We prove that a homeomorphism of a compact metric space has an expansive measure cite{ms} if and only if it has many ones with invariant support. We also study homeomorphisms for which the expansive measures are dense in the space of Borel probabilit
y measures. It is proved that these homeomorphisms exhibit a dense set of Borel probability measures which are expansive with full support. Therefore, their sets of heteroclinic points has no interior and the spaces supporting them have no isolated points.
We show that if $G$ is a finitely generated group of subexponential growth and $X$ is a Suslinian continuum, then any action of $G$ on $X$ cannot be expansive.
In this article, we show that R.H. Bings pseudo-circle admits a minimal non-invertible map. This resolves a problem raised by Bruin, Kolyada and Snoha in the negative. The main tool is the Denjoy-Rees technique, further developed by Beguin-Crovisier-
Le Roux, combined with detailed study into the structure of the pseudo-circle.
We obtain the complete conjugacy invariants of expansive Lorenz maps and for any given two expansive Lorenz maps, there are two unique sequences of $(beta_{i},alpha_{i})$ pairs. In this way, we can define the classification of expansive Lorenz maps.
Moreover, we investigate the uniform linearization of expansive Lorenz maps through periodic renormalization.
We discuss conditions for unique ergodicity of a collective random walk on a continuous circle. Individual particles in this collective motion perform independent (and different in general) random walks conditioned by the assumption that the particle
s cannot overrun each other. Additionally to sufficient conditions for the unique ergodicity we discover a new and unexpected way for its violation due to excessively large local jumps. Necessary and sufficient conditions for the unique ergodicity of the deterministic version of this system are obtained as well. Technically our approach is based on the interlacing property of the spin function which describes states of pairs of particles in coupled processes under study.