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.

  • $\begingroup$ 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. $\endgroup$ Aug 20, 2015 at 6:32
  • $\begingroup$ ... 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. $\endgroup$ Aug 20, 2015 at 6:35

1 Answer 1


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.

  • $\begingroup$ 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 $\alpha<1$). $\endgroup$ Feb 28, 2015 at 18:11
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    $\begingroup$ So you can reformulate Furstenberg's conjecture in terms of shift spaces. Indeed this was what Dan Rudolph did when he proved his positive result in the presence of positive entropy. The usefulness of the shift is that it makes it very easy to construct lots of measures using what Kalikow calls the 'monkey method' (see his book with McCutcheon). So for example, it would be easy using this book to construct lots of very bizarre measures that still satisfy your non-concentration condition. $\endgroup$ Feb 28, 2015 at 18:40
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    $\begingroup$ 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. $\endgroup$ Mar 1, 2015 at 6:04

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