LOCAL ENTROPY FUNCTION OF DYNAMICAL SYSTEM
İsmail Tok · DergiPark (Istanbul University) · 2011
In this work, we first,define the entropy function of the topological dynamical system and investigate basic properties of this function without going into details. Let (X,A,T) be a probability measure space and consider P = { pl5p2,...,pn} a finite measurable partition of all sub-sets of topological dynamical system (X,T).Then,the quantity H (P) = ^ zpt)) is called the i=1 entropy function of finite measurable partition P.Where f-1 log t if 0 < t < 1 z(t)= < \ 0 if t = 0 is a non-negative,continuous and strictly concave function.In this paper,all logarithms will be taken to be the natural base "e ". After that,we give the definition of the local entropy function of the topological dynamical system. Let P be a measurable partition of topological dynamical system (X,T) with H^(P) 0.If diam(P) < s,then the quantity L^ (T) = h^ (T) - h^ (T,P) is called a local entropy function of topological dynamical system (X,T) . In conclusion, Let (X,T) and (Y,S) be two topological dynamical system. If TxS is a transformation defined on the product space (XxY,TxS) with (TxS)(x , y) = (Tx,Sy) for all (x,y) X x Y.Then L ^^ (TxS) = L^d(T) + L (S) .and, we prove some fundamental properties of this function.