On the number of invariant measures for random expanding maps in higher dimensions
OON THE NUMBER OF INVARIANT MEASURES FOR RANDOMEXPANDING MAPS IN HIGHER DIMENSIONS
FAWWAZ BATAYNEH, CECILIA GONZ ´ALEZ-TOKMAN
Abstract.
In [26], Jab(cid:32)lo´nski proved that a piecewise expanding C multi-dimensional Jab(cid:32)lo´nski map admits an absolutely continuous invariant proba-bility measure (ACIP). In [6], Boyarsky and Lou extended this result to thecase of i.i.d. compositions of the above maps, with an on average expandingcondition. We generalize these results to the (quenched) setting of randomJab(cid:32)lo´nski maps, where the randomness is governed by an ergodic, invertibleand measure preserving transformation. We prove that the skew product as-sociated to this random dynamical system admits a finite number of ergodicACIPs. Furthermore, we provide two different upper bounds on the numberof mutually singular ergodic ACIP’s, motivated by the works of Buzzi [9] inone dimension and Gora, Boyarsky and Proppe [23] in higher dimensions. Introduction
A fundamental problem in ergodic theory is to describe the asymptotic statisticalbehavior of orbits defined by a dynamical system. In this approach, one attempts tounderstand and quantify the different invariant measures of the system, in particu-lar those which have physical relevance. This problem has been studied intensivelyfor several classes of piecewise smooth systems, starting with one dimensional de-terministic systems in the key paper [30] by Lasota and Yorke in 1973. In 2000,Buzzi [9] identified bounds on the number of physical measures for random compo-sitions of Lasota–Yorke maps. In higher-dimensional frameworks, including randomversions of [20, 38, 13, 40], understanding and, specifically, bounding the numberof physical measures is still an unsolved problem. This challenge is related to openquestions in multiplicative ergodic theory, regarding multiplicity of Lyapunov ex-ponents. The focus of this work is on investigating and bounding the number ofphysical measures for a class of higher dimensional expanding-on-average randomdynamical systems, where the randomness is driven by a rather general type ofergodic process, including but not limited to the i.i.d. case.In this paper we study a class of discrete time dynamical systems in which,at each iteration of the process, one of a collection of maps is selected and ap-plied. Ulam and von Neumann [41], Morita [32], Pelikan [35] and Buzzi [9] wereamong those who started working on such systems, which have been named timedependent, random or non autonomous dynamical systems. In general, there isno measure which is invariant under all these maps simultaneously. Therefore, weinstead consider random invariant measures which are absolutely continuous withrespect to Lebesgue (ACIPs), and their associated marginals, which give rise tophysical measures. a r X i v : . [ m a t h . D S ] F e b FAWWAZ BATAYNEH, CECILIA GONZ´ALEZ-TOKMAN
This work focuses on dynamical systems modeled by random compositions of so-called Jab(cid:32)lo´nski maps. These maps have been studied by several researchers afterthe first paper [26] by Jab(cid:32)lo´nski in 1983. In [21], G´ora and Boyarsky used Jab(cid:32)lo´nskitransformations as a model for interacting cellular systems. In [7], Boyarsky andLou presented a method for approximating the ACIPs in [26] by means of approx-imating the transfer operator by finite dimensional operators, which is a version ofUlam’s conjecture in a multidimensional setting. In [8], Boyarsky, G´ora and Louconsidered a larger class of C transformations defined on a rectangular partitionof the n dimensional cube. The authors approximated any such map by a sequenceof Jab(cid:32)lo´nski transformations and proved that the sequence of invariant densitiesassociated with these Jab(cid:32)lo´nski maps converges weakly in L to the invariant den-sity associated with that map. In [5], Bose replaced the weak approximation of theinvariant density in [7] by strong approximation using a compactness argument.The special case of random i.i.d. Jab(cid:32)lo´nski maps was studied in [6, 24].Random Jab(cid:32)lo´nski maps, introduced in Definition 2.16, are defined by a collec-tion of piecewise smooth maps ( f ω ) ω ∈ Ω defined on the state or phase space I n ,where I = [0 ,
1] and n ∈ N is the dimension, equipped with the Borel sigma algebraof measurable sets and the n dimensional Lebesgue measure m . The family of mapsis assumed to satisfy an expanding-on-average condition.Our approach relies on so-called transfer operators, acting on the space of higherdimensional functions of bounded variation. Given a nonsingular map f , its transferoperator L f encodes information about the application of f and describes howdensities, i.e. nonnegative integrable functions with integral one, evolve in time. Ifa collection of points in phase space is distributed according to a probability densityfunction h , and pushed forward by f , then the resulting collection of points will bedistributed according to a new density denoted by L f ( h ) or L f h .The first appearance of one dimensional functions of bounded variation is due toC. Jordan in 1881 in connection with Dirichlet’s test for the convergence of Fourierseries. In 1905, G. Vitali gave the first definition of bounded variation functionin two dimensions. Later on, L. Tonelli observed that Vitali’s generalization wasnot the right generalization of the one dimensional variation because it containssecond order elements related to the curvature of the graph rather than its area.In 1936, in a closer analogy to the one dimensional variation, Tonelli introducedhis generalization which measures the length of the projection of the graph ontothe vertical axis counting multiplicities at least for continuous functions. Tonelli’sdefinition is more convenient for continuous functions since the definition dependson the choice of the coordinate axes if the function is not continuous. To solvethis issue, in the same year, L. Cesari modified Tonelli’s definition by requiring theintegrals in Tonelli’s definition to be finite for functions equal almost everywhere.This definition does not depend on the coordinates even for discontinuous functions.Functions of bounded variation in this sense were called bounded variation functionsin the sense of Tonelli–Cesari. However, the point of view which is popular thesedays and adapted in most of the literature [20] as the most suitable generalizationof the one dimensional theory is due to De Giorgi and Fichera. Krickeberg andFleming independently showed that a bounded variation function in the sense ofTonelli–Cesari has a vector measure as its distributional gradient, thus obtaining theequivalence with De Giorgi’s definition. For more information on historical detailsabout higher dimensional functions of bounded variation, we refer the reader to [2]. UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 3
In the deterministic case, an early use of transfer operators in the one dimen-sional bounded variation setting is due to Lasota and Yorke, who in [30] proved theexistence of ACIPs for piecewise C transformations f on I , with the assumptionof a uniform expanding condition inf | f (cid:48) | >
1. The authors exploited the fact thatthe transfer operator corresponding to the point transformation under considera-tion has the property of keeping the variation of the functions h, L f h, . . . , L nf h, . . . under control. This result was later on referred to as Lasota-Yorke inequality. In[26], Jab(cid:32)lo´nski generalized the one dimensional work of Lasota and Yorke [30] topiecewise continuous maps on the multidimensional cube I n with similar type ofuniform expanding condition on the rectangles of a rectangular partition. The proofof this result was similar to the proof of Theorem 1 in [30], but it uses the notionof variation of functions of several variables due to Tonelli–Cesari, which we alsouse in this paper.In higher dimensions, the situation is more challenging than in one dimension.For example, in the general case, crucial difficulties come from the much richergeometry which can arise from the phase space partitions, and from the growingcomplexity of the partitions arising from the iterated dynamics. To overcome theseissues, one may impose conditions on the geometry of the partitions and, roughlyspeaking, to ensure the amount of expansion is enough to overcome the dynamicalcomplexity. See, for example, the conditions given in Theorem 4 in [23], Theorem3 . .
8) in [40].A number of authors have studied the existence of ACIPs for piecewise expandingmaps in higher dimensions. In [20], G´ora and Boyarsky proved the existence ofACIPs with densities of bounded variation for piecewise C transformations in R n for domains with piecewise C boundaries with the assumption that where the C segments of the boundaries meet, the angle subtended by the tangents to thesesegments at the point of contact is bounded away from zero. The case when theboundaries for which the angle mentioned may become zero (i.e. the boundaries ofpartitions may contain ’cusps’) is studied in [28, 1] by Keller and Adl-Zarabi. In[38], Saussol developed a Lasota-Yorke inequality for a class of piecewise expandingmaps defined on a compact subset of R n and used it to prove the existence ofa finite number of ACIPs with densities in the Quasi-H¨older space. The authoralso provided an upper bound on the number of these ACIPs. In [13], Cowiesonextended the work of G´ora and Boyarsky by establishing a simpler condition whichguarantees the existence of an ACIPs. The condition is that, the expansion mustbe greater than the cut index defined in [13, Section 2 . . . C α uniformly expanding map admits a finite number of ACIPs. Theauthor also compares his results with the work of Saussol [38] and Cowieson [13].Although no upper bounds on the number of these ACIPs are explicitly given in[40], the author mentions that the results of [38] could perhaps be adapted to hissetting.In the random one-dimensional case, in [9], Buzzi considers random expanding-on-average Lasota–Yorke maps that have neither too many branches nor too largedistortion, and proves that the associated skew product transformation possesses afinite number of mutually singular ergodic ACIPs, each giving a family of randominvariant measures with densities of bounded variation. In [3], Araujo and Solano FAWWAZ BATAYNEH, CECILIA GONZ´ALEZ-TOKMAN proved existence of ACIPs for random one dimensional dynamical systems withasymptotic expansion. Their work can be seen as a generalization of the work ofKeller [27] which proves that for maps on the interval with finite number of criticalpoints and non-positive Schwarzian derivative, existence of absolutely continuousinvariant probability is earned by positive Lyapunov exponents. They also provesimilar results for higher dimensional random systems under the assumption of slowrecurrence to the set of discontinuities and/or criticalities, which are of a certainnon-degenerate type, shown in [3, Equation (1 . C and monotonic Jab(cid:32)lo´nski maps f ,. . . , f l where l is a finite positiveinteger and f k,i ( x , . . . , x n ) = ϕ k,i,j ( x i ),for ( x , . . . , x n ) ∈ D k,j where D k,j is the j th rectangle in the partition of f k . Therandom map f is defined by choosing f i with probability p i , where the p i ’s are pos-itive and add up to one. The authors assumed an expanding-on-average conditionthat is, there exists a positive constant 0 < γ < l (cid:88) i =1 sup j p i | ϕ (cid:48) k,i,j ( x i ) | ≤ γ ,for all i = 1 , . . . , l and ( x , . . . , x n ) ∈ cl ( D k,j ) (the closure of D k,j ) and proved that f admits an ACIP with respect to the Lebesgue measure. This measure has density h which is a fixed point of the averaged transfer operator L f of f , given by(1.2) L f = l (cid:88) i =1 p i L fi ,where L fi is the transfer operator of the corresponding map f i .Our conditions on the maps are much more general than the ones in [6, 35].In both articles, the maps driving the dynamics or defining the random orbits aregiven by an i.i.d. process. Moreover, the maps must be chosen from a finite set.However, in our situation the way of selecting the maps comes from the base map σ defined on a probability measure space (Ω , F , P ). A difficulty of this setting isthat there is no known formula for an averaged transfer operator that correspondsto the one described in (1.2) in [35, 6]. The way we overcome this obstacle, as ithas been done in [9, 16, 18], is by developing a random Lasota-Yorke inequality,Equation (3.1), which we use to prove several results in this paper.Quasi-compactness is one of the concepts which has played a key role in themodern approach to investigate transport properties of random dynamical systemsthrough transfer operators [16, 18]. In the case of autonomous systems, this prop-erty was introduced in the work of Ionescu Tulcea and Marinescu [25]. A boundedlinear operator defined on a Banach space is called quasi-compact if its spectralradius is strictly larger than its essential spectral radius. The formulation of a non-autonomous analogue of the quasi-compactness property goes back to Thieullen[39]. It is now widely known that quasi-compactness can be usually derived fromLasota-Yorke type inequalities, and this is the route we pursue.The quasi-compactness theorem of Ionescu Tulcea and Marinescu [25] is used toprovide spectral decompositions and properties in the case of deterministic dynam-ical systems similar to the ones given in Section 3 in [20]. However, in the random UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 5 case one instead uses Oseledets type multiplicative ergodic theorems. In 1968, Os-eledets multiplicative ergodic theorem was first introduced by Oseledets [34] in thecontext of random multiplication of matrices. In its basic form, it describes theasymptotic behavior of a product of matrices sampled from a dynamical system.After that, different proofs were provided and different generalizations have beendeveloped and applied to transfer operator cocycles, see [16, 18]. In this paper weadapt Theorem 17 from [16] to provide an Oseledets splitting for random Jab(cid:32)lo´nskimaps.When applicable, multiplicative ergodic theorems provide existence and finite-ness of random ACIPs. However, explicit bounds do not come directly from thismachinery. Despite some progress by Buzzi [9] and Araujo–Solano [3], the questionof how to find bounds on the number of ACIPs in random dynamical systems islargely open. In [23, Theorem 2], G´ora, Boyarsky and Proppe proved that, in theirsetting, the support of absolutely continuous invariant measures is open Lebesguealmost everywhere. They used this key fact to obtain their result [23, Theorem 3]that the number of ergodic ACIP’s for deterministic dynamical systems modeledby Jab(cid:32)lo´nski transformations is at most equal to the number of crossing points. Wecombine elements of their arguments with ideas from the one dimensional work ofBuzzi on random Lasota-Yorke maps (see Section 3 in [9]) to develop a bound onthe number of mutually singular ergodic ACIP’s for a class of admissible randomJab(cid:32)lo´nski maps. Another bound is also developed, and these bounds are comparedin Section 5.3.This paper is structured as follows: in Section 2, we state the definition ofadmissible random Jab(cid:32)lo´nski maps, which involves the formulation of an expanding-on-average condition motivated from the expanding condition given in (1.1). InSection 3, in Theorem 3.1, we prove that this random map is quasi-compact and themaximal Lyapunov exponent is indeed zero. In Section 4, in Theorem 4.3, we provethat the random invariant densities of admissible random Jab(cid:32)lo´nski maps are ofbounded variation and equivariant. In Corollary 4.6, we prove these densities belongto the leading Oseledets subspace and the number of ergodic ACIPs with respect tothe associated skew product is finite. Theorem 4.8 is a probabilistic conclusion thatshows that the marginals of the measures in Theorem 4.3 are physical, which meansthat for Lebesgue almost initial condition, the asymptotic long term behaviour ofthe corresponding random orbit will be described by one of these physical measures.In Section 5, we establish upper bounds on the number of mutually singular ergodicACIPs for a class of admissible random Jab(cid:32)lo´nski maps, and present an example inSection 5.3. 2.
Terminology and background
Preliminaries.
In this subsection we state the basic definitions and tools thatwill be used throughout the paper.
Definition 2.1.
Let (Ω , F , P ) be a probability space. A measurable transformation σ : Ω (cid:9) is said to be nonsingular if P ( σ − ( A )) = 0,for all A ∈ F with P ( A ) = 0. Definition 2.2.
Let (Ω , F , P ) be a probability space. A transformation σ : Ω (cid:9) issaid to be a measure-preserving transformation or, equivalently, P is said to be a FAWWAZ BATAYNEH, CECILIA GONZ´ALEZ-TOKMAN σ − invariant measure , if P ( σ − ( A )) = P ( A ),for all A ∈ F . Definition 2.3.
Let (Ω , F , P ) be a probability space. A nonsingular transformation σ : Ω (cid:9) is said to be ergodic if for all A ∈ F , with σ − ( A ) = A, we have P ( A ) = 0or P (Ω \ A ) = 0 . Definition 2.4.
Let ( X, B , µ ) be a measure space and f : X (cid:9) a nonsingulartransformation. The unique operator L f : L ( X ) (cid:9) satisfying the dual relation (cid:90) A L f h ( x ) µ ( dx ) = (cid:90) f − A h ( x ) µ ( dx ),for every A ∈ B and h ∈ L ( X ) is called the transfer or Perron-Frobenius operator corresponding to f .For x = ( x , . . . , x n ), if A = (cid:81) ni =1 [0 , x i ] in the above definition, then differenti-ating both sides, we obtain L f h ( x ) = ∂ n ∂x . . . ∂ x n (cid:90) f − ( (cid:81) ni =1 [0 ,x i ]) h ( y ) µ ( dy ),where ∂x i is the derivative with respect to x i , i = 1 , . . . , n . This formula can beseen in Section 2 in [8]. It is well known that the transfer operator is linear, positive,contractive and L f h = h if and only if the measure ν where dν = hdµ is invariantunder f , see [4].Given sets A i , i = 1 , . . . , n , denote the Cartesian product of the sets A i by n (cid:89) i =1 A i = { ( a , . . . , a n ) : a i ∈ A i , i = 1 , . . . , n } . For i = 1 , . . . , n, let P i be theprojection of R n onto R n − given by(2.1) P i ( x , . . . , x n ) = ( x , . . . , x i − , x i +1 , . . . , x n ).We next describe the definition of the total variation of an integrable functionof several variables, due to Tonelli and Cesari, which was first used in the contextof transfer operators in [6, 26]. Definition 2.5.
Consider the n dimensional rectangle A = n (cid:89) i =1 [ a i , b i ] where a i , b i ∈ R and a i < b i and a function g : A → R . For i = 1 , . . . , n , consider a real valuedfunction A V i g of ( n −
1) variables ( x , . . . , x i − , x i +1 , . . . , x n ) given bysup a i = x i < x i < · · · < x ri = b i r ∈ N r (cid:88) k =1 (cid:16) | g ( x , . . . , x ki , . . . , x n ) − g ( x , . . . , x k − i , . . . , x n ) | (cid:17) .For a measurable function f ∈ L ( A ) and i = 1 , . . . , n , define A V i f = inf g = f a.e. A V i g is measurable (cid:90) P i ( A ) A V i gdm , UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 7
Definition 2.6. where P i is defined in (2.1)and let the total variation of f be A V f = max i =1 ,...,nA V i f .If the total variation A V f of f on A is a finite, then f is said to be of bounded variationon A , and the set of all such maps is denoted by BV ( A ). For f ∈ BV ( A ), thenorm of f is defined by (cid:107) f (cid:107) BV = (cid:107) f (cid:107) + A V f .The space BV ( A ) is a Banach space by Remark 1 .
12 in [17] and compactlyembedded in L ( A ) by Corollary 3 .
49 in [2].2.2.
Random dynamical systems.Definition 2.7.
A random dynamical system is a tuple R = (Ω , F , P , σ, X , L ),where the base σ is an invertible measure-preserving transformation of the proba-bility space (Ω , F , P ), ( X , (cid:107) · (cid:107) ) is a Banach space and L : Ω → L ( X , X ) is a familyof bounded linear maps of X , called the generator .For convenience, we let L ω := L ( ω ). A random dynamical system defines acocycle, given by ( k, ω ) (cid:55)→ L ( k ) ω := L σ k − ω ◦ · · · ◦ L σω ◦ L ω .Different regularity conditions may be imposed on the generator L . The followingconcept of P -continuity, which was first introduced by Thieullen in [39], will be usedin the sequel. Definition 2.8.
Let Ω be a topological space, equipped with a Borel probabilitymeasure P and let Y be a topological space. A mapping L : Ω → Y is said to be P -continuous if Ω can be expressed as a countable union of Borel sets such that therestriction of L to each of them is continuous.In the rest of this work, we consider random dynamical systems whose generators L : Ω → L ( X , X ), given by ω (cid:55)→ L ω , are P -continuous and that Ω is a Polish space.That is, a complete separable metric space. Definition 2.9.
The index of compactness (or
Kuratowski measure of noncompact-ness ) of a bounded linear map A : X (cid:9) is (cid:107) A (cid:107) ic ( X ) = inf { r > A ( B X ) can be covered by finitely many balls of radius r } ,where B X denotes the unit ball in X . Definition 2.10.
Let R = (Ω , F , P , σ, X , L ) be a random dynamical system. As-sume that (cid:82) Ω log + (cid:107)L ω (cid:107) d P ( ω ) < ∞ . For each ω ∈ Ω , the maximal Lyapunov expo-nent λ ( ω ) for ω is defined as λ ( ω ) = lim k →∞ k log (cid:107)L ( k ) ω (cid:107) ,whenever the limit exists. The index of compactness K ( ω ) for ω is defined as K ( ω ) = lim k →∞ k log (cid:107)L ( k ) ω (cid:107) ic ( X ) ,whenever the limit exists.The following is established in [18]. FAWWAZ BATAYNEH, CECILIA GONZ´ALEZ-TOKMAN
Remark . If a random dynamical system R has an ergodic base σ , then λ and K in the previous definition are P − almost everywhere constant. We call theseconstants λ ∗ ( R ) and K ∗ ( R ), or simply λ ∗ and K ∗ , if R is clear from the context. Itfollows from the definition that K ∗ ≤ λ ∗ . The assumption (cid:82) Ω log + (cid:107)L ω (cid:107) d P ( ω ) < ∞ implies that λ ∗ < ∞ . Definition 2.12.
A random dynamical system R with an ergodic base σ is called quasi-compact if K ∗ < λ ∗ .The next proposition relates the maximal Lyapunov exponent λ ∗ ( R ) and the in-dex of compactness K ∗ ( R ) of a random dynamical system R with the correspondingquantities for R ( n ) = (Ω , F , P , σ n , X , L ( n ) ), n ∈ N . Proposition 2.13.
Consider a random dynamical system R = (Ω , F , P , σ, X , L ) with an ergodic base σ . Then, for each n ∈ N , R ( n ) = (Ω , F , P , σ n , X , L ( n ) ) is arandom dynamical system, with a possibly non-ergodic base σ n . For each ω ∈ Ω ,let λ n ( ω ) = lim k →∞ k log (cid:107)L ( nk ) ω (cid:107) , K n ( ω ) = lim k →∞ k log (cid:107)L ( nk ) ω (cid:107) ic ( X ) .Then λ n ( ω ) = nλ ∗ ( R ) , K n ( ω ) = n K ∗ ( R ) , P − almost everywhere.Proof. For each n ∈ N , {L ( nk ) ω } ∞ k =1 is a subsequence of {L ( k ) ω } ∞ k =1 . Thus, the prooffollows from Remark 2.11. (cid:3) While the transformation σ n may be non-ergodic when σ is ergodic, the followingresult ensures that σ n is ergodic on some subset Z ⊂ Ω. This result will be used inthe proof of Proposition 4.2.
Lemma 2.14 (Gonz´alez-Tokman & Quas [19, Lemma 35]) . Let σ be an ergodic P -preserving transformation of (Ω , F , P ) and let n ∈ N . Then there exists k , a factorof n , and a σ n -invariant subset Z of Ω of measure /k such that Ω = k − (cid:91) s =0 σ − (cid:96) Z and σ n | Z is ergodic. When σ is invertible, this argument also applies to n < . Admissible random Jab(cid:32)lo´nski maps and quasi-compactness.Definition 2.15.
A partition B = { B , . . . , B q } of I n is called rectangular if foreach j = 1 , . . . , q , B j = n (cid:89) i =1 B ij ,where B ij = [ a ij , b ij ) if b ij < B ij = [ a ij , b ij ] if b ij = 1.A piecewise map f : I n (cid:9) defined on the rectangular partition given in Definition2.15 is generally written as f ( x , . . . , x n ) = ( ϕ ,j ( x , . . . , x n ) , . . . , ϕ n,j ( x , . . . , x n )), UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 9 where ( x , . . . , x n ) ∈ B j , j = 1 , . . . , q . Following [26], we next introduce Jab(cid:32)lo´nskimaps as a special case of such maps. We then define random Jab(cid:32)lo´nski maps andadmissible random Jab(cid:32)lo´nski maps. Definition 2.16 ( Jab(cid:32)lo´nski [26]) . A map f : I n (cid:9) is called a Jab(cid:32)lo´nski map if itis piecewise defined on a rectangular partition B = { B , . . . , B q } of I n and is givenby the formula f ( x , . . . , x n ) = ( ϕ ,j ( x ) , . . . , ϕ n,j ( x n )),where ( x , . . . , x n ) ∈ B j , j = 1 , , . . . , q . The vertices of the rectangles in B whichlie in the interior of I n are called the crossing points of f . The real valued maps ϕ i,j : B ij → [0 ,
1] are called the components of f . We use J to denote for the classof Jab(cid:32)lo´nski maps on I n .While the above family of Jab(cid:32)lo´nski maps may seem restrictive, in [8], Boyarsky,G´ora and Lou proved that for any piecewise C map f defined on a rectangularpartition of I n , f can be approximated by a sequence of piecewise C Jab(cid:32)lo´nskitransformations. In other words, there exists a sequence of Jab(cid:32)lo´nski maps f n that converges pointwise to f. Moreover, the corresponding sequence of invariantdensities of f n (which exists by [26]) converges weakly to an invariant density of f . Generally speaking, Jab(cid:32)lo´nski maps can be seen as the basis maps for a muchlarger class of piecewise defined maps on the n dimensional rectangle. Definition 2.17.
Let (Ω , F , P ) be a probability space and σ : Ω (cid:9) an invertible,ergodic and P − preserving transformation. A random Jab(cid:32)lo´nski map F over σ isa map F : Ω → J , where f ω := F ( ω ) : I n (cid:9) . Hence, for each ω ∈ Ω, there existsa rectangular partition B ω of I n , say, B ω = { B ω , . . . , B ωq ω } ,where q ω is a positive integer. If x = ( x , . . . , x n ) ∈ B ωj , where j ∈ { , . . . , q ω } ,then we have f ω ( x ) = ( ϕ ω, ,j ( x ) , . . . , ϕ ω,n,j ( x n )),where B ωj = n (cid:89) i =1 [ a ω,ji , b ω,ji ) and ϕ ω,i,j is a map from [ a ω,ji , b ω,ji ] into [0 , k ∈ N ,the k fold composition f ( k ) ω is defined as(2.2) f ( k ) ω := f σ k − ω ◦ · · · ◦ f σω ◦ f ω .For simplicity, we sometimes refer to the range of F , that is { f ω } ω ∈ Ω , as therandom Jab(cid:32)lo´nski map. A random Jab(cid:32)lo´nski map gives rise to a random dynamicalsystem, where X = BV ( I n ) and L ω = L f ω . The next proposition proves that forall ω ∈ Ω, L ω is a bounded operator on BV ( I n ). Proposition 2.18.
For all ω ∈ Ω , L ω is a bounded operator on BV ( I n ) . Proof.
Let ω ∈ Ω, (cid:107)L ω h (cid:107) BV = (cid:90) I n |L ω h | dm + I n V L ω h ≤ (cid:90) I n L ω | h | dm + I n V L ω h = (cid:90) I n | h | dm + I n V L ω h = (cid:107) h (cid:107) + I n V L ω h By the Lasota-Yorke inequality provided in the proof of Theorem 1 in [26] , the lastterm is less than or equal (cid:107) h (cid:107) + α (cid:107) h (cid:107) + β A V h ≤ max(1 + α, β ) (cid:107) h (cid:107) BV .for some α, β > (cid:3) We note that(2.3) L ( k ) ω = L f σk − ω ◦···◦ f σω ◦ f ω = L σ k − ω ◦ · · · ◦ L σω ◦ L ω .In what follows, we will use the notation → r = ( r , . . . , r n ) ∈ N n and denote the setof vector indices by Z → r := { → s = ( s , s , . . . , s n ) : 1 ≤ s i ≤ r i } . Remark . If the random Jab(cid:32)lo´nski map F has a finite range, then for each k ∈ N , there exists a common partition B = B ( k ) of I n into maximal rectanglessuch that the components of the maps { f ( k ) ω } ω ∈ Ω are C and monotonic on theirinterval domains. Remark . The generator L : Ω → L ( X , X ) of the random dynamical systemgenerated by a random Jab(cid:32)lo´nski map F is P -continuous if its range is at mostcountably infinite (consisting of, say, f , f , . . . ) and the preimage of each f j is ameasurable set. Our results will be valid when there exists a common partition B of I n into rectangles such that the components of the maps { f ( N ) ω } ω ∈ Ω are C andmonotonic on their interval domains, where N ∈ N satisfies the condition given in(3.2). In this case, for each i = 1 , . . . , n , there exists a partition0 = a i, < a i, < · · · < a i,r i = 1,for some r i ∈ N . Let B s i = [ a i,s i − , a i,s i ) when s i = 1 , , . . . , r i − B r i =[ a i,r i − , a i,r i ]. For each vector index → s ∈ Z → r , we denote the n dimensional rectangleby B → s = n (cid:89) i =1 B s i . The common rectangular partition is given by B = { B → s : → s ∈ Z → r } .For each ω ∈ Ω and → s ∈ Z → r , we write the map f ω with respect to B as f ω ( x ) = ( ϕ ω, , → s ( x ) , . . . , ϕ ω,n, → s ( x n )), x = ( x , . . . , x n ) ∈ B → s , UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 11 and, for each k ∈ N , the map f ( k ) ω as f ( k ) ω ( x ) = ( ϕ ω, ,k, → s ( x ) , . . . , ϕ ω,n,k, → s ( x n )), x = ( x , . . . , x n ) ∈ B → s . Remark . One can associate to the random Jab(cid:32)lo´nski map F = { f ω } ω ∈ Ω , theskew product map F on Ω × I n which encodes the dynamics of the whole system(2.4) F ( ω, x ) = ( σω, f ω ( x )).Expanding properties for dynamical systems lead to chaotic behavior of theorbits. However, they usually give rise to good ergodic properties like the existenceof absolutely continuous invariant measures. Next we introduce the admissiblerandom Jab(cid:32)lo´nski maps. This definition involves a formulation of an expanding-on-average condition. Definition 2.22.
Using the notation in Remark 2.20, a random Jab(cid:32)lo´nski map F iscalled admissible if all the components ϕ ω,i, → s are C and monotonic on [ a i,s i − , a i,s i ]and there exists a constant γ > (cid:90) Ω min i =1 ,...,n log( γ i ( ω )) d P ( ω ) > γ ,where(2.6) γ i ( ω ) := inf → s ∈ Z → r x i ∈ [ a i,si − ,a i,si ] ( | ϕ (cid:48) ω,i, → s ( x i ) | ).In addition, we assume the mapping ω (cid:55)→ L ω is P -continuous.3. Random Lasota-Yorke inequality and quasi-compactness
In the next theorem, we establish a suitable Lasota-Yorke inequality on the spaceof bounded variation BV ( I n ) and we use it to prove the quasi-compactness propertyfor admissible random Jab(cid:32)lo´nski maps. Theorem 3.1.
Let F = { f ω } ω ∈ Ω be an admissible random Jab(cid:32)lo´nski map. Then:(i) the random dynamical system generated by F is quasi-compact; and(ii) its maximal Lyapunov exponent λ ∗ is zero.Proof of Theorem 3.1 (i). The first step is to show that there are N ∈ N and posi-tive measurable functions α , α : Ω → R + such that (cid:82) Ω log α ( ω ) d P ( ω ) < I n V L ( N ) ω h ≤ α ( ω ) I n V h + α ( ω ) (cid:107) h (cid:107) ,for all h ∈ BV ( I n ), where L ( N ) ω is defined in (2.3). Let x = ( x , . . . , x n ) ∈ I n and ω ∈ Ω, choose N ∈ N such that(3.2) N γ > log(3),where γ satisfies the condition (2.5). Let → s = → s be the label of the unique rectanglein B for which x ∈ B → s and for k = 1 , , . . . , let → s k ∈ Z → r be such that f ( k ) ω ( x ) = ( ϕ ω, ,k, → s ( x ) , . . . , ϕ ω,n,k, → s ( x n ))= (cid:16) ϕ σ k − ω, , → s k − ◦ · · · ◦ ϕ ω, , → s ( x ) , . . . , ϕ σ k − ω,n, → s k − ◦ · · · ◦ ϕ ω,n, → s ( x n ) (cid:17) ∈ B → s k . Note that for any i = 1 , , . . . , n , (cid:90) Ω inf → s ∈ Z → r x i ∈ [ a i,si − ,a i,si ] log( | ϕ (cid:48) ω,i,N, → s ( x i ) | ) d P ( ω )(3.3)= (cid:90) Ω inf → s ∈ Z → r x i ∈ [ a i,si − ,a i,si ] log (cid:16) | ( ϕ σ N − ω,i, → s N − ◦ · · · ◦ ϕ σω,i, → s ◦ ϕ ω,i, → s ) (cid:48) ( x i ) | (cid:17) d P ( ω ) ≥ N − (cid:88) k =0 (cid:90) Ω inf → s ∈ Z → r x i ∈ [ a i,si − ,a i,si ] log (cid:16) | ϕ (cid:48) σ k ( ω ) ,i, → s k ( ϕ ω,i,k − , → s k − ( x i )) | (cid:17) d P ( ω ) ≥ N − (cid:88) k =0 (cid:90) Ω log( γ i ( ω )) d P ( ω ) ≥ N γ > log(3).Let E be the set of functions of the form g = M (cid:88) j =1 g j X A j , where A j = n (cid:89) i =1 [ α ji , β ji ] ⊆ I n and g j : I n → R is a C function on A j , By [26, Remark 1], E forms a densesubset of the space L ( I n ). By [26, Remark 5], E ⊂ BV ( I n ).We argue in a similar way to the proof of Theorem 1 in [26]. We provide theLasota-Yorke inequality on elements of E . Since the BV norm is a continuousfunction and by Proposition 2.18 the transfer operator is bounded, using a densityargument, the inequality can be extended to elements of BV ( I n ).Let h ∈ E be such that h ≥ i = 1 , . . . , n , let h i ∈ E be such that h i = h Lebesgue almost everywhere with the property (cid:90) P i ( I n ) I n V i h i dm = I n V i h .Let Ψ ω,i,N, → s := ϕ − ω,i,N, → s , δ ω,i,N, → s := | Ψ (cid:48) ω,i,N, → s | , I ω,N, → s := n (cid:89) i =1 ϕ ω,i,N, → s ([ a i,s i − , a i,s i ]).The transfer operator L ( N ) ω applied to h evaluated at x = ( x , . . . , x n ) ∈ I n is givenby L ( N ) ω h ( x ) = (cid:88) → s ∈ Z → r h (cid:16) Ψ ω, ,N, → s ( x ) , . . . , Ψ ω,n,N, → s ( x n ) (cid:17) n (cid:89) j =1 δ ω,j,N, → s ( x j )1 I ω,N, → s ( x ). UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 13
If we apply I n V i for the the L ( I n )-function L ( N ) ω h i and the take the integral over P i ( I n ), we get(3.4) (cid:90) P i ( I n ) I n V i L ( N ) ω h i dm ≤ I + I ,where I = (cid:88) → s ∈ Z → r (cid:90) P i ( I ω,N, → s ) I ω,N, → s V i h i (cid:16) Ψ ω, ,N, → s ( x ) , . . . , Ψ ω,n,N, → s ( x n ) (cid:17) n (cid:89) j =1 δ ω,j,N, → s ( x j ) dm , I = (cid:88) → s ∈ Z → r (cid:90) P i ( I ω,N, → s ) (cid:16) | h i (Ψ ω, ,N, → s ( x ) , . . . , Ψ ω,n,N, → s ( x n )) | δ ω,i,N, → s ( ϕ ω,i,N, → s ( a i,s i ))+ | h i (Ψ ω, ,N, → s ( x ) , . . . , Ψ ω,n,N, → s ( x n )) | δ ω,i,N, → s ( ϕ ω,i,N, → s ( a i,s i − )) (cid:17) n (cid:89) j =1 j (cid:54) = i δ ω,N, → sj ( x j ) dm .Let(3.5) ρ ω,i,N = sup → s ∈ Z → r δ ω,i,N, → s ,and K ω,i,N = sup → s ∈ Z → r δ (cid:48) ω,i,N, → s inf → s ∈ Z → r δ ω,i,N, → s + sup → s ∈ Z → r δ ω,i,N, → s .These constants are motivated from the ones given in Theorem 1 in [35] alsoTheorem 2 in [6]. By Inequality 7 in [35] adapted to our notation, we have I ≤ ρ ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( I ω,N, → s ) P i ( I ω,N, → s ) × I V i h i (cid:16) Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n ) (cid:17) n (cid:89) j =1 j (cid:54) = i δ ω,j,N, → s ( x j ) dm + K ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( I ω,N, → s ) 1 (cid:90) h i (cid:16) Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n ) (cid:17) dx in (cid:89) j =1 j (cid:54) = i δ ω,j,N, → s ( x j ) dm .Making the change of variables U = Ψ − , as in Lemma 3 in [35], then the abovesum is equal to ρ ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( B → s ) P i ( B → s ) × I V i h i ( x , . . . , x n ) dm + K ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( B → s ) 1 (cid:90) h i ( x , . . . , x n ) dx i dm .Since { B → s : → s ∈ Z → r } forms a partition for I n , the last sum is equal to2 ρ ω,i,N (cid:90) P i ( I n ) I n V i h i dm + K ω,i,N (cid:90) P i ( I n ) ( (cid:90) h i dx i ) dm ≤ ρ ω,i,N (cid:90) P i ( I n ) I n V i h i dm + K ω,i,N (cid:107) h i (cid:107) .(3.6)The expression in I is less than or equal tosup → s ∈ Z → r δ ω,i,N, → s (cid:88) → s ∈ Z → r (cid:90) P i ( I ω,N, → s ) (cid:16) | h i (Ψ ω, ,N, → s ( x ) , . . . , a i,s i , . . . , Ψ ω,n,N, → s ( x n )) | + | h i (Ψ ω, ,N, → s ( x ) , . . . , a i,s i − , . . . , Ψ ω,n,N, → s ( x n )) | (cid:17) n (cid:89) j =1 j (cid:54) = i δ ω,j,N, → s ( x j ) dm. Since h ≥
0, the argument after Equation (5) in [35], implies (cid:16) | h i (Ψ ω, ,N, → s ( x ) , . . . , a i,s i , . . . , Ψ ω,n,N, → s ( x n )) | + | h i (Ψ ω, ,N, → s ( x ) , . . . , a i,s i − , . . . , Ψ ω,n,N, → s ( x n )) | (cid:17) ≤ P i ( I ω,N, → s ) × I V i h i (Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n ))+ 2 (cid:90) h i (Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n )) dx i ,then we have the expression in I is less than or equal to ρ ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( I ω,N, → s ) (cid:16) P i ( I ω,N, → s ) × I V i h i (Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n ))+ 2 (cid:90) h i (Ψ ω, ,N, → s ( x ) , . . . , x i , . . . , Ψ ω,n,N, → s ( x n )) dx i (cid:17) n (cid:89) j =1 j (cid:54) = i δ ω,j,N, → s ( x j ) dm. Using again the change of variables U = Ψ − , we get the last sum is equal to UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 15 ρ ω,i,N (cid:88) → s ∈ Z Pi ( → r ) (cid:90) P i ( B → s ) (cid:16) P i ( B → s ) × I V i h i ( x , . . . , x n ) + 2 (cid:90) h i ( x , . . . , x n ) dx i (cid:17) dm = ρ ω,i,N (cid:90) P i ( I n ) I n V i h i dm + 2 ρ ω,i,N (cid:90) P i ( I n ) 1 (cid:90) h i dx i dm ≤ ρ ω,i,N (cid:90) P i ( I n ) I n V i h i dm + 2 ρ ω,i,N (cid:107) h i (cid:107) .(3.7)Now, combining the results from (3.6) , (3.7) and (3.4), we get (cid:90) P i ( I n ) I n V i L ( N ) ω h i dm ≤ ρ ω,i,N (cid:90) P i ( I n ) I n V i h i dm + ( K ω,i,N + 2 ρ ω,i,N ) (cid:107) h i (cid:107) .Thus, letting α ( ω ) = max i =1 ,...,n ρ ω,i,N ,(3.8) α ( ω ) = max i =1 ,...,n ( K ω,i,N + 2 ρ ω,i,N ),we have, for each i = 1 , . . . , n , (cid:90) P i ( I n ) I n V i L ( N ) ω hdm ≤ α ( ω ) (cid:90) P i ( I n ) I n V i hdm + α ( ω ) (cid:107) h (cid:107) .By [18, Lemma C.5] and Lemma 2.14, the index of compactness K N ( ω ) is lessthan (cid:90) σ − (cid:96) Z log α (¯ ω ) d P (¯ ω ),where (cid:96) is such that σ − (cid:96) Z is the ergodic component of σ N containing ω . SinceΩ = k − (cid:91) s =0 σ − (cid:96) Z and (cid:82) Ω log α ( ω ) d P ( ω ) <
0, we have (cid:82) σ − (cid:96) Z log α ( ω ) d P ( ω ) < (cid:96) = 0 , , . . . , k −
1. By Proposition 2.13, we have K ∗ = K N ( ω ) N < L ( n ) ω is a Markov operator for each ω ∈ Ω, for anydensity function h ∈ BV ( I n ), we have that (cid:107)L ( n ) ω h (cid:107) BV ≥ (cid:107)L ( n ) ω h (cid:107) = (cid:107) h (cid:107) = 1.This shows that(3.9) λ ∗ ≥ , and therefore K ∗ < λ ∗ . This finishes the proof of Theorem 3.1 (i). (cid:3) Proof of Theorem 3.1 (ii).
In the proof of Theorem 3.1 (i), we proved that thereare N ∈ N where N satisfies the condition in (3.2) and α , α : Ω → R + such that (cid:82) Ω log α ( ω ) d P ( ω ) < I n V L ( N ) ω h ≤ α ( ω ) I n V h + α ( ω ) (cid:107) h (cid:107) ,for all h ∈ BV ( I n ) and ω ∈ Ω. We also proved that λ ∗ ≥ λ ∗ ≤
0. Since (cid:107)L ω (cid:107) ≤
1, it is enough to consider the growth of the variationof the term L ( n ) ω h . Using the argument in [18, Lemma C.5] and [9, Proposition α ( ω ) and α ( ω ) can be redefined so that (3.10) holds and α ( ω ) is uniformlybounded by positive constant ˜ α , which gives a hybrid Lasota-Yorke inequality(3.11) I n V L ( N ) ω h ≤ α ( ω ) I n V h + ˜ α (cid:107) h (cid:107) .By iterating the hybrid Lasota-Yorke inequality (3.11), we get a bound on thesequence ( I n V L ( Nk ) ω h ) ∞ k =1 . Therefore,lim k →∞ N k log (cid:107)L ( Nk ) ω h (cid:107) BV ≤ ω ∈ Ω, Proposition 2.13 implies that λ ∗ ≤ (cid:3) Random invariant densities and ACIPs, skew product ACIPs andPhysical measures
The concept of random invariant measures (for random dynamical systems) is anatural generalization of the notion of invariant measures (for deterministic dynam-ical systems). In this section we introduce our main results regarding the existenceof random invariant densities and measures as well as skew product ACIPs. Afterthat, we deduce the existence of physical measures. We shall assume throughoutthe rest of the paper that (cid:82) Ω log + (cid:107)L ω (cid:107) BV d P ( ω ) < ∞ . Definition 4.1.
Let F = { f ω } ω ∈ Ω be an admissible random Jab(cid:32)lo´nski map. Afamily { µ ω } ω ∈ Ω of random invariant measures for F is a family of probabilitymeasures µ ω on I n where the map ω (cid:55)→ µ ω is measurable and f ω µ ω = µ σω , for P -a.e. ω ∈ Ω.A family { h ω } ω ∈ Ω of random invariant densities for F is a family such that h ω ≥ h ω ∈ L ( I n ), (cid:107) h ω (cid:107) = 1, the map ω (cid:55)→ h ω is measurable and(4.1) L ω h ω = h σω , for P -a.e. ω ∈ Ω. Proposition 4.2.
Let N be as in (3.2) . Then, for P -almost all ω ∈ Ω , we have lim j →∞ j j (cid:88) t =1 log( α ( σ − tN ω )) < .Proof. By Lemma 2.14, there exists k , a factor of N , and a σ − N -invariant subset Z of Ω of measure 1 /k such that Ω = k − (cid:91) s =0 σ (cid:96) Z and σ − N | Z is ergodic. In fact, since σ is invertible, ergodic and P -preserving, σ − N | σ (cid:96) Z is ergodic and P ( σ (cid:96) Z ) = k , forall (cid:96) = 0 , , . . . , k −
1. By Birkhoff ergodic theorem, we havelim j →∞ j j (cid:88) t =1 log( α ( σ − tN ω )) = k (cid:90) σ (cid:96) Z log α (¯ ω ) d P (¯ ω ),for P -almost all ω ∈ σ (cid:96) Z , and (cid:96) = 0 , , . . . , k −
1. Note that for any (cid:96) = 0 , , . . . , k − P -almost all ω ∈ σ (cid:96) Z , the definition of α ( ω ) in (3.8) and the argument in (3.3) UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 17 imply (cid:90) σ (cid:96) Z log α ( ω ) d P ( ω ) = (cid:90) σ (cid:96) Z log (cid:16) max i =1 ,...,n (cid:16) sup → s ∈ Z → r | ( ϕ − ω,i,N, → s ) (cid:48) | (cid:17)(cid:17) d P ( ω )= (cid:90) σ (cid:96) Z log (cid:16) max i =1 ,...,n (cid:16) sup → s ∈ Z → r | (cid:16) ( ϕ σ N − ω,i, → s N − ◦ · · · ◦ ϕ σω,i, → s ◦ ϕ ω,i, → s ) − (cid:17) (cid:48) | (cid:17)(cid:17) d P ( ω ) ≤ (cid:90) σ (cid:96) Z log (cid:16) max i =1 ,...,n (cid:16) | N − (cid:89) t =0 sup → s ∈ Z → r ϕ (cid:48) σ t ( ω ) ,i, → s t ( ϕ ω,i,t − , → s t − ( x i )) | (cid:17)(cid:17) d P ( ω ).By definition of γ i in Equation (2.6), we have (cid:90) σ (cid:96) Z log α ( ω ) d P ( ω ) ≤ (cid:90) σ (cid:96) Z log 3 − min i =1 ,...,n log (cid:16) N − (cid:89) t =0 γ i ( σ t ω ) (cid:17) d P ( ω )= log(3) k − N − (cid:88) t =0 (cid:90) σ (cid:96) Z min i =1 ,...,n log( γ i ( σ t ω )) d P ( ω ).Since σ is measure preserving, a change of variables makes the last term equal tolog(3) k − N − (cid:88) t =0 (cid:90) Z min i =1 ,...,n log( γ i ( σ t − (cid:96) ω )) d P ( ω ) = log(3) k − Nk (cid:90) Ω min i =1 ,...,n log( γ i ( ω )) d P ( ω )= 1 k (log(3) − N Γ) < k (log(3) − N γ ) < (cid:3) Theorem 4.3.
Consider an admissible random Jab(cid:32)lo´nski map F . For each ω ∈ Ω and k = 1 , , . . . , we define h kω = ( L σ − ω ◦ · · · ◦ L σ − ( k − ω ◦ L σ − k ω )1 ,where ∈ BV ( I n ) is the constant function and for each s = 1 , , . . . , we define H sω = 1 s s (cid:88) k =1 h kω .Then, for P -a.e. ω ∈ Ω :(i) the sequence { H sω } s ∈ N is relatively compact in L ; and(ii) the following limit exists, (4.2) lim s →∞ H sω =: h ω ∈ BV ( I n ) in L .Moreover, { h ω } ω ∈ Ω is a family of random invariant densities for F .Proof. Recall from the proof of Theorem 3.1 (ii), there are N ∈ N where N satisfiesthe condition in (3.2), a constant ˜ α and a positive measurable function α : Ω → R + such that (cid:82) Ω log α ( ω ) d P ( ω ) < I n V L ( N ) ω h ≤ α ( ω ) I n V h + ˜ α (cid:107) h (cid:107) , for all h ∈ BV ( I n ) and ω ∈ Ω. For k = 1 , , . . . , and P -almost all ω ∈ Ω, thefollowing holds, h Nkω = L ( Nk ) σ − Nk ω Applying (3.11) to upper bound the variation of h Nkω on I n yields I n V h Nkω = I n V L ( Nk ) σ − Nk ω ≤ α ( σ − N ω ) I n V ( L σ − ( N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1+ ˜ α (cid:107) ( L σ − ( N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1 (cid:107) ≤ α ( σ − N ω ) α ( σ − N ω ) I n V ( L σ − (2 N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1+ ˜ α (cid:107) ( L σ − ( N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1 (cid:107) + α ( σ − N ω )˜ α (cid:107) ( L σ − (2 N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1 (cid:107) ≤ · · · ≤ α ( σ − N ω ) α ( σ − N ω ) . . . α ( σ − kN ω ) I n V
1+ ˜ α (cid:107) ( L σ − ( N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1 (cid:107) + α ( σ − N ω )˜ α (cid:107) ( L σ − (2 N +1) ( ω ) ◦ · · · ◦ L σ − N ( ω ) ◦ L σ − Nk ( ω ) )1 (cid:107) + · · · + α ( σ − N ω ) α ( σ − N ω ) . . . α ( σ − kN ω )˜ α (cid:107) (cid:107) ,and since I n V (cid:107) (cid:107) = 1 and the transfer operator is contractive, we have I n V h Nkω ≤ ˜ α (cid:16) α ( σ − N ω ) + α ( σ − N ω ) α ( σ − N ω ) + . . . + α ( σ − N ω ) α ( σ − N ω ) . . . α ( σ − kN ω ) (cid:17) = ˜ α (1 + k (cid:88) j =1 α ( j )1 ( σ − jN ω )),where for j = 1 , , . . . , we let α ( j )1 ( σ − jN ω ) = α ( σ − N ω ) α ( σ − N ω ) . . . α ( σ − jN ω ).By Proposition 4.2, there exists 0 < ˆ α ( ω ) < j log α ( j )1 ( σ − jN ω )converge to log(ˆ α ( ω )) <
0. Choose α ( ω ) such that 0 < ˆ α ( ω ) < α ( ω ) <
1. For suffi-ciently large j ( ω ), we have that α ( j )1 ( σ − jN ω ) < α ( ω ) j , for all j ≥ j ( ω ).Let c ( ω ) be defined as c ( ω ) = max ≤ j ≤ j ( ω ) ( α ( j )1 ( σ − jN ω ) α ( ω ) j , j , we have that α ( j )1 ( σ − jN ω ) < c ( ω ) α ( ω ) j .Taking the sum over j , we get that˜ α (1 + k (cid:88) j =1 α ( j )1 ( σ − jN ω )) ≤ ˜ α (1 + c ( ω ) ∞ (cid:88) j =0 α ( ω ) j )= ˜ α (1 + c ( ω )˜ α ( ω )),where ˜ α ( ω ) = − α ( ω ) . Let c ( ω ) = ˜ α (1 + c ( ω )˜ α ( ω )), UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 19 then we have proven that for every k ∈ N I n V h Nkω ≤ c ( ω ).From this inequality, it follows { I n V h Nkω } k ∈ N is bounded. The same holds for thewhole sequence { I n V h kω } k ∈ N , and indeed for the averages { I n V H sω } s ∈ N . Hence, { H sω } s ∈ N is relatively compact in L by [31, Lemma A.1]. This establishes (i).Then, the random mean ergodic theorem [33, Theorem B] shows that { H sω } s ∈ N converges in the strong sense to a random invariant density h ω , as in (4.2). Thefact that h ω ∈ BV ( I n ) follows once again from the relative compactness of BV ( I n )in L . This establishes (ii). (cid:3) We can think of the above random invariant densities h ω as asymptotic distribu-tions arrived at by running the dynamics of a uniform distribution from the distantpast. Returning to the present setting of random compositions of Jab(cid:32)lo´nski maps,a family of random invariant measures with densities of bounded variation will alsodefine a measure that is invariant with respect to the associated skew product, asdescribed in the following remark. Remark . For P -a.e. ω ∈ Ω, define µ ω on the fiber { ω } × I n ⊂ Ω × I n , as dµ ω dm = h ω ,where h ω is given by (4.2). Then µ ω is a random invariant ACIP and the measure µ defined on P × m -measurable sets A ⊆ Ω × I n by µ ( A ) = (cid:90) Ω µ ω ( A ) d P ( ω ),is an ACIP for the associated skew product F defined in (2.4).Multiplicative ergodic theorems are concerned with random dynamical systems R = (Ω , F , P , σ, X , L ). They give rise to an ω -dependent hierarchical decompo-sition of X into equivariant subspaces, called Oseledets spaces. In the literature,multiplicative ergodic theorems are divided into two types, according to the invert-ibility of the base map σ and the operators L ω . In [16], Froyland, Lloyd and Quasshow a semi-invertible multiplicative ergodic theorem, where the base is assumedto be invertible, but there is no assumption about invertibility of the operators L ω .We will apply this theorem to show that the random invariant densities h ω foundin Theorem 4.3 belong to the leading Oseledets subspace. Moreover, we will deducethe finiteness of the number of ergodic ACIPs in Corollary 4.6.An Oseledets splitting for a random dynamical system R = (Ω , F , P , σ, X , L )consists of • A sequence of isolated (exceptional) Lyapunov exponents ∞ > λ ∗ = λ > λ > · · · > λ l > K ∗ ≥ −∞ ,where the index l ≥ • A family of ω -dependent splittings,(4.3) X = Y ( ω ) ⊕ · · · ⊕ Y l ( ω ) ⊕ V ( ω ),where for j = 1 , . . . , l , d j := dim( Y j ( ω )) < ∞ and V ( ω ) ∈ G ( X ) where G ( X ) is the Grassmannian of X . For all j = 1 , . . . , l and P -a.e. ω ∈ Ω, we have L ω Y j ( ω ) = Y j ( σω ),(4.4) L ω V ( ω ) ⊆ V ( σω ),(4.5)and lim s →∞ s log (cid:107)L ( s ) ω y (cid:107) = λ j , ∀ y ∈ Y j ( ω ) \{ } ,(4.6) lim s →∞ s log (cid:107)L ( s ) ω v (cid:107) ≤ K ∗ , ∀ v ∈ V ( ω ).(4.7) Theorem 4.5 (Froyland, Lloyd and Quas [16, Theorem 17]) . Let Ω be a Borelsubset of a separable complete metric space, F the Borel sigma-algebra and P aBorel probability measure. Let X be a Banach space. Consider a random dynam-ical system R = (Ω , F , P , σ, X , L ) with base transformation σ : Ω (cid:9) an ergodichomeomorphism, and suppose that the generator L : Ω → L ( X , X ) is P -continuousand satisfies (cid:90) Ω log + (cid:107)L ω (cid:107) d P ( ω ) < ∞ .If R is quasi-compact, that is, if K ∗ < λ ∗ , then R admits a unique P -continuousOseledets splitting. By Theorem 3.1, admissible random Jab(cid:32)lo´nski maps give rise to quasi-compactrandom dynamical systems with λ = 0. Therefore, Theorem 4.5 implies the fol-lowing. Corollary 4.6.
For P -a.e. ω ∈ Ω , the random invariant density h ω given in (4.2) belongs to the Oseledets space Y ( ω ) given in (4.3) . Moreover, the number r of ergodic ACIPs µ , . . . , µ r with respect to the associated skew product is finite;indeed, we have (4.8) r ≤ d = dim( Y ( ω )) .Proof. Let ω ∈ Ω, by the equivariance property given in (4.1), we have L ( m ) ω h ω = h σ m ω , for m ∈ N . To show that h ω ∈ Y ( ω ), we verify the limit condition given in(4.6) for j = 1. Note thatlim m →∞ m log (cid:107)L ( m ) ω h ω (cid:107) BV = lim m →∞ m log (cid:107) h σ m ω (cid:107) BV ≥ lim m →∞ m log (cid:107) h σ m ω (cid:107) = 0 = λ ∗ ,on the other hand lim m →∞ m log (cid:107)L ( m ) ω h ω (cid:107) BV ≤ lim m →∞ m log (cid:107)L ( m ) ω (cid:107) BV = 0 = λ ∗ ,by Theorem 3.1. Since the splitting in Theorem 4.5 is unique, this gives that h ω ∈ Y ( ω ). By the finite dimensionality of the leading Oseledets subspace Y ( ω ),we get the bound given in (4.8). (cid:3) UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 21
Next, we define physical measures and show how the measures given in Corollary4.6 are physical measures.
Definition 4.7.
Consider the tuple (Ω , F , P , σ, f ) where (Ω , F , P ) is a probabilityspace, σ : Ω (cid:9) an invertible, ergodic and P − preserving transformation and f = { f ω : M → M } ω ∈ Ω where M ⊆ R n . A probability measure ν on M is called physical if for P -a.e. ω ∈ Ω, the Lebesgue measure of the random basin RB ω ( ν ) of ν at ω is positive where RB ω ( ν ) = { x ∈ M : 1 s s − (cid:88) k =0 δ f ( k ) ω ( x ) → ν } ,where δ x is the Dirac measure at a point x .The convergence in Definition 4.7 is in the weak convergence sense. In the casewhere f ω is independent of ω , this reduces to the definition of physical measurefor a deterministic dynamical system. The next probabilistic result due to Buzziapplies in our setting. Theorem 4.8 (Buzzi [9, Proposition 4 . . Let µ i be one of the measures µ i : i = 1 , . . . r given in Corollary 4.6. Then, the marginal measure of µ i on I n , denotedby ν i , is a physical measure on I n . The union of all basins of the of the physical measures ν i coming from themarginals of µ i on I n , i = 1 , . . . r has full Lebesgue measure, which means Lebesguealmost everywhere, the asymptotic long term behaviour of the random orbits willbe described by one of these physical measures. Another immediate consequenceof the proof of Theorem 4.8 is the following. Corollary 4.9.
There exists a constant b > such that for P -a.e. ω ∈ Ω and i = 1 , . . . r , m ( RB ω ( ν i )) > b . Bounds on the number of ergodic skew product ACIPs
A difficulty in the general study of ACIPs of piecewise expanding maps in higherdimensions is that the geometric complexity around discontinuities or interior cross-ing points might grow rapidly as the dynamical partitions are refined [10]. This isin contrast to one-dimensional maps, where the geometry is much simpler and sucha complexity growth can not happen. However, this complication does not occur inthe context of random Jab(cid:32)lo´nski maps. In [23], G´ora, Boyarsky and Proppe provedthat for a class of deterministic dynamical systems modeled by Jab(cid:32)lo´nski transfor-mations, the number of crossing points gives an upper bound for the number ofergodic ACIPs.In this section, we establish bounds on the number of ergodic ACIPs for randomJab(cid:32)lo´nski maps. The first bound, presented in Section 5.1, is motivated by the workof Buzzi [9] in the one dimensional case of random Lasota-Yorke maps. The secondbound, presented in Section 5.2 is inspired by the work of G´ora, Boyarsky andProppe on absolutely continuous invariant measures for deterministic dynamicalsystems given by multidimensional expanding maps [23]. An example is presentedin Section 5.3.Let F = { f ω } ω ∈ Ω be an admissible random Jab(cid:32)lo´nski map. Suppose that thereexist r mutually singular ergodic ACIPs µ , . . . , µ r for the associated skew productmap F . Fix i ∈ { , . . . , r } and ω ∈ Ω, then the fiber measure µ iω is a measure on I n . By Theorem 2 in [23], the support Supp( µ iω ) of µ iω is open Lebesgue almosteverywhere. This fact was before introduced in Keller’s thesis [28]. Let I i,ω (0) ⊆ Supp( µ iω ) be a nontrivial rectangle lying inside one of the rectangles of B ω . Definethe sequence(5.1) I i,ω ( s + 1) = f σ s ω ( I i,ω ( s )) ∩ J , s ∈ N ∪ { } ,where J is the open rectangle in the partition B σ s +1 ω of the Jab(cid:32)lo´nski map f σ s +1 ω which maximizes the Lebesgue measure of I i,ω ( s +1). For s ∈ N ∪{ } , define c i,ω ( s )to be the number of crossing points in the partition B σ s +1 ω lying inside the image f σ s ω ( I i,ω ( s )). Let(5.2) M ( ω ) = max z ∈ R max d =1 ,...,n { number of rectangles B ∈ B σω s.t. H ( d ) n − ( z ) ∩ Int( B ) (cid:54) = φ } ,where H ( d ) n − ( z ) is the ( n −
1) dimensional hyperplane given by the equation x d = z .This definition of M is motivated by a deterministic analogue, Definition 3 in [23].For i = 1 , . . . , r , denote by D i = { ω ∈ Ω : Supp( µ iω ) has a crossing point in its interior } .Also, let(5.3) γ ( ω ) = n (cid:89) i =1 γ i ( ω ),and γ i ( ω ) is defined in equation (2.6).5.1. Multidimensional bound `a la Buzzi.
In this section, we assume the fol-lowing.(5.4) δ := (cid:90) Ω log( γ ( ω ) M ( ω ) ) d P ( ω ) > Lemma 5.1.
Let F = { f ω } ω ∈ Ω be an admissible random Jab(cid:32)lo´nski map and as-sume that (5.4) is satisfied. Then, the number r of mutually singular ergodic ACIPsfor the associated skew product map F satisfies (5.5) (cid:90) Ω log (cid:0) n − ( c t ( ω ) r + 1) (cid:1) d P ( ω ) ≥ δ .Proof. First we show that at least one of the sets in(5.6) f σ s ω ( I i,ω ( s )), s ∈ N ∪ { } has a crossing point in its interior. The argument proceeds by contradiction. Sup-pose that for none of the sets in (5.6) has a crossing point in the interior. Then, m ( I i,ω ( s + 1)) ≥ γ ( σ s ω ) M ( σ s ω ) m ( I i,ω ( s )) ≥ γ ( σ s ω ) M ( σ s ω ) . . . . . γ ( ω ) M ( ω ) m ( I i,ω (0)). UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 23
By (5.4), we have δ = (cid:82) Ω log( γ ( ω ) M ( ω ) ) d P ( ω ) >
0. Hence, Birkhoff ergodic theoremimplies that m ( I i,ω ( s + 1)) → ∞ as s → ∞ , and this is a contradiction. Hence, atleast one of the sets in (5.6) has a crossing point in its interior.For k = 0 , , , , . . . and ω ∈ Ω, define g i,k ( ω ) = γ ( σ k ω )2 n − ( c i,ω ( k )+1) : σ k ω ∈ D iγ ( σ k ω ) M ( σ k ω ) : σ k ω ∈ Ω \ D i .By equation (5.1), for s ∈ N , I i,ω ( s ) comes from evolving I i,ω ( s −
1) by the map f σ s − ω and then taking the largest intersection of its image with one of the partitionrectangles of B σ s ω . Therefore, the volume of I i,ω ( s ) depends on whether the set f σ s − ω ( I i,ω ( s − I i,ω ( s ) is bounded below by the volumeof I i,ω ( s −
1) expanded by γ ( σ s − ω ) and scaled by 2 n − ( c i,ω ( s −
1) + 1). Thislast scaling term is an upper bound on the number of rectangles of B σ s ω meeting f σ s − ω ( I i,ω ( s − f σ s − ω ( I i,ω ( s − I i,ω ( s ) is bounded below by the volume of I i,ω ( s − γ ( σ s − ω ) and scaled by M ( σ s − ω ). Thus, in general, m ( I i,ω ( s )) ≥ g i,s − ( ω ) m ( I i,ω ( s − m ( I i,ω ( s )) ≥ g i,s − ( ω ). . . . . g i, ( ω ) m ( I i,ω (0)).Since m ( I i,ω ( s )) ≤
1, for all s = 1 , , , . . . , we have(5.8) s − (cid:88) k =0 log( 1 g i,k ( ω ) ) ≥ log( m ( I i,ω (0)).By summing over i = 1 , . . . , r and dividing by r , we get s − (cid:88) k =0 r r (cid:88) i =1 log( 1 g i,k ( ω ) ) ≥ ξ ,where ξ := r (cid:80) ri =1 log( m ( I i,ω (0)). This gives that(5.9) s − (cid:88) k =0 log( 1( g ,k ( ω ). . . . . g r,k ( ω )) r ) ≥ ξ .Since the measures µ i are mutually singular, for all ω ∈ Ω, we have c ,ω ( k ) + · · · + c r,ω ( k ) ≤ c t ( σ k ω ),where we recall that c t ( ω ) is the total number of interior crossing points in thepartition B σω of f σω . By adding r to both sides, dividing by r and using thearithmetic-geometric mean inequality, we get (cid:16) ( c ,ω ( k ) + 1). . . . .( c r,ω ( k ) + 1) (cid:17) r ≤ c t ( σ k ω ) + rr . Therefore, (5.9) and the definition of g i,k ( ω ) yield1 s s − (cid:88) k =0 log (cid:16) n − ( c t ( σ k ω )+ rr ) M ( σ k ω ) γ ( σ k ω ) (cid:17) ≥ s s − (cid:88) k =0 log (cid:16) g ,k ( ω ). . . . . g r,k ( ω )) r (cid:17) ≥ ξs .(5.10)Applying Birkhoff ergodic theorem, we get (cid:90) Ω log (cid:16) n − ( c t ( ω ) r + 1) M ( ω ) γ ( ω ) (cid:17) d P ( ω ) ≥ (cid:90) Ω log(2 n − ( c t ( ω ) r + 1)) d P ( ω ) + (cid:90) Ω log( M ( ω ) γ ( ω ) ) d P ( ω ) ≥ (cid:90) Ω log (cid:0) n − ( c t ( ω ) r + 1) (cid:1) d P ( ω ) ≥ δ . (cid:3) Lemma 5.1 may be used to obtain explicit bounds on r . Lemma 5.2.
Suppose (5.4) holds, for P -a.e. ω ∈ Ω , c t ( ω ) ≤ c and log(2 n − ) < δ .Then (5.5) gives an explicit bound on r , that is (5.12) r ≤ c exp( δ )2 n − − .Proof. Since for P -a.e. ω ∈ Ω, c t ( ω ) ≤ c , we get (cid:90) Ω log(2 n − ( c t ( ω ) r + 1)) d P ( ω ) ≤ log(2 n − ( cr + 1)) . By (5.5), we have log(2 n − ( cr + 1)) ≥ δ which implies(5.13) cr + 1 ≥ exp( δ )2 n − .Since log(2 n − ) < δ , we have exp( δ )2 n − > r . By solving (5.13) for r , we get the upper bound given in (5.12). (cid:3) The next corollary shows another way of getting finiteness of the number ofmeasures r , previously obtained in Corollary 4.6 using multiplicative ergodic theory. Corollary 5.3.
Consider the assumptions in Lemma 5.2. Then the number ofmeasures r in Corollary 4.6 is finite.Proof. The integrand in (5.5) is a non-increasing function of r . Hence, as r → ∞ ,we get (cid:82) Ω log(2 n − ) d P ( ω ) ≥ δ ,which contradicts the assumption. (cid:3) Another immediate consequence of Lemma 5.2 is the following.
Corollary 5.4. If c exp( δ )2 n − − < , then there exists a unique ergodic ACIP for theskew product. UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 25
Another bound on r . We recall that γ ( ω ), introduced in (5.3), quantifies theexpansion in the random system. The geometry of the partitions {B ω } ω ∈ Ω is relatedto the quantities q ω , the number of rectangles in the partition B ω ; M ( ω ), defined in(5.2); and c t ( ω ), the total number of interior crossing points in the partition B σω . Lemma 5.5.
Assume for P -a.e. ω ∈ Ω , M ( ω ) ≤ M , c t ( ω ) ≤ c and q ω ≤ q . Then, (5.14) r ≤ c (log( q ) − log( M )) (cid:82) Ω log( γ ( ω )) d P ( ω ) − log( M ) .Proof. Recall that
M < q , by the definition of M ( ω ) in (5.2). For i = 1 , , . . . r ,define g i ( ω ) = γ ( ω ) q ω : ω ∈ D iγ ( ω ) M ( ω ) : ω ∈ Ω \ D i .In a similar argument to (5.7), note that for all ω ∈ Ω and s = 1 , , , . . . , wehave m ( I i,ω ( s )) ≥ g i ( σ s − ω ). . . . . g i ( ω ) m ( I i,ω (0)).Then, for all s = 1 , , , . . . , we have(5.15) 1 s s − (cid:88) k =0 log( g i ( σ k ω )) ≤ log( m ( I i,ω (0)) ) s .It is also clear that(5.16) g i ( ω ) ≥ γ ( ω ) q : ω ∈ D iγ ( ω ) M : ω ∈ Ω \ D i .Using (5.16) and Birkhoff ergodic theorem, from (5.15), we get (cid:90) D i log( γ ( ω ) q ) d P ( ω )+ (cid:90) Ω \ D i log( γ ( ω ) M ) d P ( ω ) ≤ (cid:90) Ω log( γ ( ω )) d P ( ω ) − log( M ) + m i log( Mq ) ≤ m i = m ( D i ). Therefore, for all i = 1 , . . . , r , we have(5.17) m i ≥ (cid:82) Ω log( γ ( ω )) d P ( ω ) − log( M )log( q ) − log( M ) .For i = 1 , . . . , r , define a i ( ω ) = (cid:26) ω ∈ D i ω ∈ Ω \ D i ,then for P -a.e. ω ∈ Ω, (cid:80) ri =1 a i ( ω ) ≤ c . Note that m i = (cid:82) Ω a i ( ω ) d P ( ω ). Taking thesum over all i = 1 , . . . , r , we get r ≤ c min( m ,...,m r ) . By (5.17), we get the boundgiven in (5.14). (cid:3) Since m i ≤ i = 1 , . . . , r , an immediate consequence of (5.17) is thefollowing. Figure 1. I parti-tioned into 25 equalsquares. Figure 2.
Bounds in (5.18)(solid) and (5.19) (dashed).
Corollary 5.6.
We have (cid:82) Ω log( γ ( ω )) d P ( ω ) ≤ log( q ) , where q is defined in Lemma5.5. Corollary 5.7. If c (log( q ) − log( M )) (cid:82) Ω log( γ ( ω )) d P ( ω ) − log( M ) < , then there exists a unique ergodicACIP for the skew product. Example.
Consider an admissible random Jab(cid:32)lo´nski map where the commonpartition is taken to be the equally sized 25 squares partition shown in Figure (1).For this partition, we have M = 5, c = 16 and q = 25.Let γ , γ > ω ∈ Ω, γ ( ω ) ≥ γ and γ ( ω ) ≥ γ where γ i ( ω ) is defined in (2.6). By (5.3), we have γ ( ω ) = γ ( ω ) γ ( ω ) ≥ γ γ ,for all ω ∈ Ω. Note that γ and γ can not take values such that γ γ >
25, becausethe rectangles of the partition would be mapped outside I . The constant δ definedin (5.4) is δ = (cid:90) Ω log( γ ( ω ) M ( ω ) ) d P ( ω ) ≥ log( γ γ γ and γ can notbe such that γ γ ≤
10. Since this contradicts the condition in Lemma 5.2 thatlog(2 n − ) < δ , we can make the restriction that10 < γ γ ≤ . Then, (5.12) implies that(5.18) r ≤ γ γ − . The bound in (5.14), implies that(5.19) r ≤
16 log(5)log( γ γ ) .Figure (2) shows the dependence of the two bounds on γ γ and the regions onwhich each of the bounds is sharper. The bounds from Sections 5.1 and 5.2 areshown in black/solid and orange/dashed, respectively. UMBER OF INVARIANT MEASURES FOR RANDOM EXPANDING MAPS 27 Acknowledgments
The authors are thankful to the referees for helpful suggestions and corrections.The authors are thankful to Prof. Luigi Ambrosio for valuable comments regardingthe space of bounded variation defined in different equivalent forms, Prof. ChrisBose for useful conversations and Prof. Sandro Vaienti for bringing reference [33]to our attention. The authors have been partially supported by the AustralianResearch Council (DE160100147). Fawwaz Batayneh acknowledges the support ofthe University of Queensland through an Australian Government Research TrainingProgram Scholarship.
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