Rotation sets of invariant separating continua of annular homeomorphisms

2010/11/14 by Matsumoto, Shigenori
#37E30 #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1011.3176

Abstract

Let f be a homeomorphism of the closed annulus A isotopic to the identity, and let X⊂ \rm IntA be an f-invariant continuum which separates A into two domains, the upper domain U₊ and the lower domain U_-. Fixing a lift of f to the universal cover of A, one defines the rotation set ρ(X) of X by means of the invariant probabilities on X. For any rational number p/q∈ ρ(X), f is shown to admit a p/q periodic point in X, provided that (1) X consists of nonwandering points or (2) X is an attractor and the frontiers of U_- and U₊ coincides with X. Also the Carathéodory rotation numbers of U_± are shown to be in ρ(X) for any separating invariant continuum X.

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