Positively expansive dynamical systems
This paper proposes two generalizations of the positively expansive notion on discrete dynamical systems. Let \(X\) be a metrizable space and \(f:X\rightarrow X\) a continuous map then \(f\) is said to be positively expansive if any two points are separated by more than a fixed distance in some iterate. The authors claim that if \(f\) is positively expansive and is topologically conjugate to \(g:Y\rightarrow Y\) then \(g\) is not necessarily positively expansive, i.e. the positively expansive notion is not a dynamical property. The generalizations introduced are those of weakly and strongly positively expansive dynamical systems (the strongly notion is only introduced for homeomorphisms), for their definitions we need some terminology. Let \(f:X\to X\) be a continuous map and \(F_f:X\times X\to X\times X\) be such that \(F_f(x,y)=(f(x),f(y))\). An expansivity neighbourhood for \(f\) is a closed neighbourhood \(N\subset X\times X\) such that for any distinct \(x,y\in X\) there exists \(n\geq 0\) such that \(F^n(x,y)\not\in N\). A set \(N\) is called overflowing if \(N\subset f(N)\) and \(M\subset X\times X\) is proper if for any compact subset \(A\subset X\) then \(M(A):=\bigcup_{x\in A}\{y\in X: (x,y)\in M\}\) is compact. Let \(f:X\to X\) be a continuous map then \(f\) is weakly positively expansive if it has a expansivity neighbourhood. If \(f\) is a homeomorphism then \(f\) is strongly positively expansive if it has a proper overflowing expansivity neighbourhood. The following main results are proved: {\parindent=6mm \begin{itemize}\item[[1]] If \(X\) is compact metric and \(f:X\rightarrow X\) is an homeomorphism then the notions strongly positively expansive, weakly positively expansive and positively expansive are equivalent (if \(X\) is not compact it is also proved some relations). \item[[2]] Weakly and strongly positively expansive are dynamical properties. \item[[3]] Strongly positively expansive notion implies weakly positively expansive one. \end{itemize}} Some restrictions on the dynamics of positively expansive (resp. weakly and strongly positive expansive) homeomorphisms are remarked. And finally two classifications of strongly positively expansive homeomorphisms on locally compact metrizable spaces are proved.
scientific article; zbMATH DE number 5121452
scientific article