On notions of determinism in topological dynamics
In this interesting paper the author investigates relations between determinism in topological dynamics and entropy. A dynamical system \((X,f)\) (i.e., a continuous map \(f: X\to X\) on a compact metric space) is said to be topologically predictable (TP) if every topological factor of \((X,f)\) is invertible. NEWLINENEWLINENEWLINE NEWLINEThe main results of the paper can be summarized as follows: NEWLINE{\parindent=6mmNEWLINE\begin{itemize}\item[1.]For every ergodic system \((X,\mu,\mathcal{B},T)\) with zero entropy there are a dynamical system \((Y,S)\) and an invariant measure \(\nu\) on \(Y\) such that \((Y,\nu,S)\) is isomorphic to \((X,\mu,\mathcal{B},T)\) and every point in the system \((Y\times Y, S\times S)\) is recurrent (in particular \((Y,S)\) is TP). NEWLINE\item[2.]Every TP system has zero topological entropy. (This was known before; the author provides a new and more direct proof which works also for \(\mathbb{Z}^d\) actions, with appropriately extended definition of TP). NEWLINE\item[3.]Every measure-preserving system with entropy zero is isomorphic to a shift-invariant Borel measure on a uniquely ergodic subshift \(X\subset \{0,1\}^{\mathbb{Z}}\) with the property that every sequence infinite to the left \(x\in \{0,1\}^{-\mathbb{N}}\) has at most two bi-infinite extensions in \(X\).NEWLINENEWLINE\end{itemize}}
scientific article; zbMATH DE number 6038227