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What are SRB measures, and which dynamical systems have them?
"... This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures ..."
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This is a slightly expanded version of the text of a lecture I gave in a conference at Rutgers University in honor of David Ruelle and Yasha Sinai. In this lecture I reported on some of the main results surrounding an invariant measure introduced by Sinai, Ruelle and Bowen in the 1970s. SRB measures, as these objects are called, play an important role in the ergodic theory of dissipative dynamical systems with chaotic behavior. Roughly speaking, • SRB measures are the invariant measures most compatible with volume when volume is not preserved; • they provide a mechanism for explaining how local instability on attractors can produce coherent statistics for orbits starting from large sets in the basin. An outline of this paper is as follows. The original work of Sinai, Ruelle and Bowen was carried out in the context of Anosov and Axiom A systems. For these dynamical systems they identified and constructed an invariant measure which is uniquely important from several different points of view. These pioneering works are reviewed in Section 1. Subsequently, a nonuniform, almosteverywhere notion of hyperbolicity expressed in terms of Lyapunov exponents was developed. This notion provided a new framework for the ideas in the last paragraph. While not all of the previous characterizations are equivalent in this broader setting, the central ideas have remained intact, leading to a more general notion of SRB measures. This is discussed in Section 2. The extension above opened the door to the possibility that the dynamics on many attractors are described by SRB measures. Determining if this is (or is not) the case, however, let alone proving it, has turned out to be very challenging. No genuinely nonuniformly hyperbolic examples were known until the early 1990s, when SRB measures were constructed for certain Hénon maps. Today we still do not have a good understanding of which dynamical systems admit SRB measures, but some progress has been made; a sample of it is reported in Section 3.
Numeration systems and Markov partitions from self similar tilings
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the int ..."
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Abstract. Using self similar tilings we represent the elements of R n as digit expansions with digits in R n being operated on by powers of an expansive linear map. We construct Markov partitions for hyperbolic toral automorphisms by considering a special class of self similar tilings modulo the integer lattice. We use the digit expansions inherited from these tilings to give a symbolic representation for the toral automorphisms. Fractals and fractal tilings have captured the imaginations of a wide spectrum of disciplines. Computer generated images of fractal sets are displayed in public science centers, museums, and on the covers of scientific journals. Fractal tilings which have interesting properties are finding applications in many areas of mathematics. For example, number theorists have linked fractal tilings of R 2 with numeration systems for R 2 in complex bases [16], [8]. We will see that fractal self similar tilings of R n provide natural building blocks for numeration systems of R n. These numeration systems generalize the 1dimensional cases in [14],[10],[11] as well as the 2dimensional cases mentioned above.
Ruelle's Transfer Operator for Random Subshifts of Finite Type
 Ergod. Th. & Dynam. Sys
, 1995
"... We consider a RuellePerronFrobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibb ..."
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We consider a RuellePerronFrobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures. Key words: random dynamical system, random subshift of finite type, transfer operator, Gibbs measures, equilibrium states. 1991 Mathematics Subject Classification. Primary 58F03, 58F15; Secondary 60J10, 54H20. Introduction Random dynamical systems (RDS) generalize dynamical systems by allowing the dependence on an additional parameter evolving in time and describing a stochastic influence (cf. Arnold and Crauel [3]). The latter is modelled by an abstract dynamical system(\Ome...
Algebraic coding of expansive group automorphisms and twosided betashifts
 Monatsh. Math
"... Abstract. Let α be an expansive automorphisms of compact connected abelian groupX whose dual group X ̂ is cyclic w.r.t. α (i.e. X ̂ is generated by {χ · αn: n ∈ Z} for some χ ∈ X̂). Then there exists a canonical group homomorphism ξ from the space `∞(Z,Z) of all bounded twosided sequences of intege ..."
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Abstract. Let α be an expansive automorphisms of compact connected abelian groupX whose dual group X ̂ is cyclic w.r.t. α (i.e. X ̂ is generated by {χ · αn: n ∈ Z} for some χ ∈ X̂). Then there exists a canonical group homomorphism ξ from the space `∞(Z,Z) of all bounded twosided sequences of integers onto X such that ξ · σ = α · ξ, where σ is the shift on `∞(Z,Z). We prove that there exists a sofic subshift V ⊂ `∞(Z,Z) such that the restriction of ξ to V is surjective and almost onetoone. In the special case where α is a hyperbolic toral automorphism with a single eigenvalue β> 1 and all other eigenvalues of absolute value < 1 we show that, under certain technical and possibly unnecessary conditions, the restriction of ξ to the twosided betashift Vβ ⊂ `∞(Z,Z) is surjective and almost onetoone. The proofs are based on the study of homoclinic points and an associated algebraic construction of symbolic representations in [13] and [7]. Earlier results in this direction were obtained by Vershik ([24]–[26]), Kenyon and Vershik ([10]), and Sidorov and Vershik ([20]–[21]). 1.
Fifty years of entropy in dynamics: 1958–2007
 Journal of Modern Dynamics
"... These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical ” nature including allusions to certain episodes and discussion of attitudes and contributions o ..."
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These notes combine an analysis of what the author considers (admittedly subjectively) as the most important trends and developments related to the notion of entropy, with information of more “historical ” nature including allusions to certain episodes and discussion of attitudes and contributions of various participants. I directly participated in many of those developments for the last forty three or forty four years of the fiftyyear period under discussion and on numerous occasions was fairly close to the center of action. Thus, there is also an element of personal recollections with all attendant peculiarities of this genre. These notes are meant as “easy reading ” for a mathematically sophisticated reader familiar with most of the concepts which appear in the text. I recommend the book [59] as a source of background reading and the survey [44] (both written jointly with Boris Hasselblatt) as a reference source for virtually all necessary definitions and (hopefully) illuminating discussions. The origin of these notes lies in my talk at the dynamical systems branch of the huge conference held in Moscow in June of 2003 on the occasion of Kolmogorov’s
Lyapunov exponents and rates of mixing for onedimensional maps
, 2002
"... We show that one dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive some power of f is mixing and in particular the correlation of Hölder continuous observables decays to zero. The main obj ..."
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We show that one dimensional maps f with strictly positive Lyapunov exponents almost everywhere admit an absolutely continuous invariant measure. If f is topologically transitive some power of f is mixing and in particular the correlation of Hölder continuous observables decays to zero. The main objective of this paper is to show that the rate of decay of correlations is determined, in some situations, to the average rate at which typical points start to exhibit exponential growth of the derivative.
The structure of C*algebras associated with hyperbolic dynamical systems
 J. Funct. Anal
, 1999
"... The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of sresolving, finitetoone fa ..."
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The paper considers a mixing Smale space, the relations of stable and unstable equivalence on such a space and the C*algebras which are constructed from them. In general, these associations are not functorial. However, it is shown that, if one restricts to the class of sresolving, finitetoone factor maps, then the construction of the stable C*algebra is contravariant, while that of the unstable C*algebra is covariant. The paper also discusses the constructions of these C*algebras for Smale spaces which are not mixing. 1. Introduction and
Stochasticlike behaviour in nonuniformly expanding maps Handbook of Dynamical Systems
, 2006
"... 1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3 ..."
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1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3
Symbolic representations of nonexpansive group automorphisms
 Israel J. Math
"... Abstract. If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spi ..."
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Abstract. If α is an irreducible nonexpansive ergodic automorphism of a compact abelian group X (such as an irreducible nonhyperbolic ergodic toral automorphism), then α has no finite or infinite state Markov partitions, and there are no nontrivial continuous embeddings of Markov shifts in X. In spite of this we are able to construct a symbolic space V and a class of shiftinvariant probability measures on V each of which corresponds to an αinvariant probability measure on X. Moreover, every αinvariant probability measure on X arises essentially in this way. The last part of the paper deals with the connection between the twosided betashift Vβ arising from a Salem number β and the nonhyperbolic ergodic toral automorphism α arising from the companion matrix of the minimal polynomial of β, and establishes an entropypreserving correspondence between a class of shiftinvariant probability measures on Vβ and certain αinvariant probability measures on X. This correspondence is much weaker than, but still quite closely modelled on, the connection between the twosided betashifts defined by Pisot numbers and the corresponding hyperbolic ergodic toral automorphisms. 1.
Generalized zetafunctions for Axiom A basic sets
 Bull. Amer. Math. Soc
, 1976
"... Let X be a set, ƒ: X \—> X a map, <p: X \—> C a complexvalued function. We write formally ^(^=exp[f: i z "n */**) L n=l n CGFixf " k = 0 J Taking if constant, i.e. replacing <p by z G C, we can interpret l/D(z) as a zetafunction proved to be rational for Axiom A diffeomorphi ..."
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Let X be a set, ƒ: X \—> X a map, <p: X \—> C a complexvalued function. We write formally ^(^=exp[f: i z "n */**) L n=l n CGFixf " k = 0 J Taking if constant, i.e. replacing <p by z G C, we can interpret l/D(z) as a zetafunction proved to be rational for Axiom A diffeomorphisms by Guckenheimer and Manning [6]. Similarly, if ( ƒ *) is a flow on X, we write formally where the product extends over the periodic orbits 7 of the flow, X(7) is the prime period of 7 and x a point of 7. In this note we indicate analyticity properties of A — • D(eA) or A — • d(A) for diffeomorphisms or flows satisfying Smale's Axiom A, assuming only that A is Holder continuous. Our results hold in particular for Anosov diffeomorphisms and flows, and when A is Cl. Stronger properties of meromorphy hold under suitable assumptions of realanalyticity and will be published elsewhere by P. Cartier and the author. Let A be a basic set for a C1 diffeomorphisms or flow satisfying Smale's Axiom A (see [13]). Choosing a Riemann metric d, and a G (0, 1) we let Ca be the Banach space of real Holder continuous functions of exponent a, with the norm ( \A(y)A(x) \) \\A\\a = sup {1,4001 + 77, 7Z—'x, ye A and * *y> I (d(x, y))a) We denote by CQ the corresponding space of complex functions. 1. THEOREM. Let the Axiom A diffeomorphism f restricted to the basic set A be topologically mixing. We denote by P(A) the {topological) pressure of a real continuous function A on A (see [8] , [14], [4]). There is a continuous real function R on CQ satisfying