Friedland’s Entropy And Directional Entropy For Z_+~K—Actions

In the study of dynamical system, entropy is one of the most important invariants to measure the complexity of the system. Compared with the classical dynamical system, i.e., Z1-ction or Z+1-ction, the study of Z-k-action or Z+k-action (k≥2) is more complex and difficult. To describe the complexity of Zk-actions or Z+k-actions from different points of view, several kinds of different forms of entropies have been introduced. Among them, Friedland’s entropy and directional entropy, introduced by Friedland in [10] and Milnor in [32], respectively, are the important ones. The main aim of this thesis is to give further research for these two types of entropies for Z+k-actions.In the first part, Friedland’s entropy for Z+k-actions is studied. The main result is that we give entropy formula or upper bounds for several types of classical systems using different techniques. Firstly, starting from the definition of Friedland’s entropy, we give an upper bound of this entropy for the Lipschitz Z+k-action. Secondly, the upper bounds of Friedland’s entropy for two types of Z+k-actions:Z+k-actions on finite graphs and expanding Z+k-actions on closed Riemannian manifolds, are obtained via the preimage entropies, which are entropy-like invariants depending on the "inverse orbits" structure of the system. Finally, a formula of Friedland’s entropy for Z+k-actions on tori generated by commuting endomorphisms is given via the entropy of a skew product transformation with the fiber maps taken from the generators of the Z+k-action.In the second part, we consider the directional entropies for Z+k-ctions. Using the techniques of "coding" and "shading" which were given by Boyle and Lind in [6], we obtain several properties of directional entropy. Moreover, applying some known results relating entropies and other invariants (such as preimage entropies and the entropies of nonautonomous dynamical systems), we also give the formulas of directional entropies for some classic examples. Firstly, applying the techniques of "coding" and "shading" we show that topological and measure-theoretic directional entropies are both contin-uous at positively expansive directions. Secondly, we relate the directional entropies of a Z+k-action at a ray L to the entropies of a nonautonomous dynamical system which is induced by the compositions of a sequence of maps along L and hence the variational principle relating topological and measure-theoretic directional entropies is given at pos-itively expansive directions. Finally, applying some known results relating entropies and other invariants (such as preimage entropies, degrees and Lyapunov exponents), we obtain the formulas of directional entropies for some classic examples, such as the Z+k-subshift actions on (Z/2Z)Z+, Z+k-ctions on finite graphs and C1+a(a>1) smooth Z+k-actions on Riemannian manifolds and hence get some results on continuity of the directional entropy.