# Is there a minimal, topologically mixing but not positively expansive dynamical system?

Is there a compact metric space $X$ and a function $f:X\to X$ such that the dynamical system $(X, f)$ has the following three properties?

1. minimal

2. topologically mixing (a map $f$ is topologically mixing if for all nonempty open subsets $A$, $B$, there exists $N$ such that for every $n \geq N$ we have $f^n(A) \cap B \neq \emptyset$)

3. not positively expansive (a map $f$ is positively expansive if there exists an $c>0$ such that for all $x\neq y$ there exists an $n \geq 0$ such that $d(f^n(x), f^n(y))>c$)

Somewhat of relevance:

• There is no minimal continuous self-map on an even-dimensional sphere. Indeed, any self-homeomorphism of $\mathbb{S}^{2n}$ with degree $\neq -1$ has a fixed point (see Theorem 5.4 in Granas, Dugundji, Fixed Point Theory). So for every continuous self-map on $\mathbb{S}^{2n}$, $f^2$ has a fixed point. – YCor Oct 26 '16 at 20:54
• @YCor: yes, I'm asking whether there exist a compact metric $X$ and $f:X\to X$. I edited the question. Thank you for pointing this out. – user Oct 27 '16 at 9:40

E. Lehrer, Topological mixing and uniquely ergodic systems, Israel J. Math. 57 (1987), no. 2, 239-255 proved that every ergodic measure preserving system $(X,\mathcal{X}, \mu, T)$ has a topologically mixing, strictly ergodic topological model.
Recall that a measurable system is a quadruple $(X,\mathcal{X}, \mu, T)$, where $(X,\mathcal{X}, \mu)$ is a Lebesgue probability space and $T \colon X \to X$ is an invertible measure preserving transformation. A topological dynamical system is a pair $(X, T)$, where $X$ is a compact metric space and $T \colon X \to X$ is a homeomorphism. Let $(X,\mathcal{X}, \mu, T)$ be an ergodic measurable system. We say that $(\hat{X},\hat{\mathcal{X}}, \hat{\mu}, \hat{T})$ is a topological model for $(X,\mathcal{X}, \mu, T)$ if $(\hat{X} , \hat{T})$ is a topological dynamical system, $\hat{\mu}$ is an invariant Borel probability measure on $\hat{X}$, $\hat{\mathcal{X}}$ denotes the Bore $\sigma$-algebra on $\hat{X}$ and the measure preserving systems $(X,\mathcal{X}, \mu, T)$ and $(\hat{X},\hat{\mathcal{X}}, \hat{\mu}, \hat{T})$ are measure theoretically isomorphic.
Lehrer's result implies in particular that if you start with a measure preserving system where $X=[0,1]^\infty$ with the product topology (the Hilbert cube), $T=\sigma$ the shift transformation, Borel $\sigma$-algebra on $X$ and the product $\lambda^\infty$ of Lebesgue measures then its topological model will be minimal and will have infinite topological entropy hence it will not be positively expansive (since positively expansive systems have finite entropy). I believe that simpler examples can be constructed by considering minimal subsets of $X=[0,1]^\infty$.