# Piecewise linear expanding maps

Let $$(I_{n})$$ be a countable infinite disjoint partition of $$[0,1)$$ into half-open intervals. Let $$f:[0,1)\to [0,1)$$ be the piecewise linear expanding map with $$f(I_{n})=[0,1)$$ for all $$n$$. I suppose that the Lebesgue measure is ergodic with respect to $$f$$, as in the case of $$2x$$ mod $$1$$. I think this result should be know, but I do not find a reference. Any help is appreciated.

• Symbolically, it's a countable Bernoulli shift $(X, \mu, \sigma)$, where $\mu([k])=|I_k|$. Its ergodicity is a well known fact. It follows from it being mixing, for example. For any two cylinders $A$ and $B$, there exists $N\ge1$ such that $\mu(\sigma^n A\cap B)=\mu(A)\mu(B)$ for all $n\ge N$. – Nikita Sidorov Mar 14 at 19:39
• Dear Nikita, what reference would You use in a paper? – Jörg Neunhäuserer Mar 14 at 20:02