Orbit complexity and computable Markov partitions

    Research output: ThesisDoctoral Thesis

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    Abstract

    Markov partitions provide a 'good' mechanism of symbolic dynamics for uniformly hyperbolic systems, forming the classical foundation for the thermodynamic formalism in this setting, and remaining useful in the modern theory. Usually, however, one takes Bowen's 1970's general construction for granted, or restricts to cases with simpler geometry (as on surfaces) or more algebraic structure. This thesis examines several questions on the algorithmic content of (topological) Markov partitions, starting with the pointwise, entropy-like, topological conjugacy invariant known as orbit complexity. The relation between orbit complexity de nitions of Brudno and Galatolo is examined in general compact spaces, and used in Theorem 2.0.9 to bound the decrease in some of these quantities under semiconjugacy. A corollary, and a pointwise analogue of facts about metric entropy, is that any Markov partition produces symbolic dynamics matching the original orbit complexity at each point. A Lebesgue-typical value for orbit complexity near a hyperbolic attractor is also established (with some use of Brin-Katok local entropy), and is technically distinct from typicality statements discussed by Galatolo, Bonanno and their co-authors. Both our results are proved adapting classical arguments of Bowen for entropy. Chapters 3 and onwards consider the axiomatisation and computable construction of Markov partitions. We propose a framework of 'abstract local product structures' - in principle generalising de nitions of Fried ( nitely presented systems) and Adler - supplemented by shadowing properties for our focus on locally maximal hyperbolic sets. Our approach follows computable analysis via representations (TTE), though even on LMHSs it must be seen as partial; we do not treat e.g. topological transitivity, discontinuity results, positive computational information for local (un)stable manifolds, or partition construction after Fried (1987). Nevertheless, under one characterisation of computability for the coding, we observe a 'natural' effective version of the subdivision used by Bowen assumes more information on interiors of initial rectangles than is known to us from shadowing. Chapter 5 describes positive and negative results on the construction of single rectangles with such information. These have a simple geometric avour not involving dynamical invariance, but may help computable study of the local product across di erent classes of hyperbolic set. Finally, we follow a construction, found in Palis & Takens (1993), of Markov partitions from stable & unstable segment families for topologically mixing basic sets on surfaces. Such partitions are always computable, but more careful parametrization of the construction is desirable for uniform questions. All details can straightforwardly be made constructive with geometric estimates, but how canonical are these? Instead of a more topological approach, we try to extract 'good' geometric arguments: the estimates are selected to allow relatively coarse partitions, and often countable segment families.
    Original languageEnglish
    QualificationDoctor of Philosophy
    Publication statusUnpublished - 2008

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