Abstract:
We consider polygonal billiards on surfaces of constant curvature. We show that the usual phase space of the billiard flow can be extended to a compact three dimensional manifold without boundary that is a circle bundle. We extend the dynamics $C^\infty$ to the manifold defined. We relate the expansiveness of the billiard flow with the existence of periodic orbits and with the injectivity of the itinerary map for billiard tables in the hyperbolic disc, the Euclidean plane and the sphere.

Abstract:
In the hyperspace of subcontinua of a compact surface we consider a second order Hausdorff distance. This metric space is compactified in such a way that the stable foliation of a pseudo-Anosov map is naturally identified with a hypercontinuum. We show that negative iterates of a stable arc converges to this hypercontinuum in the considered metric. Some dynamical properties of pseudo-Anosov maps, as topological mixing and the density of stable leaves, are generalized for cw-expansive homeomorphisms of pseudo-Anosov type on compact metric spaces.

Abstract:
It is known that if a compact metric space X admits a minimal expansive homeomorphism then X is totally disconnected. In this note we give a short proof of this result and we analyze its extension to expansive flows.

Abstract:
A homeomorphism on a compact metric space is said hyper-expansive if every pair of different compact sets are separated by the homeomorphism in the Hausdorff metric. We characterize such dynamics as those with a finite number of orbits and whose non-wandering set is the union of the repelling and the attracting periodic orbits. We also give a characterization of compact metric spaces admiting hyper-expansive homeomorphisms.

Abstract:
We study the relationship between homoclinic orbits associated to repellors, usually called snap-back repellors, and expanding sets of smooth endomorphisms. Critical homoclinic orbits constitutes an interesting bifurcation that is locally contained in the boundary of the set of maps having homoclinic orbits. This and other possible routes to the creation of homoclinic orbits are considered in low dimensions.

Abstract:
In this paper we study kinematic expansive flows on compact metric spaces, surfaces and general manifolds. Different variations of the definition are considered and its relationship with expansiveness in the sense of Bowen-Walters and Komuro is analyzed. We consider continuous and smooth flows and robust kinematic expansiveness of vector fields is considered on smooth manifolds.

Abstract:
In this paper we present a technique for constructing Lyapunov functions based on Whitney's size functions. Applications to asymptotically stable equilibrium points, isolated sets, expansive homeomorphisms and continuum-wise expansive homeomorphisms are given.

Abstract:
Let $X$ be a compact metric space and let $|A|$ denote the cardinality of a set $A$. We prove that if $f\colon X\to X$ is a homeomorphism and $|X|=\infty$ then for all $\delta>0$ there is $A\subset X$ such that $|A|=4$ and for all $k\in Z$ there are $x,y\in f^k(A)$, $x\neq y$, such that $dist(x,y)<\delta$. An observer that can only distinguish two points if their distance is grater than $\delta$, for sure will say that $A$ has at most 3 points even knowing every iterate of $A$ and that $f$ is a homeomorphism. We show that for hyper-expansive homeomorphisms the same $\delta$-observer will not fail about the cardinality of $A$ if we start with $|A|=3$ instead of $4$. Generalizations of this problem are considered via what we call $(m,n)$-expansiveness.