We obtain geometric upper bounds on the topological entropy of generalized polygon exchange transformations. As an application of our results, we show that billiards in polygons and rational polytops have zero topological entropy.

We use ending laminations for Weil-Petersson geodesics to establish that bounded geometry is equivalent to bounded combinatorics for WeilPetersson geodesic segments, rays, and lines. Further, a more general notion of non-annular bounded combinatorics, which allows arbitrarily large Dehn-twisting, corresponds to an equivalent condition for Weil-Petersson geodesics. As an application, we show the...

For a pair of points x, y in a compact, riemannian manifold M let nt(x, y) (resp. st(x, y)) be the number of geodesic segments with length ≤ t joining these points (resp. the minimal number of point obstacles needed to block them). We study relationships between the growth rates of nt(x, y) and st(x, y) as t → ∞. We derive lower bounds on st(x, y) in terms of the topological entropy h(M) and it...

Certain classes of automorphisms of reduced amalgamated free products of C∗– algebras are shown to have Brown–Voiculescu topological entropy zero. Also, for automorphisms of exact C∗–algebras, the Connes–Narnhofer–Thirring entropy is shown to be bounded above by the Brown–Voiculescu entropy. These facts are applied to generalize Størmer’s result about the entropy of automorphisms of the II1–fac...

We introduce a concept of independence entropy for symbolic dynamical systems. This notion of entropy measures the extent to which one can freely insert symbols in positions without violating the constraint defined by the shift space. We show that for a certain class of one-dimensional shift spaces X, the independence entropy coincides with the limiting, as d tends to infinity, topological entr...

We prove that the topological entropy of subshifts having decidable language is uncomputable in the following sense: For no error bound less than 1/4 does there exists a program that, given a decision procedure for the language of a subshift as input, will approximate the entropy of the subshift within the error bound. In addition, we prove that not only is the topological entropy of sofic shif...

It is known that for n ≥ 3 centres and positive energies the ncentre problem of celestial mechanics leads to a flow with a strange repellor and positive topological entropy. Here we consider the energies above some threshold and show: Whereas for arbitrary g > 1 independent integrals of Gevrey class g exist, no real-analytic (that is, Gevrey class 1) independent integral exists.

Let G be a discrete group which admits an amenable action on a compact space and γ ∈ Aut(G) be an automorphism. We define a notion of entropy for γ and denote the invariant by ha(γ). This notion is dual to classical topological entropy in the sense that if G is abelian then ha(γ) = hTop(γ̂) where hTop(γ̂) denotes the topological entropy of the induced automorphism γ̂ of the (compact, abelian) dual...

We establish lower bounds for the entropy of the Hard Core Model on a few 2d lattices L. In this model the allowed configurations inside {0,1} are the one’s in which the nearest neighbor 1’s are forbidden. Our method which is based on a sequential fill-in scheme is unbiassed and thereby yields in principle arbitrarily good estimates for the topological entropy. The procedure also gives some det...

We prove that for a certain class of Z shifts of nite type with positive topological entropy there is always an invariant measure with entropy arbitrarily close to the topological entropy, that has strong metric mixing properties. With the additional assumption that there are dense periodic orbits, one can ensure that this measure is Bernoulli.