# a mixing property on a tower

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?

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.

• 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. Commented Nov 24, 2021 at 21:27