The Study Of Entropies In Dynamical Systems And Pressures For Entropy-expansive Systems

There are four main parts in this paper.In the first part (Chapter 2), we study the preimage entropies of nonautonomous dynamical systems. four entropy-like invariants for nonautonomous discrete dynamical systems given by a sequence of continuous selfmaps of a compact space are introduced and studied. We Proved that these entropies are all invariant with respect to equiconjugacy, and they all satisfy subadditivity and submultiplicativity. The relations between these entropies are established. We get that for expansive nonautonomous systems, two types of pointwise preimage entropies are equal, and the preimage branch entropy and the preimage relation entropy are equal too. We also get that two classes of nonautonomous systems: (a). a sequence of small C¹-perturbations of an expanding map on a closed Riemmanian manifold, and (b). a sequence of equicontinuous maps defined on a finite graph, have zero preimage branch entropy.In the second part(Chapter 3), we consider the preimage entropies for continuous semi-flow of a compact metric space. We prove that most of these entropies are invariant in a certain sense under conjugate when the semi-flows under consideration are free of fixed points, and get an inequality relating these entropies. We also show that most of these entropies for semi-flow are consistent with that for its time-1 mapping. As applications, the relation between the entropies for a continuous map and for its suspension is given.In the third part (Chapter 4), we study the topological entropy of a sequence of monotone maps on circles. We prove that the topological entropy of a sequence ofequicontinual monotone maps f₁,∞ = {f_i}_i=1^∞ is h(f₁,∞) = lim sup.As applications, we give the estimation of the entropies for some skew product on anular and Torus. And also show that a C¹ diffeomophism / on a 2-dimmensional smooth and closed Riemannian Manifold and its extension on the ball-bundle have the same entropy.In the fourth part (Chapter 5), we study the pressure of the entropy-expansive...