In this note we prove that a homeomorphism is countably-expansive if and only if it is measure-expansive. This result is applied for showing that the C 1 -interior of the sets of expansive, measure-expansive and continuum-wise expansive C 1 -diffeomorphisms coincide.

In this paper we study the polynomial entropy of homeomorphism on compact metric space. We construct a homeomorphism on a compact metric space with vanishing polynomial entropy that it is not equicontinuous. Also we give examples with arbitrarily small polynomial entropy. Finally, we show that expansive homeomorphisms and positively expansive maps of compact metric spaces with infinitely many points have polynomial entropy greater or equal than 1.

In this paper we define and study the topological entropy of a setvalued dynamical system. Actually, we obtain two entropies based on separated and spanning sets. Some properties of these entropies resembling the singlevalued case will be obtained.

We study the dynamics of a family of perturbed three-degreesof-freedom (3-DOF) Hamiltonian systems which are in 1:1:1 resonance. The perturbation consists of axially symmetric cubic and quartic arbitrary polynomials. Our analysis is performed by normalisation, reduction and KAM techniques. Firstly, the system is reduced by the axial symmetry and then, periodic solutions and KAM 3-tori of the full system are determined from the relative equilibria. Next, the oscillator symmetry is extended by normalisation up to terms of degree 4 in rectangular coordinates; after truncation of higher orders and reduction to the orbit space, some relative equilibria are established and periodic solutions and KAM 3-tori of the original system are obtained. As a third step, the reduction of the two symmetries leads to a one-degrees-offreedom system that is completely analysed in the twice reduced space. All the relative equilibria, together with the stability and parametric bifurcations are determined. Moreover the invariant 2-tori (related to the critical points of the twice reduced space), some periodic solutions and the KAM 3-tori, all corresponding to the full system, are established. Additionally, the bifurcations of equilibria occurring in the twice reduce space are reconstructed as quasi-periodic bifurcations involving 2-tori and periodic solutions of the full system.

We extend the concept of expansive measure [2] defined for homeomorphism to flows. We obtain some properties for such measures including abscense of singularities in the support, aperiodicity, expansivity with respect to time-T maps, invariance under flow-equivalence, negligibleness of orbits, characterization of expansive measures for expansive flows and naturallity under suspensions. As an application we obtain a new proof of the well known fact that there are no continuous expansive flows on surfaces (e.g. [12]).2010 Mathematics Subject Classification. 54H20, 34C35.