An Introduction to Ergodic Theory

Rotation, entropy, and equilibrium states

“…Let M be a compact orientable Riemannian surface of genus g, with corresponding real homology space One of the goals of this paper is to sharpen and extend these results in various directions. We first consider the very general case where the function f is merely continuous, and the dynamical system (X, T ) has upper semi-continuous entropy map h. This guarantees that every continuous function g : X → R has at least one equilibrium state (see [55], p. 224).…”

“…The upper semi-continuity of h is guaranteed, for example, if T is (topologically) expansive (see [55]), or more generally if T admits a finite generating partition α (i.e. for all µ ∈ M, the limiting refinement ∞ i=0 T −i α agrees with the Borel σ-algebra up to sets of µ-measure zero) such that boundaries of sets in α have zero measure for every invariant measure (see [23], Cor.…”

“…Our general approach is motivated by thermodynamic formalism, in particular the theory of pressure and equilibrium states developed by Ruelle [48], [49] and Walters [54], [55]. After briefly recalling some notions from convex geometry in section 2, we develop some thermodynamic ideas in section 3.…”

“…If f is locally constant (the case treated by [14], [59]), then (after a standard re-coding to ensure f only depends on two coordinates) each equilibrium state m v.f is a Markov measure, whose explicit form is given in [42], p. 27. Consequently there are simple formulae for both f * (m v.f ) and h(m v.f ) (see [55], p. 103), and we can obtain an explicit formula for the entropy function H. For example suppose X is the full shift on the alphabet {0, 1}, and f = (χ [1] , χ [11] ) (i.e. the coordinate functions are characteristic functions of the cylinders [1] and [11]).…”

“…Consequently any ergodic measure supported on Y α will be directional, and in the fibre f −1 * ( ). Moreover, the variational principle (see Walters [54], [55]) means we can find an ergodic measure µ α , with supp(…”