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43
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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Cited by 28 (6 self)
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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...
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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Cited by 24 (0 self)
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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.
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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Cited by 10 (7 self)
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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.
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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Cited by 10 (1 self)
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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
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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Cited by 9 (1 self)
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1.1. Determinism versus randomness 2 1.2. Nonuniform expansivity 2 1.3. General overview of the notes 3
SpatioTemporal Chaos. 1. Hyperbolicity, Structural Stability, SpatioTemporal Shadowing and Symbolic Dynamics
"... In a series of three papers, we study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinitedimensional analogues of Axiom A systems. Our main result is the existence of a natural spatiotemporal measure which is the spatiotemporal a ..."
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Cited by 8 (3 self)
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In a series of three papers, we study the geometrical and statistical structure of a class of coupled map lattices with natural couplings. These are infinitedimensional analogues of Axiom A systems. Our main result is the existence of a natural spatiotemporal measure which is the spatiotemporal analogue of the SRB measure. In this paper we develop a stable manifold theory for such systems as well as spatiotemporal shadowing, Markov partitions and symbolic dynamics. In the second, we treat in general terms the question of the existence and uniqueness of Gibbs states for the associated higherdimensional symbolic systems. The final paper contains the proof of the main theorem which asserts the existence and uniqueness of a natural spatiotemporal measure for certain weakly coupled circle map lattices with a natural coupling. Key words: spatially extended system, coupled map lattice, Axiom A, hyperbolicity, structural stability, stable manifold, shadowing lemma, Markov partition, Gibb...
ON THE CONTINUITY OF THE SRB ENTROPY FOR ENDOMORPHISMS
"... Abstract. We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of nonuni ..."
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Cited by 7 (1 self)
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Abstract. We consider classes of dynamical systems admitting Markov induced maps. Under general assumptions, which in particular guarantee the existence of SRB measures, we prove that the entropy of the SRB measure varies continuously with the dynamics. We apply our result to a vast class of nonuniformly expanding maps of a compact manifold and prove the continuity of the entropy of the SRB measure. In particular, we show that the SRB entropy of Viana maps varies continuously with the map.
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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Cited by 7 (3 self)
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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.
A finitary classification of topological Markov chains and sofic systems
 Bull. London Math. Soc
, 1977
"... In [9] a classification theory for topological Markov chains (shifts of finite type, intrinsic Markov chains) appears which, although incomplete [10], provides algebraic criteria for deciding when two such chains are topologically conjugate. Numerous invariants effectively exclude topological conjug ..."
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Cited by 5 (1 self)
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In [9] a classification theory for topological Markov chains (shifts of finite type, intrinsic Markov chains) appears which, although incomplete [10], provides algebraic criteria for deciding when two such chains are topologically conjugate. Numerous invariants effectively exclude topological conjugacy for many interesting examples.
Arithmetic Dynamics
, 2002
"... This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepres ..."
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Cited by 5 (1 self)
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This survey paper is aimed to describe a relatively new branch of symbolic dynamics which we call Arithmetic Dynamics. It deals with explicit arithmetic expansions of reals and vectors that have a “dynamical” sense. This means precisely that they (semi) conjugate a given continuous (or measurepreserving) dynamical system and a symbolic one. The classes of dynamical systems and their codings considered in the paper involve: • Betaexpansions, i.e., the radix expansions in noninteger bases; • “Rotational ” expansions which arise in the problem of encoding of irrational rotations of the circle; • Toral expansions which naturally appear in arithmetic symbolic codings of algebraic toral automorphisms (mostly hyperbolic). We study ergodictheoretic and probabilistic properties of these expansions and their applications. Besides, in some cases we create “redundant” representations (those whose space of “digits ” is a priori larger than necessary)