Sequence Entropy Of Induced Dynamic System And The Related Problems

Topolgical entropy is very important in the study of topological dynamical systems, it reflect the complexity of dynamical systems. In order to have a better understanding of zero entropy systems, the notion of topological sequence entropy was introduced and studied. The relationship between a topological dynamical system and its induced system is also an interesting problem in the study of dynamical systems. Recently, Auslander, Kolyada and Snoha introduced a new induced sytem—the functional envelope. In this thesis, we study the toplogical sequence entropy, and for some one dimensional dynamical system, we study the relationship of the topological sequence entropy of dynamical systems and their induced functional envelop. More precisely,1. For unit circle and unit interval, we studied the relationship between topological sequence entropy of interval map (circle map) and its induced functional envelope, which is an analoge of Kolyada’s result on topological entropy. Let X be a compact metric space, f:X→X be a continuous map and S(X) is the set of all the continuous map from X to itself. We proved that when X is unit interval or unit circle, the topological sequence entropy of (S(X),F) can only be 0 or+∞. And when the topological sequence entropy of (X,f) is positive, then the topological sequence entropy of (S(X),F) is+∞.2. For finite tree T, we studied the relationship between topological sequence entropy of tree map and its induced system, which is an analoge of Matviichuk’s result on topological entropy. We proved that when the topological sequence entropy of (T, f) is positive, then the toplogical sequence entropy of (SH (T), F) is+∞.