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Bulletin of the American Mathematical Society

Published by the American Mathematical Society, the Bulletin of the American Mathematical Society (BULL) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.47.

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Equilibrium states in dynamical systems via geometric measure theory
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by Vaughn Climenhaga, Yakov Pesin and Agnieszka Zelerowicz PDF
Bull. Amer. Math. Soc. 56 (2019), 569-610 Request permission

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
  • MR Author ID: 852541
  • Email: climenha@math.uh.edu
  • Yakov Pesin
  • Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 138355
  • Email: pesin@math.psu.edu
  • Agnieszka Zelerowicz
  • Affiliation: Department of Mathematics, Pennsylvania State University, University Park, Pennsylvania 16802
  • MR Author ID: 1224879
  • Email: axz157@psu.edu
  • 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.
  • © Copyright 2018 American Mathematical Society
  • 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
  • MathSciNet review: 4007162