(t,r)-Entropy Of Continuous Maps

In this paper, we introduce the (t, r)-entropy of continuous maps on a compact metric space. Entropy is an important invariant to describe the complexity of dynamical systems. In order to study actions for groups, Jacob Feldman introduced the r-entropy, it is especially useful for the classification of systems. So it is quite necessary for us to study r-entropy. As we all know, for the positive entropy systems, the larger the r-entropy is, the more complex the system is. In order to study the complexity of the dynamical systems of zero r-entropy, we apply (t, r)-entropy and r-entropy dimension to describe their complexity, and investigate the properties of them. Besides, we also discuss the local homeomorphism on finite graphs, and we get some interesting results. The main results are as follows:Firstly, we introduce the concepts of topological (t, r)-entropy, topological r-entropy dimension, measure-theoretic (t, r)-entropy and measure-theoretic r-entropy dimension, and discuss some fundamental properties. Meanwhile, we get the results as follows:the limit of measure-theoretic (t, r)-entropy (r â†' 0) is no more than the measure-theoretic t-entropy.Secondly, for the special r= 0, we introduce two kinds of preimage (t, r)-entropy, discuss some essential properties, and get relations between them. And also show that for:(1)forward-expansive maps, (2)local homeomorphisms on finite graphs, the preimage branch (t,0)-entropy is zero. Then, their (t,0)-entropy and pointwise preimage (t,0)-entropy are the same. And in particular, the entropy dimension of the homeomorphism on finite graphs is zero.