How little is little enough?
Let \(X\) be a topological space and \(f:X\to X\) be a continuous map. If there are \(n\geq 2\) compact nonempty sets \(A_1,\dots ,A_n\) such that \(f(A_i)\supset \cup_{j=1}^n A_j\) and \(\cap_{j=1}^n A_k=\emptyset\), then it is said that \(f\) has a weak \(n\)-horseshoe. The main theorem of the present paper is the following. Theorem. Let \(h(f)\) be the topological entropy of \(f\), and suppose that a power of \(f\) admits a weak horseshoe. Then \(h(f)>0\) and there exists a set \(B\) and a power \(g\) of \(f\) such that \(B\) is \(g\)-invariant and \(g|B\) is semiconjugate to a full shift. The above result gives a positive answer to a question asked by \textit{A. Lasota} and \textit{G. Pianigiani} [Boll. Un. Mat. Ital., V. Ser. 14-B, 592-603 (1977; Zbl 0372.28019)] on the existence of an invariant nonatomic ergodic measure \(\mu\) of \(f\) such that \(h_\mu(f)>0\).
scientific article; zbMATH DE number 1941559