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An answer to Furstenberg’s problem on topological disjointness
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems / Volume 40 / Issue 9 / September 2020
- Published online by Cambridge University Press:
- 10 April 2019, pp. 2467-2481
- Print publication:
- September 2020
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- Article
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In this paper we give an answer to Furstenberg’s problem on topological disjointness. Namely, we show that a transitive system
$(X,T)$ is disjoint from all minimal systems if and only if 
$(X,T)$ is weakly mixing and there is some countable dense subset 
$D$ of 
$X$ such that for any minimal system 
$(Y,S)$, any point 
$y\in Y$ and any open neighbourhood 
$V$ of 
$y$, and for any non-empty open subset 
$U\subset X$, there is 
$x\in D\cap U$ such that 
$\{n\in \mathbb{Z}_{+}:T^{n}x\in U,S^{n}y\in V\}$ is syndetic. Some characterization for the general case is also given. By way of application we show that if a transitive system 
$(X,T)$ is disjoint from all minimal systems, then so are 
$(X^{n},T^{(n)})$ and 
$(X,T^{n})$ for any 
$n\in \mathbb{N}$. It turns out that a transitive system 
$(X,T)$ is disjoint from all minimal systems if and only if the hyperspace system 
$(K(X),T_{K})$ is disjoint from all minimal systems.
