Indeed, in [3] they showed that, if one of the invariant circles of f is missing, then for some p and q with gcd(p; q) = 1 the map must have a periodic orbit of type (p; q) which is not a Birkhoff orbit. On the other hand, Boyland showed in [2] that a twistmap with a non Birkhoff periodic orbit of type (p; q) (gcd(p; q) = 1 must have positive topological entropy, which clearly implies the theorem.

Let Y be the space {z € C: z3 £ [0, 1]} with a metric defined by the arc length. Suppose that / is a continuous map from Y to itself and P is a finite /-invariant subset. In this paper we construct a continuous map C/> from Y to itself satisfying Cp\p = f\p which achieves the infimum topological entropies of continuous maps from Y to itself which agree with / on P .

In this paper an example is given of a C2 map g from the circle onto itself, which permits an ß-explosion. It is shown that topological entropy (considered as a map from C2(Sl, S1) to the nonnegative real numbers) is continuous at g.

In this article, we prove two formulas for the topological entropy of an F-optical Hamiltonian flow induced by H ∈ C∞(M, R), where F is a Lagrangian distribution. In these formulas, we calculate the topological entropy as the exponential growth rate of the average of the determinant of the differential of the flow, restricted to the Lagrangian distribution or to a proper modification of it.

We consider the topological entropy of state space and quasi-state space homeomorphisms induced from C∗-algebra automorphisms. Our main result asserts that, for automorphisms of separable exact C∗-algebras, zero Voiculescu-Brown entropy implies zero topological entropy on the quasi-state space (and also more generally on the entire unit ball of the dual). As an application we obtain a simple de...

Applying a theorem according to Rhemtulla and Formanek, we partially solve an open problem raised by Hochman with an affirmative answer. Namely, we show that if G is a countable torsion-free locally nilpotent group that acts by homeomorphisms on X, and S ⊂ G is a subsemigroup not containing the unit of G such that f ∈ 〈1, s f : s ∈ S〉 for every f ∈ C(X), then (X, G) has zero topological entropy.

It is well known that a continuous piecewise monotone interval map with positive topological entropy is semiconjugate to a map of a constant slope and the same entropy, and if it is additionally transitive then this semiconjugacy is actually a conjugacy. We generalize this result to piecewise continuous piecewise monotone interval maps, and as a consequence, get it also for piecewise monotone g...

We present a method for estimating the minimal periodic orbit structure, the topological entropy and a fat representative of the homeomorphism associated to the existence of a finite collection of periodic orbits of an orientation preserving homeomorphism of the disk D. The method focuses on the concept of fold and recurrent bogus transition and is more direct than existing techniques. In parti...

For noninvertible maps, mainly subshifts of finite type and piecewise monotone interval maps, we investigate what happens if we follow backward trajectories, random in the sense that at each step every preimage can be chosen with equal probability. In particular, we ask what happens if we try to compute the entropy this way. It tuns out that instead of the topological entropy we get the metric ...

We introduce a functor which associates to every measure preserving system (X,B, μ, T ) a topological system (C2(μ), T̃ ) defined on the space of 2-fold couplings of μ, called the topological lens of T . We show that often the topological lens “magnifies” the basic measure dynamical properties of T in terms of the corresponding topological properties of T̃ . Some of our main results are as follow...