Ergodic Theory and Dynamical Systems

Publisher: Cambridge University Press

Description

  • Impact factor
    0.87
  • 5-year impact
    1.02
  • Cited half-life
    0.00
  • Immediacy index
    0.14
  • Eigenfactor
    0.01
  • Article influence
    1.21
  • Other titles
    Ergodic theory and dynamical systems (Online), Ergodic theory and dynamical systems
  • ISSN
    1469-4417
  • OCLC
    41949087
  • Material type
    Document, Periodical, Internet resource
  • Document type
    Internet Resource, Computer File, Journal / Magazine / Newspaper

Publisher details

Cambridge University Press

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    • Author can archive a pre-print version
  • Post-print
    • Author can archive a post-print version
  • Conditions
    • Author's Pre-print on author's personal website, departmental website, social media websites, institutional repository, non-commercial subject-based repositories, such as PubMed Central, Europe PMC or arXiv
    • Author's post-print for HSS journals, on author's personal website, departmental website, institutional repository, non-commercial subject-based repositories, such as PubMed Central, Europe PMC or arXiv, on acceptance of publication
    • Author's post-print for STM journals, on author's personal website on acceptance of publication
    • Author's post-print for STM journals, on departmental website, institutional repository, non-commercial subject-based repositories, such as PubMed Central, Europe PMC or arXiv, after a 6 months embargo
    • Publisher's version/PDF cannot be used
    • Published abstract may be deposited
    • Pre-print to record acceptance for publication
    • Publisher copyright and source must be acknowledged with set statement, for deposit of Authors Post-print or Publisher's version/PDF
    • Must link to publisher version
    • Publisher last reviewed on 07/10/2014
  • Classification
    ​ green

Publications in this journal

  • [Show abstract] [Hide abstract]
    ABSTRACT: Bratteli–Vershik systems have been widely studied. In the context of general zero-dimensional systems, Bratteli–Vershik systems are homeomorphisms that have Kakutani–Rohlin refinements. Bratteli diagrams are well suited to analyzing such systems. Besides this approach, general graph covers can be used to represent any zero-dimensional system. Indeed, all zero-dimensional systems can be described as certain kinds of sequences of graph covers that may not be brought about by Kakutani–Rohlin partitions. In this paper, we follow the context of general graph covers to analyze the relations between ergodic measures and circuits of graph covers. First, we formalize the condition for a sequence of graph covers to represent minimal Cantor systems. In constructing invariant measures, we deal with general compact metrizable zero-dimensional systems. In the context of Bratteli diagrams with finite rank, it has previously been mentioned that all ergodic measures should be limits of some combinations of towers of Kakutani–Rohlin refinements. We demonstrate this for the general zero-dimensional case, and develop a theorem that expresses the coincidence of the time average and the space average for ergodic measures. Additionally, we formulate a theorem that signifies the old relation between uniform convergence and unique ergodicity in the context of graph circuits for general zero-dimensional systems. Unlike previous studies, in our case of general graph covers there arises the possibility of the linear dependence of circuits. We give a condition for a full circuit system to be linearly independent. Previous research also showed that the bounded combinatorics imply unique ergodicity. We present a lemma that enables us to consider unbounded ranks of winding matrices. Finally, we present examples that are linked with a set of simple Bratteli diagrams having the equal path number property.
    Ergodic Theory and Dynamical Systems 10/2014;
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    ABSTRACT: In this paper some aspects on chaotic behavior and minimality in planar piecewise smooth vector fields theory are treated. The occurrence of non-deterministic chaos is observed and the concept of orientable minimality is introduced. It is also investigated some relations between minimality and orientable minimality and observed the existence of new kinds of non-trivial minimal sets in chaotic systems. The approach is geometrical and involve the ordinary techniques of non-smooth systems.
    Ergodic Theory and Dynamical Systems 04/2014;
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    ABSTRACT: We consider subshifts of the full shift of bi-infinite sequences with alphabet \$\{ 0, 1, \ldots , n- 1\} \$ defined by not allowing the sum of finite words to exceed a value depending on its length. These shifts we call bounded density shifts. We study these shifts in detail and make a comparison on the similarities to but also differences from the well-known \$\beta \$-shifts.
    Ergodic Theory and Dynamical Systems 12/2013; 33(06).
  • Ergodic Theory and Dynamical Systems 12/2013;
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    ABSTRACT: We demonstrate essential coexistence of hyperbolic and non-hyperbolic behavior in the continuous-time case by constructing a smooth volume preserving flow on a five-dimensional compact smooth manifold that has non-zero Lyapunov exponents almost everywhere on an open and dense subset of positive but not full volume and is ergodic on this subset while having zero Lyapunov exponents on its complement. The latter is a union of three-dimensional invariant submanifolds, and on each of these submanifolds the flow is linear with Diophantine frequency vector.
    Ergodic Theory and Dynamical Systems 12/2013; 33(06).
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    ABSTRACT: We study dynamics of continuous maps on compact metrizable spaces containing a free interval (i.e. an open subset homeomorphic to an open interval). Special attention is paid to relationships between topological transitivity, weak and strong topological mixing, dense periodicity and topological entropy as well as to the topological structure of minimal sets. In particular, a trichotomy for minimal sets and a dichotomy for transitive maps are proved.
    Ergodic Theory and Dynamical Systems 12/2013; 33(06).
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    ABSTRACT: Leaves of laminations can be symbolically represented by deforming them into paths of labeled embedded carrier graphs, including train tracks. Here, we describe and characterize the languages of two-way infinite words coming from this kind of coding, called lamination languages, first, by using carrier graph sequences, and second, by using word combinatorics. These characterizations generalize those existing for interval exchange transformations. We also show that lamination languages have ultimately affine factor complexity, and we present effective techniques to build these languages.
    Ergodic Theory and Dynamical Systems 12/2013; 33(06).
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    ABSTRACT: Let Ω be a class of unital C * -algebras. Then any simple unital C * -algebra A∈TA(TAΩ) is a TAΩ algebra. Let A∈TAΩ be an infinite-dimensional α-simple unital C * -algebra with the property SP. Suppose that α:G→Aut(A) is an action of a finite group G on A which has a certain non-simple tracial Rokhlin property. Then the crossed product algebra C * (G,A,α) belongs to TAΩ.
    Ergodic Theory and Dynamical Systems 10/2013; 33(05).
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    ABSTRACT: The author extends her earlier results from [J. Lond. Math. Soc., II. Ser. 84, No. 3, 785–806 (2011; Zbl 1263.37012)] to include geometrically infinite surfaces: a vector whose full horocyclic orbit is dense on a non-elementary geometrically infinite oriented hyperbolic surface has both half-orbits (forward and backward) dense if the geodesic flow intersects infinitely many closed geodesic of bounded length at an angle bounded away from zero (Theorem 1.2). The necessity of this assumption is exhibited in Theorem 1.3, a construction in which the bound on the lengths of the closed geodesics is removed and the resulting horocycle is only dense in one direction.
    Ergodic Theory and Dynamical Systems 08/2013; 33(04).
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    ABSTRACT: The Ruelle operator has been studied extensively both in dynamical systems and iterated function systems (IFSs). Given a weakly contractive IFS \$(X, \{w_j\}_{j=1}^m)\$ and an associated family of positive continuous potential functions \$\{p_j\}_{j=1}^m\$, a triple system \$(X, \{w_j\}_{j=1}^m, \{p_j\}_{j=1}^m)\$is set up. In this paper we study Ruelle operators associated with the triple systems. The paper presents an easily verified condition. Under this condition, the Ruelle operator theorem holds provided that the potential functions are Dini continuous. Under the same condition, the Ruelle operator is quasi-compact, and the iterations sequence of the Ruelle operator converges with a specific geometric rate, if the potential functions are Lipschitz continuous.
    Ergodic Theory and Dynamical Systems 08/2013; 33(04).
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    ABSTRACT: This paper contains results on the geometric and ergodic properties of a class of strange attractors introduced by Wang and Young [Towards a theory of rank one attractors. Ann. of Math. (2) 167 (2008), 349–480]. These attractors can live in phase spaces of any dimension, and have been shown to arise naturally in differential equations that model several commonly occurring phenomena. Dynamically, such systems are chaotic; they have controlled non-uniform hyperbolicity with exactly one unstable direction, hence the name rank-one. In this paper we prove theorems on their Lyapunov exponents, Sinai–Ruelle–Bowen (SRB) measures, basins of attraction, and statistics of time series, including central limit theorems, exponential correlation decay and large deviations. We also present results on their global geometric and combinatorial structures, symbolic coding and periodic points. In short, we build a dynamical profile for this class of dynamical systems, proving that these systems exhibit many of the characteristics normally associated with ‘strange attractors’.
    Ergodic Theory and Dynamical Systems 08/2013; 33(04).
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    ABSTRACT: If a partially hyperbolic diffeomorphism on a torus of dimension \$d\geq 3\$ has stable and unstable foliations which are quasi-isometric on the universal cover, and its centre direction is one-dimensional, then the diffeomorphism is leaf conjugate to a linear toral automorphism. In other words, the hyperbolic structure of the diffeomorphism is exactly that of a linear, and thus simple to understand, example. In particular, every partially hyperbolic diffeomorphism on the 3-torus is leaf conjugate to a linear toral automorphism.
    Ergodic Theory and Dynamical Systems 06/2013; 33(03).
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    ABSTRACT: We give an almost self-contained group theoretic proof of Furstenberg’s structure theorem as generalized by Ellis: each minimal compact distal flow is the result of a transfinite sequence of equicontinuous extensions, and their limits, starting from a flow consisting of a singleton. The groups that we use are CHART groups, and their basic properties are recalled at the beginning of the paper.
    Ergodic Theory and Dynamical Systems 06/2013; 33(03).
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    ABSTRACT: Beyond the uncoupled regime, the rigorous description of the dynamics of (piecewise) expanding coupled map lattices remains largely incomplete. To address this issue, we study repellers of periodic chains of linearly coupled Lorenz-type maps which we analyze by means of symbolic dynamics. Whereas all symbolic codes are admissible for sufficiently small coupling intensity, when the interaction strength exceeds a chain length independent threshold, we prove that a large bunch of codes is pruned and an extensive decay follows suit for the topological entropy. This quantity, however, does not immediately drop off to 0. Instead, it is shown to be continuous at the threshold and to remain extensively bounded below by a positive number in a large part of the expanding regime. The analysis is firstly accomplished in a piecewise affine setting where all calculations are explicit and is then extended by continuation to coupled map lattices based on \$C^1\$-perturbations of the individual map.
    Ergodic Theory and Dynamical Systems 06/2013; 33(03).
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    ABSTRACT: Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.
    Ergodic Theory and Dynamical Systems 04/2013; 33(02).
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    ABSTRACT: We give a complete characterization of the compact metric dynamical systems that appear as boundaries of the canonical compactification of a locally compact countable state mixing Markov shift. Consider such a compact metric dynamical system. Then there is a pair of non-conjugate Markov shifts with conjugate canonical compactifications, one of which has the given compact system as canonical boundary.
    Ergodic Theory and Dynamical Systems 04/2013; 33(02).