Hausdorff Metric Entropy And Friedland Entropy To Multiple Mappings

Entropy is an important concept in Mathematics and Science, in classical discrete topologically dynamical systems, which is a pair of a compact metric space (X, d) and a continuous transformation f:Xâ†' X, topological entropy and metric entropy in the sense of Bowen have been defined. We have see that they are equivalent, and then denote the entropy by h(f). More generally, considering a set of continuous self-maps F={f1,f2,...,fn} on a compact metric space (X, d), researcher-s have proposed many different definitions, such as Friedland entropy, topological entropy of a semigroup, invers image entropy of semigroup, preimage relation entropy and point entropy.In this paper, we discuss two concepts of entropy for multiple map-pings:Hausdorff metric entropy and Friedland entropy. The major is Hausdorff metric entropy, which is a new concept of entropy firstly in-troduced in the present article and is different from other definitions of entropy for a set of continuous maps such as Friedland entropy, topolog-ical entropy of a semigroup, inverse image entropy of semigroup and so on. For x∈X, F(x)={f1(x), f2(x)..., fn(x)} is a compact subset of X, and the compact metric space (X, d) induces a compact metric space (K(X),dH), where K(X) is the set of all nonempty compact subsets in X, dH is the Hausdorff metric on K(X). Then Fn can be regard-ed as a continuous mapping from X to K(X). Furthermore, following from the manner to define metric entropy by Bowen, we can define (n, ε)-generating sets and (n, ε)-separated sets in the sense of Hausdorff metric. Then we obtain a new entropy named Hausdorff metric entropy from a view of set values. Essentially, we only need to consider two continuous self-mappings on a compact metric space as an example to study multi-ple maps from a point of set values, related definitions and properties can be simply extended to arbitrary finiteness situation. For convenience, it is without loss to let the cardinary of F be 2.Following from the way to study the classical Bowen’s metric en-tropy, we discuss the properties of Hausdorff metric entropy.(1) If F={f}, then EntH(F)= h(f).(2) EntH(F)> 0.(3) Hausdorff metric entropy is a topologically conjugate invariant for a set of continuous self-mappings.(4) If Y (?) X is an invariant closed subset of X, then Entu{F|Y)≤ EntH(F).(5) If K(?)K1∪K2 U... U Km are all compact subsets of X, then(6) Let (X,F={f1,f2}) and (Y,G={91,92}) be two systems of multiple continuous self-mappings on compact metric spaces. Then(7) We study the relation between Hausdorff metric entropy and Friedland entropy, and give exact examples such that EntH(F)= h(F) or EntH(F)<h(F).From the point of view of set values, we try to extend other dy-namical properties in the sense of Hausdorff, such as Li-Yorke chaos, distributional chaos and non-wandering set. In particular, we prove that the non-wandering set Ω(F) must be a closed subset but do not have to be an invariant set.At the end, we discuss F={×p, xq} consisting of two linear map on R1 U{∞}, and give an estimation of its Friedland entropy, Moreover, we try and give a complete idea to obtain the exact value of the Friedland entropy.