# Classification of ergodic measures for circle expanding maps

Let us consider the classical self-covering of the circle $S^1=\mathbb{R}/\mathbb{Z}$ given by $$\times_d(x) = dx \mod 1$$ where the degree $d$ is any integer greater than $1$.

There are a wealth of ergodic invariant measures; I know at least of uniform measures on periodic orbits, the Lebesgue measure, Gibbs measures for Hölder potentials, Bernoulli and Markov measures.

My question is:

To which extend do we know a kind of classification of all ergodic measures of $\times_d$? For example, is there a precise classification of ergodic measure of positive entropy?

I am pretty sure no complete, very explicit classification is known, because it would probably answer Furstenberg's conjecture (Lebesgue measure is the only non-atomic measure which is invariant under both $\times_2$ and $\times_3$). Any classification, however rough, or examples of invariant measures not cited above would be welcome.

• I'd like to add at Riesz product measures to this list, and suggest $g$-measures as a way of describing $X_d$-invariant measures. I believe (perhaps someone can confirm this?) that if $h_\mu(X_d) > 0$ then $\mu$ can be written as a $g$-measure. – Daniel Mansfield Aug 20 '15 at 6:32
• ... which is why nobody has used $g$-measures to solve Furstenberg's conjecture. Although it might be nice to reprove Dan Rudolph's result using $g$-measures. – Daniel Mansfield Aug 20 '15 at 6:35

You're right. The set of measures for these maps is a zoo! There is the obvious map (base $d$ expansion) $\pi$ from $\{0,1\ldots,d-1\}^{\mathbb N}$ to $[0,1)$ which is a bijection off a countable set. $\pi$ is then a factor map from the full one-sided $d$-shift to $\times_d$. There are only two ergodic invariant measures that give positive measure to the set where $\pi$ fails to be 1-1, namely the $\delta$-measures at $\overline 0$ and $\overline{d-1}$. This means that you can study invariant measures for $\times_d$ by studying invariant measures for the full shift on $d$ symbols, and there is a 1-1 correspondence between these sets of measures except for the additional fixed point that I mentioned.
There is also a natural bijection between one-sided and two-sided invariant measures on the full $d$-shift.
Maybe a nice way to dramatize the fact that there are lots of measures is to quote the Krieger generator theorem: for any invertible ergodic process at all (say on a Lebesgue space) with entropy less than $\log d$, there exists a generator: a finite partition such that the set of points that have a `twin' (another point whose entire forward and backward orbit follows the same sequence of partition elements) has measure 0. There is then a measure-theoretic isomorphism between your arbitrarily chosen ergodic transformation with entropy less than $\log d$ and $\times_d$ equipped with an appropriate ergodic invariant measure.
This property of $\times_d$, that it contains copies of all processes with entropy less than its own topological entropy is called universality. Some other universal processes are known.
• I was aware of the relation with the shift, but wanted to stress Furstenberg's conjecture (which cannot be stated for shifts as they act on different spaces). Universality I didn't know, this sure shows how hopeless it is to imagine a precise classification; but maybe there is something to be hoped when using the metric on the circle; for example maybe one could say something about not-too concentrated measures (e.g. $\mu([a,b])\le C|b-a|^\alpha$ for some $C$ and $\alpha<1$). – Benoît Kloeckner Feb 28 '15 at 18:11
• To be more specific about producing enormous numbers of measures all satisfying your positive entropy condition, given any measure on $\{0,1\ldots,d-1\}^{\mathbb Z}$, take its product with a product measure on $\{0,1\}^{\mathbb Z}$ (where 1 has measure $\epsilon$ and 0 has measure $1-\epsilon$). You can think of the new measure as a measure with $2d$ symbols $\{(0,0),\ldots,(1,d-1)\}$. All of these measures satisfy your concentration condition. As explained before, these measures have any factor with entropy $<\log d$. The randomization part adds $|\epsilon\log\epsilon|$ entropy. – Anthony Quas Mar 1 '15 at 6:04