Measures of expanding maps of the circle and their coding

Question

It is well known that the dynamics of linear examples $f(x)=mx(mod1)$ for natural $m\geq 2$ is semi conjugated to the full shift on the space of one-sided sequences of digits $\{0,…,m−1\}.$ Is it true that the semi conjugated map measures theoretical isomorphism for all measures? If not, under what (minimum assumption on maps) that is. Thanks in advance.

Answer 1

You can start by noting the obvious:

The semiconjugacy is not a conjugacy because there are points with more than one coding.

For example, for $m=2$ these are the points in $[0,1]$ having a base-$2$ representation with infinitely many consecutive $1$'s. Note that all of them are of the form $k/2^n$ for some integers $k$ and $n$. They form a Borel set.

So you obtain an isomorphism if and only if this Borel set has measure zero.

Continuing the example, since the set is countable, it has zero Lebesgue measure. So Lebesgue measure induces a measure on the shift (in this case the Bernoulli measure with probabilities $1/2$, $1/2$) for which we have an isomorphism.

You can make a similar reasoning for any $m$.

Answer 2

To talk about a measure theoretic isomorphism you first need measures on both spaces. The most classic example is if you give $[0,1]$ Lebesgue measure and give sequence space $\{0,1,\dots,m-1\}^{\mathbb{N}}$ the infinite product of the uniform measure on $\{0,1,\dots,m-1\}$. In this case the semiconjugacy map is a measure theoretic isomorphism between the two systems. There are also many other choices of measures you can do this with.