The K-Property for Some Unique Equilibrium States in Flows and Homeomorphisms

Call, Benjamin

doi:10.48550/arxiv.2007.00035

Cited by 4 publications

(8 citation statements)

References 22 publications

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“…With the pressure gap condition for such potentials in hand we briefly note a second class of potentials for which it holds. Proposition 4.7 of [Cal20] notes that if the pressure gap P (Sing, φ) < P (φ) holds for φ, then for any function sufficiently close to φ (specifically with 2 φ − ψ < P (φ) − P (Sing, φ)) and any constant c, P (Sing, ψ + c) < P (ψ + c). Applying this to the locally constant functions φ discussed in this section gives us a further class of potentials with a pressure gap.…”

Section: Note That E ′′mentioning

confidence: 99%

“…For such a 'λ-decomposition', P = S and, roughly speaking, orbit segments in P and S have small average values of λ wheareas any initial or terminal segment of an element of G has average value of λ which is not small. Furthermore, by utilizing a λ-decomposition, we are able to appeal to the following result, which can be seen by combining [CT19, Theorem 6.3] and [Cal20, Theorem 2.21 & Corollary 4.3]: Cal20]). Let F be a continuous flow on a compact metric space X, and let φ : X → R be continuous.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Unique equilibrium states for geodesic flows on flat surfaces with singularities

Call¹,

Constantine²,

Erchenko³

et al. 2021

Preprint

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Consider a compact surface of genus ≥ 2 equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than 2π. Following the technique in the work of Burns, Climenhaga, Fisher, and Thompson, we prove that sufficiently regular potential functions have unique equilibrium states if the singular set does not support the full pressure. Moreover, we show that the pressure gap holds for any potential which is locally constant on a neighborhood of the singular set. Finally, we establish that the corresponding equilibrium states are weakly mixing and closed regular geodesics equidistribute.1.1. Outline of the argument. A general scheme for proving that unique equilibrium states exist was developed by Climenhaga and Thompson in [CT16], building on ideas of Bowen in [Bow75] which were extended to flows in [Fra77]. To prove that there are unique equilibrium states for a flow {f t } and a potential φ on a compact metric space X, Climenhaga and Thompson ask for the following (see [CT16, Theorems A & C]):• The pressure of obstructions to expansivity, P ⊥ exp (φ), is smaller than P (φ), and • There are three collections of orbit segments P, G, S, that we call collections of prefixes, good orbit segments, and suffixes, respectively, such that each orbit segment can be decomposed into a prefix, a good part, and a suffix (see [BCFT18, Definition 2.3]), satisfying

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Section: Note That E ′′mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Unique equilibrium states for geodesic flows on flat surfaces with singularities

Call¹,

Constantine²,

Erchenko³

et al. 2021

Preprint

Self Cite

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show abstract

“…20 We could also define the class of one-sided λ-decompositions by taking the longest initial segment in B(η), declaring what is left over to be good, and setting S = ∅, or conversely by putting S = B(η) and P = ∅. This formalism is defined in [Cal20]: the decompositions in §9.2 are examples of one-sided λ-decompositions.…”

Section: Part III Geodesic Flowsmentioning

confidence: 99%

“…The original result is for discrete-time systems. We state here a version of it for flows; the proof is given in [CT19a], and in more detail in [Cal20].…”

Section: Part III Geodesic Flowsmentioning

confidence: 99%

Beyond Bowen's Specification Property

Climenhaga¹,

Thompson²

2020

Preprint

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A classical result in thermodynamic formalism is that for uniformly hyperbolic systems, every Hölder continuous potential has a unique equilibrium state. One proof of this fact is due to Rufus Bowen and uses the fact that such systems satisfy expansivity and specification properties. In these notes, we survey recent progress that uses generalizations of these properties to extend Bowen's arguments beyond uniform hyperbolicity, including applications to partially hyperbolic systems and geodesic flows beyond negative curvature. We include a new criterion for uniqueness of equilibrium states for partially hyperbolic systems with 1-dimensional center.

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“…However, it is worth mentioning recent work of Call and Thompson[CT19,Cal20] that uses the specification approach to establish the K property, and even Bernoullicity in some cases.…”

mentioning

confidence: 99%

SRB and equilibrium measures via dimension theory

Climenhaga

2020

Preprint

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It is well-known that SRB and equilibrium measures for uniformly hyperbolic flows admit a product structure in terms of measures on stable and unstable leaves with scaling properties given by the potential function. We describe a construction of these leaf measures analogous to the definition of Hausdorff measure, relying on the Pesin-Pitskel' description of topological pressure as a dimensional characteristic using Bowen balls. These leaf measures were constructed for discrete-time systems by the author, Ya. Pesin, and A. Zelerowicz. In the continuous-time setting here, the description of the scaling properties is more complete, and we use a similar procedure with two-sided Bowen balls to directly produce the equilibrium measure itself.For Tolya, who taught me to learn things a second time.

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The K-Property for Some Unique Equilibrium States in Flows and Homeomorphisms

Cited by 4 publications

References 22 publications

Unique equilibrium states for geodesic flows on flat surfaces with singularities

Unique equilibrium states for geodesic flows on flat surfaces with singularities

Beyond Bowen's Specification Property

SRB and equilibrium measures via dimension theory

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