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Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Equilibrium states in dynamical systems via geometric measure theory


Authors: Vaughn Climenhaga, Yakov Pesin and Agnieszka Zelerowicz
Journal: Bull. Amer. Math. Soc. 56 (2019), 569-610
MSC (2010): Primary 37D35, 37C45; Secondary 37C40, 37D20
DOI: https://doi.org/10.1090/bull/1659
Published electronically: December 10, 2018
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Abstract: Given a dynamical system with a uniformly hyperbolic (chaotic) attractor, the physically relevant Sinaĭ-Ruelle-Bowen (SRB) measure can be obtained as the limit of the dynamical evolution of the leaf volume along local unstable manifolds. We extend this geometric construction to the substantially broader class of equilibrium states corresponding to Hölder continuous potentials; these states arise naturally in statistical physics and play a crucial role in studying stochastic behavior of dynamical systems. The key step in our construction is to replace leaf volume with a reference measure that is obtained from a Carathéodory dimension structure via an analogue of the construction of Hausdorff measure. In particular, we give a new proof of existence and uniqueness of equilibrium states that does not use standard techniques based on Markov partitions or the specification property; our approach can be applied to systems that do not have Markov partitions and do not satisfy the specification property.


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Additional Information

Vaughn Climenhaga
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204
Email: climenha@math.uh.edu

Yakov Pesin
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: pesin@math.psu.edu

Agnieszka Zelerowicz
Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
Email: axz157@psu.edu

DOI: https://doi.org/10.1090/bull/1659
Received by editor(s): March 28, 2018
Received by editor(s) in revised form: October 23, 2018
Published electronically: December 10, 2018
Additional Notes: The first author was partially supported by NSF grants DMS-1362838 and DMS-1554794.
The second and third authors were partially supported by NSF grant DMS-1400027.
Article copyright: © Copyright 2018 American Mathematical Society