Rotation sets for subshifts of finite type

Abstract

For a dynamical system (X, f ) and a function ϕ : X → R N the rotation set is defined. The case when (X, f ) is a transitive subshift of finite type and ϕ depends on the cylinders of length 2 is studied. Then the rotation set is a convex polyhedron. The rotation vectors of periodic points are dense in the rotation set. Every interior point of the rotation set is a rotation vector of an ergodic measure.

Rotation Sets of Billiards with One Obstacle

“…Note that if we use the discrete time (the number of reflections) rather than continuous time, we would get all good properties of the admissible rotation set from the description of the admissible sequences via the graph G and the results of [11]. Since we are using continuous time, the situation is more complicated.…”

“…The last statement of the lemma follows from the convexity of the balls in R m . Now we can follow the methods of [8] and [11]. We assume that our billiard has a small obstacle.…”

“…In other words, there are infinite trajectories without reflections, so when considering billiards as maps, we would have to divide by zero. Although this is not so bad by itself (infinity exists), we lose compactness and cannot apply nice general machinery of the rotation theory (see, e.g., [11]).…”

Symbol ratio minimax sequences in the lexicographic order

“…When X is a subshift of finite type, a theorem of Ziemian [17] states that ρ(X) is a convex set with finitely many extreme points, given by the rotation vectors of the minimal loops of the transition diagram. While this result is useful, subshifts of finite type are rather special, and are often ill-suited to understand dynamical behaviour in parameterised families, since Markov partitions can change dramatically under small changes in the map.…”

A horseshoe with a discontinuous entropy spectrum

“…For higher dimensional spectra the situation is more complicated, in part, since concave and upper semi-continuous functions may have in theory discontinuities at the boundary of the domain unless the domain is polyhedral [6]. We note that the case R L (f ) being a polyhedron occurs for example if f is piecewise linear [14], in which case H f is continuous on R L (f ). In this paper we show that the entropy spectrum may be discontinuous even when the polyhedron property fails only at one point.…”