Statistical stability for Barge-Martin attractors derived from tent maps

Abstract

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ \{f_t\}_{t\in(1,2]} $\end{document}</tex-math></inline-formula> be the family of core tent maps of slopes <inline-formula><tex-math id="M2">\begin{document}$ t $\end{document}</tex-math></inline-formula>. The parameterized Barge-Martin construction yields a family of disk homeomorphisms <inline-formula><tex-math id="M3">\begin{document}$ \Phi_t\colon D^2\to D^2 $\end{document}</tex-math></inline-formula>, having transitive global attractors <inline-formula><tex-math id="M4">\begin{document}$ \Lambda_t $\end{document}</tex-math></inline-formula> on which <inline-formula><tex-math id="M5">\begin{document}$ \Phi_t $\end{document}</tex-math></inline-formula> is topologically conjugate to the natural extension of <inline-formula><tex-math id="M6">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>. The unique family of absolutely continuous invariant measures for <inline-formula><tex-math id="M7">\begin{document}$ f_t $\end{document}</tex-math></inline-formula> induces a family of ergodic <inline-formula><tex-math id="M8">\begin{document}$ \Phi_t $\end{document}</tex-math></inline-formula>-invariant measures <inline-formula><tex-math id="M9">\begin{document}$ \nu_t $\end{document}</tex-math></inline-formula>, supported on the attractors <inline-formula><tex-math id="M10">\begin{document}$ \Lambda_t $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>We show that this family <inline-formula><tex-math id="M11">\begin{document}$ \nu_t $\end{document}</tex-math></inline-formula> varies weakly continuously, and that the measures <inline-formula><tex-math id="M12">\begin{document}$ \nu_t $\end{document}</tex-math></inline-formula> are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures <inline-formula><tex-math id="M13">\begin{document}$ \rho_t $\end{document}</tex-math></inline-formula>. <p style='text-indent:20px;'>Similar results are obtained for the family <inline-formula><tex-math id="M14">\begin{document}$ \chi_t\colon S^2\to S^2 $\end{document}</tex-math></inline-formula> of transitive sphere homeomorphisms, constructed in a previous paper of the authors as factors of the natural extensions of <inline-formula><tex-math id="M15">\begin{document}$ f_t $\end{document}</tex-math></inline-formula>.