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Statistical stability for Barge-Martin attractors derived from tent maps

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

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  • 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 $.

    Mathematics Subject Classification: 37A10, 37B45, 37C75, 37E05.


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