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?



a mixing property on a tower
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.


