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Benjamin Call Thesis v3.pdf (809.2 KB)

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Uniqueness and Mixing Properties of Equilibrium States

Call, Benjamin
http://orcid.org/0000-0003-2964-6342
http://rave.ohiolink.edu/etdc/view?acc_num=osu1656082212974144

Abstract Details

2022, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we study uniqueness and mixing properties of equilibrium states for systems beyond uniform hyperbolicity. The main results are contained in Chapters 3-7. The results in Chapter 3 are focused on abstract thermodynamic formalism. We formalize a type of decomposition which one can use to apply the Climenhaga-Thompson machinery for establishing uniqueness of equilibrium states, termed λ-decompositions. In this chapter, we establish various properties of these decompositions and prove that they can be used to show higher mixing properties of equilibrium states, such as weak mixing and the K-property. Then in Chapters 4 and 5, we discuss applications of these results to geometrically interesting systems. In particular, we establish the K-property for equilibrium states for the geodesic flow on rank one nonpositively curved manifolds. We also sketch the proof of uniqueness and the K-property for the geodesic flow on translation surfaces, which are a class of CAT(0) spaces. This is the first result on uniqueness of equilibrium states for non-zero potentials in this setting, and the full proof is given in [18]. The K-property for measures is often used as a stepping stone to the Bernoulli property, the strongest qualitative mixing property a dynamical system can have. Ornstein and Weiss, followed by Chernov and Haskell, provided a blueprint for how to bootstrap from the K-property to the Bernoulli property for smooth K-systems with some amount of hyperbolicity. In Chapter 6, we show how to apply these arguments to non-smooth measures, requiring only a local product structure property of the measure in question. In this chapter, we build off of the result from Chapter 4 to show that the measure of maximal entropy for the geodesic flow on rank one nonpositively curved manifolds is Bernoulli. Chapter 7 concerns subadditive thermodynamic formalism. In this chapter, we establish the K-property for equilibrium states for some subadditive potentials on shift spaces, as well as discussing more general results on compact metric spaces. This builds on work of Morris, and provides a new way of showing higher mixing properties of subadditive equilibrium states.
Daniel Thompson (Advisor)
Vitaly Bergelson (Committee Member)
Andrey Gogolyev (Committee Member)
Steven MacEachern (Committee Member)
161 p.
Mathematics
Ergodic theory, thermodynamic formalism, K-property, Bernoulli property, hyperbolic dynamics

Recommended Citations

Citations

  • Call, B. (2022). Uniqueness and Mixing Properties of Equilibrium States [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1656082212974144

    APA Style (7th edition)

  • Call, Benjamin. Uniqueness and Mixing Properties of Equilibrium States. 2022. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1656082212974144.

    MLA Style (8th edition)

  • Call, Benjamin. "Uniqueness and Mixing Properties of Equilibrium States." Doctoral dissertation, Ohio State University, 2022. http://rave.ohiolink.edu/etdc/view?acc_num=osu1656082212974144

    Chicago Manual of Style (17th edition)

osu1656082212974144
258
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This open access ETD is published by The Ohio State University and OhioLINK.