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?