Preimage Entropies of Semi-Flows

In this paper, several preimage entropies for semi-flows on compact metric spaces are introduced and studied. We prove that most of these entropies are invariant in a certain sense under conjugacy when the semi-flows under consideration are free of fixed points. The relation between these entropies is studied and an inequality relating them is given. It is also shown that most of these entropies for semi-flow are consistent with that for the time-1 mapping.

On an Entropy of Zk＋-Actions

In this paper,a definition of entropy for Z＋k（k≥2）-actions due to Friedland is studied.Unlike the traditional definition,it may take a nonzero value for actions whose generators have finite（even zero） entropy as single transformations.Some basic properties are investigated and its value for the Z＋k-actions on circles generated by expanding endomorphisms is given.Moreover,an upper bound of this entropy for the Z＋k-actions on tori generated by expanding endomorphisms is obtained via the preimage entropies,which are entropy-like invariants depending on the ＂inverse orbits＂ structure of the system.

Forward Expansiveness and Entropies for Subsystems of Z^(k)_(+)-actions

In this paper,forward expansiveness and entropies of"subsystems"2)of Z^(k)_(+)-actions are investigated.Letαbe a Z^(k)_(+)-action on a compact metric space.For each 1≤j≤k-1,denote G^(j)_(+)={V+:=V∩R^(k)_(+):V is a j-dimensional subspace of R^(k)}.We consider the forward expansiveness and entropies forαalong V+∈G^(j)_(+).Adapting the technique of"coding",which was introduced by M.Boyle and D.Lind to investigate expansive subdynamics of Z^(k)-actions,to the Z^(k)_(+)cases,we show that the set E^(j)_(+)(α)of forward expansive j-dimensional V_(+)is open in G^(j)_(+).The topological entropy and measure-theoretic entropy of j-dimensional subsystems ofαare both continuous in E^(j)_(+)(α),and moreover,a variational principle relating them is obtained.For a 1-dimensional ray L∈G^(+)_(1),we relate the 1-dimensional subsystem ofαalong L to an i.i.d.random transformation.Applying the techniques of random dynamical systems we investigate the entropy theory of 1-dimensional subsystems.In particular,we propose the notion of preimage entropy(including topological and measure-theoretical versions)via the preimage structure ofαalong L.We show that the preimage entropy coincides with the classical entropy along any L∈E1+(α)for topological and measure-theoretical versions respectively.Meanwhile,a formula relating the measure-theoretical directional preimage entropy and the folding entropy of the generators is obtained.