Basins of measures on inverse limit spaces for the induced homeomorphism

JUDY KENNEDY, BRIAN E. RAINES, DAVID R. STOCKMAN
2009 Ergodic Theory and Dynamical Systems  
Let f : X → X be continuous and onto, where X is a compact metric space. Let Y := lim ←− (X, f ) be the inverse limit and F : Y → Y the induced homeomorphism. Suppose that µ is an f -invariant measure, and let m be the measure induced on Y by (µ, µ, . . .). We show that B is a basin of µ if and only if π −1 1 (B) is a basin of m. From this it follows that if µ is an SRB measure for f on X , then the induced measure m on Y is an inverselimit SRB measure for F. Conversely, if m is an
more » ... is an inverse-limit SRB measure for F on Y , then the induced measure µ on X is an SRB measure for f .
doi:10.1017/s0143385709000388 fatcat:ttudoyesizb4jd4p6owrrbnl2a