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Gibbs measures for hyperbolic attractors defined by densities
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Parmenter, David and Pollicott, Mark (2022) Gibbs measures for hyperbolic attractors defined by densities. Discrete and Continuous Dynamical Systems. Series A, 42 (8). pp. 3953-3977. doi:10.3934/dcds.2022038 ISSN 1078-0947.
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WRAP-Gibbs-measures-hyperbolic-attractors-defined-densitites-2022.pdf - Accepted Version - Requires a PDF viewer. Download (586Kb) | Preview |
Official URL: https://doi.org/10.3934/dcds.2022038
Abstract
In this article we will describe a new construction for Gibbs measures for hyperbolic attractors generalizing the original construction of Sinai, Bowen and Ruelle of SRB measures. The classical construction of the SRB measure is based on pushing forward the normalized volume on a piece of unstable manifold. By modifying the density at each step appropriately we show that the resulting measure is a prescribed Gibbs measure. This contrasts with, and complements, the construction of Climenhaga-Pesin-Zelerowicz who replace the volume on the unstable manifold by a fixed reference measure. Moreover, the simplicity of our proof, which uses only explicit properties on the growth rate of unstable manifold and entropy estimates, has the additional advantage that it applies in more general settings.
| Item Type: | Journal Article | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
| Library of Congress Subject Headings (LCSH): | Probabilities, Differentiable dynamical systems, Differential equations, Partial, Geometry, Hyperbolic, Measure theory | |||||||||
| Journal or Publication Title: | Discrete and Continuous Dynamical Systems. Series A | |||||||||
| Publisher: | American Institute of Mathematical Sciences | |||||||||
| ISSN: | 1078-0947 | |||||||||
| Official Date: | August 2022 | |||||||||
| Dates: |
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| Volume: | 42 | |||||||||
| Number: | 8 | |||||||||
| Page Range: | pp. 3953-3977 | |||||||||
| DOI: | 10.3934/dcds.2022038 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Reuse Statement (publisher, data, author rights): | This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Discrete and Continuous Dynamical Systems. Series A following peer review. The definitive publisher-authenticated version [insert complete citation information here] is available online at: xxxxxxx [insert URL that the author will receive upon publication here]. | |||||||||
| Access rights to Published version: | Restricted or Subscription Access | |||||||||
| Date of first compliant deposit: | 7 March 2022 | |||||||||
| Date of first compliant Open Access: | 7 March 2022 | |||||||||
| RIOXX Funder/Project Grant: |
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