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Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems

Wang, Tianyu
http://rave.ohiolink.edu/etdc/view?acc_num=osu1618937837609911

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2021, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we concentrate on the study of different properties of non-uniformly hyperbolic systems. From section 1 to section 3, we introduce our main results, preliminaries and main techniques. The core of the thesis is in section 4 - section 6, where we illustrate the detailed proofs of our main results. The first result is on thermodynamic formalism. In section 4, we work with a C^1 non-uniformly hyperbolic diffeomorphism on the 2-torus, known as the Katok map. We prove for a Holder continuous potential with one additional condition, or geometric t-potential with t < 1, the equilibrium state exists and is unique. Motivated by the `Orbit Decomposition' technique we used in the derivation of the thermodynamic formalism, we also obtain a weak version of Gibbs property and the level-2 large deviation principle for the equilibrium state from above. In section 5, we prove an asymptotic version of the Central Limit Theorem for the unique measure of maximal entropy of the geodesic flows on rank-one non-positively curved manifold with Holder observables. We generalize an approach of Denker, Senti and Zhang from the discrete case with uniform expansiveness and specification to the continuous flow where only partial specification holds, and simplify the condition so that only Lindeberg condition and a weak positive variance condition are required. Moreover, we show that the Lindeberg condition follows from a strong positive variance condition which parallels the one used in the classic study of Central Limit Theorem in dynamics. We also extend our results to dynamical arrays of Holder observables, and to weighted periodic orbit measures which converge to a unique equilibrium state. Finally, in section 6, we investigate how results in thermodynamic formalism of non-uniformly hyperbolic systems can benefit the study of dimension theory and multifractal analysis in those cases. The main example we study here is the topological entropy and Hausdorff dimension of Lyapunov level sets in the case of the geodesic flow on rank-one surfaces with no focal points.
Daniel Thompson, Dr. (Advisor)
Jean-Francois Lafont, Dr. (Committee Member)
Nimish Shah, Dr. (Committee Member)
Linda Mizejewski, Dr. (Committee Member)
123 p.
Mathematics
non-uniformly hyperbolic system, thermodynamic formalism, Katok map, large deviation principle, Central limit theorem, Lyapunov exponent, multifractal analysis

Recommended Citations

Citations

  • Wang, T. (2021). Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618937837609911

    APA Style (7th edition)

  • Wang, Tianyu. Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems. 2021. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1618937837609911.

    MLA Style (8th edition)

  • Wang, Tianyu. "Thermodynamic formalism, statistical properties and multifractal analysis of non-uniformly hyperbolic systems." Doctoral dissertation, Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618937837609911

    Chicago Manual of Style (17th edition)

osu1618937837609911
243
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This open access ETD is published by The Ohio State University and OhioLINK.