Statistical Stability for Barge-Martin attractors derived from tent maps

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 ${\Phi }_t:D^2\to D^2$, 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 ${\chi }_t:S^2\to S^2$ of transitive sphere homeomorphisms, constructed in [17] as factors of the natural extensions of $f_t$.

Author accepted version