We describe the homotopy classes of self-homeomorphisms of solenoids and Knaster continua. In particular, we demonstrate that homeomorphisms within one homotopy class have the same (explicitly given) topological entropy and that they are actually semi-conjugate to an algebraic homeomorphism in the case when the entropy is positive.
For a continuous map f preserving orbits of an aperiodic -action on a compact space, its displacement function assigns to x the “time” it takes to move x to f(x). We show that this function is continuous if the action is minimal. In particular, f is homotopic to the identity along the orbits of the action.
This article is devoted to the study of the translation flow on self-similar tilings associated with a substitution of Pisot type. We construct a geometric representation and give necessary and sufficient conditions for the flow to have pure discrete spectrum. As an application we demonstrate that, for certain beta-shifts, the natural extension is naturally isomorphic to a toral automorphism.
Page 1