u = u ( z u ). If u + u < n we take u = u ( z u + u ). We proceed in this way until we find the first s ≤ n such that n ≤ u + · · · + u s . Note that u s = u ( z u + ··· + u s − ), and by thechoice of s we must have x ∈ C ( z u + ··· + u s − ). Our assumption impliesthat u ( z u + ··· + u s − ) ≤ N , and so u + · · · + u s ≤ n + N . By construction J. F. ALVES, ARMANDO CASTRO, AND VILTON PINHEIRO we have z ∈ U u , f u ( z ) = z u ∈ U u , f u + u ( z ) = z u + u ∈ U u , . . . ,f u + ··· u s − ( z ) = z u + ··· u s − ∈ U u s By the definition of a concatenated collection we conclude that z ∈ U u + u + ··· + u s . 3. Proofs of main results
Let us now prove Theorem 1.2. Suppose that h ∈ L p (Leb), for some p >
3. This implies that P n ≥ n p Leb( h − ( n )) < ∞ , and so there existssome constant K > h − ( n )) ≤ Kn − p for every n ≥ < γ < ( p − / ( p −
1) we have for some K ′ > ∞ X n =1 n ∞ X k = n Leb( h − ( k )) ! − γ ≤ ∞ X n =1 n ( K ′ /n p − ) − γ < ∞ . Defining U n = { x ∈ M : | det Df n ( x ) | ≥ b n } , then we have that { U , U , . . . } is a concatenated collection with respect to the Lebesguemeasure. Moreover, setting U ∗ n = U n \ ( U ∪ ... ∪ U n − ) one observesthat U ∗ n ⊂ ∪ m ≥ n h − ( m ), for otherwise there would be x ∈ U ∗ n ∩ h − ( m )with m < n , and so a m ≥ b m > | det Df m ( x ) | ≥ a m , which is notpossible. As | det Df j ( x ) | < b j for every x ∈ U ∗ n and j < n , we getLeb( f j ( U ∗ n )) ≤ b j Leb( U ∗ n ) for each j < n . Hence ∞ X n = n +1 n − X j =0 Leb( f j ( U ∗ n )) ≤ ∞ X n = n +1 n − X j =0 b j Leb( U ∗ n ) ≤ ∞ X n = n +1 n − X j =0 b j Leb( U ∗ n ) + ∞ X n = n +1 n − X j = n b j Leb( U ∗ n ) ≤ n − X j =0 b j + ∞ X n = n +1 n − X j = n b j Leb( U ∗ n ) ACKWARD VOLUME CONTRACTION FOR ENDOMORPHISMS WITH EVENTUAL VOLUME EXPANSION5
Now, we just have to check that the last term in the sum above is finite.Indeed, ∞ X n = n +1 n − X j = n b j Leb( U ∗ n ) ≤ ∞ X n = n +1 n − X j = n b j ∞ X k = n Leb( h − ( k )) ≤ ∞ X n = n +1 nb n ∞ X k = n Leb( h − ( k )) ≤ ∞ X n = n +1 n ∞ X k = n Leb( h − ( k ) ! − γ ∞ X k = n Leb( h − ( k ))= ∞ X n = n +1 n ∞ X k = n Leb( h − ( k )) ! − γ < ∞ . Applying Lemmas 2.1 and 2.2, we get for each generic point x ∈ M a positive integer number N x such that if y ∈ f − n ( x ) then y ∈ U n + s for some 0 ≤ s ≤ N x . Therefore, | det Df n + s ( y ) | > b n + s ≥ b n . Tak-ing C x = K − N x , where K = sup {| det Df ( z ) | : z ∈ M } , we obtainTheorem 1.1: | det Df n ( y ) | = | det Df n + s ( y ) || det Df s ( x ) | > C x b n . Now we explain how we use Theorem 1.1 to prove Corollary 1.2.Recall that in Corollary 1.2 we have a n = e λn for each n ∈ N . Assumefirst that Leb(Γ n ) ≤ O ( e − c ′ n ) for some c ′ >
0. Then it is possible tochoose c > b n = e cn , for n ≥ n . The other two cases areobtained under similar considerations.4. Examples: non-uniformly expanding maps
An important class of dynamical systems where we can immediatelyapply our results is the class of non-uniformly expanding dynamicalmaps introduced in [2]. As particular examples of this kind of sys-tems we present below one-dimensional quadratic maps and the higherdimensional Viana maps.Quadratic maps. Let f a : [ − , → [ − ,
1] be given by f a ( x ) = 1 − ax ,for 0 < a ≤
2. Results in [6, 9] give that for a positive Lebesguemeasure set of parameters f a in non-uniformly expanding. Ongoingwork [8] gives that for a positive Lebesgue measure set of parametersthere are C, c > n ) ≤ Ce − cn for every n ≥ J. F. ALVES, ARMANDO CASTRO, AND VILTON PINHEIRO
Thus, it follows from Corollary 1.2 that there exists β > such forLebesgue almost every x ∈ I there is C x > such that | ( f n ) ′ ( y ) | >C x e βn for every y ∈ f − n ( x ) . Viana maps. Let a ∈ (1 ,
2) be such that the critical point x = 0 ispre-periodic for the quadratic map Q ( x ) = a − x . Let S = R / Z and b : S → R given by b ( s ) = sin(2 πs ). For fixed small α >
0, considerthe map ˆ f from S × R into itself given by ˆ f ( s, x ) = (cid:0) ˆ g ( s ) , ˆ q ( s, x ) (cid:1) ,where ˆ q ( s, x ) = a ( s ) − x with a ( s ) = a + αb ( s ), and ˆ g is the uniformlyexpanding map of S defined by ˆ g ( s ) = ds (mod Z ) for some integer d ≥
2. For α > I ⊂ ( − ,
2) forwhich ˆ f ( S × I ) is contained in the interior of S × I . Thus, any map f sufficiently close to ˆ f in the C topology has S × I as a forwardinvariant region. Moreover, there are C, c > n ) ≤ Ce − c √ n for every n ≥
1; see [1, 7, 10].Thus, it follows from Corollary 1.2 that there exists β > suchfor Lebesgue almost every X ∈ S × I there is C X > such that | det Df n ( Y ) | > C X e β √ n for every Y ∈ f − n ( X ) . References [1] J. F. Alves, V. Ara´ujo,
Random perturbations of nonuniformly expanding maps ,Ast´erisque (2003), 25-62.[2] J. F. Alves, C. Bonatti, M. Viana,
SRB measures for partially hyperbolic sys-tems whose central direction is mostly expanding , Invent. Math. (2000),351-398.[3] A. Castro,
Backward inducing and exponential decay of correlations for par-tially hyperbolic attractors with mostly contracting central direction , PhD. the-sis, IMPA, 1998.[4] A. Castro,
Backward inducing and exponential decay of correlations for par-tially hyperbolic attractors , Israel J. Math. (2002), 29-75.[5] A. Castro,
Fast mixing for attractors with mostly contracting central direction ,Ergodic Th. Dynam. & Syst., (2004), 17-44.[6] M. Benedicks, L. Carleson, On iterations of − ax on ( − , (1985), 1-25.[7] J. Buzzi, O. Sester, M. Tsujii, Weakly expanding skew-products of quadraticmaps , Ergodic Theory Dynam. Systems (2003), no. 5, 1401–1414[8] J. Freitas, in preparation.[9] M. Jakobson, Absolutely continuous invariant measures for one-parameterfamilies of one-dimensional maps , Comm. Math. Phys. (1981), 39-88.[10] M. Viana, Multidimensional non-hyperbolic attractors , Publ. Math. IHES (1997), 63-96. ACKWARD VOLUME CONTRACTION FOR ENDOMORPHISMS WITH EVENTUAL VOLUME EXPANSION7
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