## Some questions on ergodic measures

This post will be a brief set of notes recording some thoughts from my talk at the Penn State dynamics seminar today, since I did not produce slides but rather spoke more informally about some results I’ve recently learned (which I’ve written about in the previous two posts).

Let ${X}$ be a compact metric space and ${f\colon X\rightarrow X}$ a continuous map. Write ${{\mathcal{M}_f}}$ for the space of ${f}$-invariant Borel probability measures and ${{\mathcal{M}_f^e}}$ for the space of ergodic measures in ${{\mathcal{M}_f}}$. Say that the system has property (D) if ${{\mathcal{M}_f^e}}$ is dense in ${{\mathcal{M}_f}}$, and property (C) if ${{\mathcal{M}_f^e}}$ is path-connected. We saw in the last two posts that (D) implies (C), and that both properties hold for systems with the specification property.

In addition to being of interest in their own right, properties (D) and (C) are related to various results on large deviations and multifractal analysis. We will not go into these here, but mention briefly another important related property: the system ${(X,f)}$ has entropy density of ergodic measures if for every ${\nu\in {\mathcal{M}_f}}$ there exists a sequence ${\nu_n \in {\mathcal{M}_f^e}}$ such that ${\nu_n \rightarrow \nu}$ in the weak* topology and in addition, ${h(\nu_n) \rightarrow h(\nu)}$. Thus entropy density (which was introduced by Orey in 1986 and Föllmer and Orey in 1988) is an a priori stronger condition than (D).

Another condition that is used in certain proofs of large deviations and multifractal results is the follows: we say that ${(X,f)}$ satisfies condition (E) if there exists a subspace ${V\subset C(X)}$ such that

1. ${V}$ is dense in the uniform norm;
2. every ${\varphi\in V}$ has a unique equilibrium state ${\mu_\varphi \in {\mathcal{M}_f^e}}$ — that is, a unique measure maximising the quantity ${h(\mu) + \int \varphi\,d\mu}$.

This condition was used by Kifer in 1990 and more recently by Comman and Rivera–Letelier in 2011 to obtain large deviations results. It also appears in an arXiv preprint from last year in which I use it to derive various multifractal results.

Condition (E) holds for systems with specification, and also for certain classes of systems without specification, such as the ${\beta}$-shifts. It gives a natural way of selecting paths in ${{\mathcal{M}_f^e}}$: if ${t\mapsto \varphi_t \in V}$ is a continuous path, then so too is ${t\mapsto \mu_{\varphi_t} \in {\mathcal{M}_f^e}}$.

There are a number of natural questions regarding the relationships between the various properties introduced so far, and I am unaware of the answer to a number of them.

1. Does condition (E) imply either (D) or (C)?
2. Is there a transitive map that satisfies (C) but not (D)? (I originally asked this question without the requirement of topological transitivity, and Misha Guysinsky pointed out that for the identity map, the ergodic measures are precisely the point masses, which are path-connected as long as the phase space is, but are not dense if phase space has more than one point.)
3. What is the relationship between (E) and entropy density?
4. Fix ${0 < h < h_{\mathrm{top}}(X,f)}$ and let ${{\mathcal{M}_f}(h) = \{ \nu\in {\mathcal{M}_f} \mid h(\nu) \geq h\}}$. Then ${{\mathcal{M}_f}(h)}$ is a metrisable simplex: is it the Poulsen simplex? Entropy density implies that each measure in ${{\mathcal{M}_f}(h)}$ can be approached by ergodic measures with entropies converging to at least ${h}$, but these measures may not lie in ${{\mathcal{M}_f}(h)}$ themselves. Thus it is not clear whether ${{\mathcal{M}_f^e}(h)}$ is dense in ${{\mathcal{M}_f}(h)}$ even if the system has entropy density. Similarly, is ${{\mathcal{M}_f^e}(h)}$ path-connected? 