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A tower $\Delta_g:=\{(x,n)\in X \times \{0,1,2,\cdots\}: n < R(x)\}$

where $R:X \to \{1,2,3,\cdots\}$ is a $L^1$ function on a probability space $(X,\mu)$, $g: X \to X$ is mixing and $\gcd \{R\}=1$,

A map $f: \Delta \to \Delta$ is defined as: $f(x,n)=(x,n+1)$ if $n < R(x)-1$ and $f(x,n)=(g(x),0)$ if $n=R(x)-1$.

An invariant probability on $\mu_{\Delta}$ is $\mu_{\Delta}=(\int R d\mu)^{-1}\sum_{i\ge 0}f^i_{*}(\mu|_{R>i})$.

Is $(\Delta, f, \mu_{\Delta})$ mixing? or have a counter-example?

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1 Answer 1

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There are counterexamples. The easiest way to “cheat” is to let the height function be cohomologous to a constant.

As an example, let $T$ be an ergodic transformation of a space $X$. Let $A$ be a subset of $X$ such that $A$ and $T^{-1}A$ are disjoint. Now define $g(x)=1$ if $x\in A$, $g(x)=3$ if $x\in T^{-1}A$ and $2$ otherwise.

Then $T_g$ has an eigenfunction: $h(x,0)=1$ if $x\in A$, $h(x,0)=h(x,2)=-1$ and $h(x,1)=1$ if $x\in T^{-1}A$, and $h(x,0)=1$ and $h(x,1)=-1$ for other $x$’s. Then $h\circ T_g=-h$.

The way that this works is that $g$ is cohomologous to the constant function 2: the difference $g-2$ can be expressed as $j\circ T-j$ where $j(x)=1$ if $x\in A$ and 0 otherwise.

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  • $\begingroup$ It is great! If the cohomology is excluded, is it possible to be mixing? My question comes from the paper by BM Gurevich in 1967 and 1969 which considered K-flow: if the base is K and the roof function satisfies some conditions like gcd=1, then the suspension flow is also K. $\endgroup$
    – jason
    Commented Nov 24, 2021 at 21:27

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