Statistical Stability for Barge-Martin Attractors Derived from Tent Maps



Hall, Toby, Boyland, Philip and de Carvalho, André
(2020) Statistical Stability for Barge-Martin Attractors Derived from Tent Maps. Discrete and Continuous Dynamical Systems Series A.

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Abstract

Let $\{f_t\}_{t\in(1,2]}$ be the family of core tent maps of slopes~$t$. The parameterized Barge-Martin construction yields a family of disk homeomorphisms $\Phi_t\colon D^2\to D^2$, having transitive global attractors $\Lambda_t$ on which $\Phi_t$ is topologically conjugate to the natural extension of $f_t$. The unique family of absolutely continuous invariant measures for $f_t$ induces a family of ergodic $\Phi_t$-invariant measures $\nu_t$, supported on the attractors~$\Lambda_t$. We show that this family $\nu_t$ varies weakly continuously, and that the measures $\nu_t$ are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures $\rho_t$. Similar results are obtained for the family $\chi_t\colon S^2\to S^2$ of transitive sphere homeomorphisms, constructed in a previous paper of the authors as factors of the natural extensions of~$f_t$.

Item Type: Article
Additional Information: Author accepted manuscript
Depositing User: Symplectic Admin
Date Deposited: 11 Dec 2019 09:27
Last Modified: 19 Jan 2023 00:13
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URI: https://livrepository.liverpool.ac.uk/id/eprint/3065881