The <i>K</i>-property for some unique equilibrium states in flows and homeomorphisms

Call, Benjamin

doi:10.1017/etds.2021.61

Cited by 2 publications

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“…These generalizations are not pursued here since it would distract from the main ideas necessary for our analysis. We note that some further applications of our approach in both continuous and discrete‐time have been explored in the preprint [10] by the first‐named author, which is a sequel to this paper.…”

Section: Introductionmentioning

confidence: 99%

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

Call

Thompson

2022

Journal of London Math Soc

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Equilibrium states for geodesic flows over closed rank 1 manifolds were studied recently in Burns, Climenhaga, Fisher, and Thompson [Geom. Funct. Anal. 28 (2018), no. 5, 1209–1259]. For sufficiently regular potentials, it was shown that if the singular set does not carry full pressure, then the equilibrium state is unique. The main result of this paper is that these equilibrium states have the Kolmogorov property. In particular, these measures are mixing of all orders and have positive entropy. For the Bowen‐Margulis measure, we go further and obtain the Bernoulli property from the Kolmogorov property using classic arguments from Ornstein theory. Our argument for the Kolmogorov property is based on an idea due to Ledrappier. We prove uniqueness of equilibrium states on the product of the system with itself. To carry this out, we develop techniques for uniqueness of equilibrium states which apply in the presence of the 2‐dimensional center direction which appears for a product of flows. This is a key technical challenge of this paper.

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Section: Introductionmentioning

confidence: 99%

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

Call

Thompson

2022

Journal of London Math Soc

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Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities

Call

Constantine

Erchenko

et al. 2022

International Mathematics Research Notices

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Consider a compact surface of genus $\geq 2$ equipped with a metric that is flat everywhere except at finitely many cone points with angles greater than $2\pi $. 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 that is locally constant on a neighborhood of the singular set. Finally, we establish that the corresponding equilibrium states have the $K$-property and closed regular geodesics equidistribute.

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The K-property for some unique equilibrium states in flows and homeomorphisms

Cited by 2 publications

References 28 publications

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

Equilibrium states for self‐products of flows and the mixing properties of rank 1 geodesic flows

Unique Equilibrium States for Geodesic Flows on Flat Surfaces with Singularities

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