Diagonal points having dense orbit

T. K. Subrahmonian Moothathu¹

¹ Department of Mathematics and Statistics University of Hyderabad Hyderabad 500 046, India

Colloquium Mathematicum, Tome 120 (2010) no. 1, pp. 127-138

Résumé

Let $f:X\to X$ be a topologically transitive continuous map of a compact metric space $X$. We investigate whether $f$ can have the following stronger properties: (i) for each $m\in \mathbb{N}$, $f\times f^2\times \cdots \times f^m:X^m\to X^m$ is transitive, (ii) for each $m\in \mathbb{N}$, there exists $x\in X$ such that the diagonal $m$-tuple $(x,x,\ldots, x)$ has a dense orbit in $X^m$ under the action of $f\times f^2\times \cdots \times f^m$. We show that (i), (ii) and weak mixing are equivalent for minimal homeomorphisms, that all mixing interval maps satisfy (ii), and that there are mixing subshifts not satisfying (ii).

Détail

DOI : 10.4064/cm120-1-9

Keywords: topologically transitive continuous map compact metric space investigate whether have following stronger properties nbsp each mathbb times times cdots times m transitive each mathbb there exists diagonal m tuple ldots has dense orbit under action times times cdots times weak mixing equivalent minimal homeomorphisms mixing interval maps satisfy there mixing subshifts satisfying