Shadowing with chain transitivity
Let \(f:X\to X\) be a continuous surjective map on a compact metric space \(X\). A \(\delta\)-chain from \(x\) to \(y\) of length \(n\) is a finite sequence \(x_0=x\), \(x_1, \dots,x_n=y\) such that \(d(f(x_i),x_{i+1}) \leq\delta\) for \(i=0,\dots,n-1\). A sequence \(\xi=\{x_i\}^\infty_{i=0}\) is called a \(\delta\)-pseudo orbit if for any \(i\geq 0\), \(d(f(x_i),x_{i+1})\leq \delta\). A \(\delta\)-pseudo orbit is said to be \(\varepsilon\)-shadowed by a true orbit if there exists \(z\in X\) such that for \(i\geq 0\), \(d(f^i(z),x_i) \geq\varepsilon\). The map \(f\) is said to have the shadowing property if for any \(\varepsilon>0\) there exists \(\delta>0\) such that any \(\delta\)-pseudo orbit is \(\varepsilon\)-shadowed by a true orbit. The system is said to be chain transitive if any two points can be jointed by a \(\delta\)-chain for any \(\delta>0\). The sensitivity of the system is defined by the formula: \(\exists\varepsilon:\forall x,\forall\delta,\exists B(x,\delta),\exists n>0\), \(d(f^n(x),f^n(y))\geq\varepsilon\). The authors' main result is: a chain transitive system with the shadowing property is either sensitive or equicontinuous.
scientific article; zbMATH DE number 5597560