# American Institute of Mathematical Sciences

May  2020, 40(5): 2903-2915. doi: 10.3934/dcds.2020154

## Statistical stability for Barge-Martin attractors derived from tent maps

 1 Department of Mathematics, University of Florida, 372 Little Hall, Gainesville, FL 32611-8105, USA 2 Departamento de Matemática Aplicada, IME-USP, Rua Do Matão 1010, Cidade Universitária, 05508-090 São Paulo SP, Brazil 3 Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK

Received  August 2019 Revised  November 2019 Published  March 2020

Fund Project: The authors are grateful for the support of FAPESP grant 2016/25053-8 and CAPES grant 88881.119100/2016-01.

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 $f_t$.

Citation: Philip Boyland, André de Carvalho, Toby Hall. Statistical stability for Barge-Martin attractors derived from tent maps. Discrete & Continuous Dynamical Systems, 2020, 40 (5) : 2903-2915. doi: 10.3934/dcds.2020154
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