2018/12/02 by Boyland, Philip, de Carvalho, André, Hall, Toby
#37A10 #37B45 #37C75 #37E05 #Dynamical Systems (math.DS) #FOS: Mathematics
paper · doi:10.48550/arxiv.1812.00453
Let \ft\t∈(1,2] be the family of core tent maps of slopes t. The parameterized Barge-Martin construction yields a family of disk homeomorphisms Φt\colon D2→ D2, having transitive global attractors Λt on which Φt is topologically conjugate to the natural extension of ft. The unique family of absolutely continuous invariant measures for ft induces a family of ergodic Φt-invariant measures νt, supported on the attractors Λt. We show that this family νt varies weakly continuously, and that the measures νt are physical with respect to a weakly continuously varying family of background Oxtoby-Ulam measures ρt. Similar results are obtained for the family χt\colon S2→ S2 of transitive sphere homeomorphisms, constructed in [17] as factors of the natural extensions of ft.