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Let $M$ be a Riemannian manifold with volume measure $\lambda$, let $f \colon M \to M$ be a continuous map, and let $\mu$ be a probability measure on $M$ with compact support.

Definition. The measure $\mu$ is a physical measure of $f$ if there is a set $V \subset M$ with [PropertyX] such that for all $x \in V$, the sequence $\frac{1}{n}\sum_{i=0}^{n-1} \delta_{f^i(x)}$ converges weakly to $\mu$ as $n \to \infty$.

Different definitions give different versions of Property X; three that I've seen are:

  • $V$ is a $\lambda$-positive measure set;
  • $V$ includes $\lambda$-almost all points in some neighbourhood of $\mathrm{supp}\,\mu$;
  • $V$ is a neighbourhood $\mathrm{supp}\,\mu$.

Example. Take $M=\mathbb{S}^1$ with $\lambda=\mathrm{Lebesgue}$, and let $f \colon M \to M$ be a homeomorphism with a unique fixed point $p$. (E.g. $f(x)=x+\varepsilon(1-\cos(2\pi x))$, with $p=0$.) Since all trajectories of $f$ converge to $p$, we clearly have that $\delta_p$ is a physical measure (under any version of Property X).

However, in the above example, the fixed point $p$ is not stable (not even stable in the sense of Lyapunov), and therefore it seems intuitively strange to me to consider $\delta_p$ a "physical" measure. I feel like there should be a stronger stability requirement, such as what I will now suggest.

Definition. Given a topological space $T$, we say that a sequence of sets $S_n \subset T$ converges to a point $x^\ast \in T$ if for every neighbourhood $U$ of $x^\ast$ there exists $N \in \mathbb{N}$ such that for all $n \geq N$, $S_n \subset U$.

Definition. The measure $\mu$ is a stable physical measure of $f$ if there is a neighbourhood $V$ of $\mathrm{supp}\,\mu$ such that the sequence of sets $$ S_n = \left\{ \frac{1}{n}\sum_{i=0}^{n-1} \delta_{f^i(x)} : x \in V \right\} $$ converges to $\mu$ with respect to the topology of weak convergence.

Remark. The sequence $S_n$ converges to $\mu$ with respect to the topology of weak convergence if and only if for every bounded continuous function $g \colon M \to \mathbb{R}$, $\frac{1}{n}\sum_{i=0}^{n-1}g(f^i(x)) \to \mu(g)$ uniformly across $x \in V$ as $n \to \infty$.

Is there an existing term for what I have called a "stable physical measure", or for something conceptually similar to this? Are there any references that simply define the term "physical measure" along similar lines to how I have defined a "stable physical measure"?


Further thoughts. I suppose the issue I'm raising is somewhat philosophical -- actually there's a very natural sense in which it is reasonable to consider $\delta_p$ in my example as being "stable": For all $\varepsilon>0$ there exists $\delta>0$ such that given any $\delta$-pseudo-orbit $(x_n)$ of $f$, for sufficiently large $n$ we have $\frac{1}{n}\sum_{i=0}^{n-1}\mathbf{1}_{B_\varepsilon(p)}(x_i)>1-\varepsilon$.

I wonder whether some analogous statement holds for more general physical measures -- i.e. whether for a large class of "physical measures" under the definition where $V$ is simply a $\lambda$-positive set, "typical" pseudo-orbits starting in $V$ will eventually have their empirical measures close to $\mu$.

I suppose my intuition for being "physical" was about the physical realisticness of being able to model a time-series $(X_n)_{n=0,\ldots,N}$ recorded at a "random" time from a process that has been running since "indefinitely long into the past" as a stochastic process whose law is the image measure of $\mu$ under $x \mapsto (x,f(x),\ldots,f^N(x))$. The conventional definition of a "physical" measure (with $V$ simply being $\lambda$-positive) seems to describe exactly this property if one ignores the potential for small perturbations. That's why I wondered whether there is a version that describes this property and also takes into account the potential for small perturbations. Perhaps the best way to achieve this is simply to specify that the measure $\mu$ is not only physical but also has the property that every trajectory in $\mathrm{supp}\,\mu$ is stable in the sense of Lyapunov.

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  • $\begingroup$ I've never seen this definition before. It looks a bit too strong to me, though; not because of your addition, but because you require $V$ to be a neighborhood of the support. Once you go beyond delta measures, the most important examples of physical measures are the SRB measures, and for uniformly hyperbolic systems these certainly do not satisfy even your first definition if you require $V$ to be a neighborhood of the support; there are large sets of "irregular points" whose empirical measures do not converge to $\mu$. $\endgroup$ – Vaughn Climenhaga Jul 11 at 3:00
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I realize this doesn't directly answer the "reference request" part of the question, but I believe that if you require $V$ to be full (Lebesgue) measure in a neighborhood of the support of $\mu$, then this definition is too strong to be satisfied by any of the usual examples that one would want, going beyond delta measures to SRB measures where the expanding direction is nontrivial.

Here's why: Suppose $\Lambda$ is a topologically mixing Axiom A attractor for a diffeo $f$, and $\mu$ is the SRB measure. Let $\xi(x) = \log \det Df|_{E^u(x)}$, so that $\mu$ satisfies the following lower Gibbs bound: $$\mu \{ y : d(f^k y, f^k x) \leq \epsilon\} \geq C(\epsilon) e^{-S_n\xi(x)}. $$ Theorem 1 (part 3) of Lai-Sang Young's 1990 TAMS paper "Large deviation results for dynamical systems" gives, for each continuous $\phi\colon \Lambda\to \mathbb{R}$, the estimate $$ \varliminf_{n\to\infty} \frac 1n \log \mu \{ x : \frac 1n S_n\phi(x) > c \} \geq \sup \{ h_\nu(f) - \int \xi\,d\nu : \nu \text{ is $f$-invariant and } \int\phi\,d\nu > c\}. $$ Write the supremum as $\alpha(\phi,c)$, so that for each $\beta < \alpha(\phi,c)$ we have some constant $K$ such that $$ \mu \{x : \frac 1n S_n\phi(x) > c \} \geq K e^{\beta n}. $$ Choose any continuous $\phi$ such that there is an $f$-invariant $\nu$ with $\int\phi\,d\nu > \int\phi\,d\mu$ (such a $\phi$ exists since $f$ is not uniquely ergodic), and choose $c$ between these two integrals. Then taking $U = \{ m : \int \phi \,dm < c\}$, we see that $U$ is an open set containing $\mu$ for which we have $$ \mu \{ x : \frac 1n \sum_{k=0}^{n-1} \delta_{f^k x} \notin U \} \geq K(\beta) e^{\beta n} $$ whenever $\beta < \alpha(\phi,c)$; here $\alpha>-\infty$ since we can take $\nu=m$ in the supremum. I expect (though I didn't check details) that a similar bound would hold replacing $\mu$ by $\lambda$ restricted to a neighborhood. In particular if $V$ contains a full measure set in a neighborhood of the support of $\mu$, then the sets $S_n$ that you define would never be contained in $U$, so the sequence of sets would not converge to $\mu$.

This doesn't rule out obtaining this behavior for a positive measure set of $V$, and indeed you could probably get it for sets $V$ whose complements intersect the neighborhood of the support with arbitrarily small Lebesgue measure. But at this point I've strayed quite far from your actual request for a reference... I just had the thought about large deviations results creating problems for some possible definitions, and then got carried away writing down the details.

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  • $\begingroup$ Thank you for this! I suppose the basic point you're getting at is that the very thing we want from the study of "physical measures" is the ability to observe invariant measures on sets whose dynamics is not so nice and stable as to make the observability of these measures a trivial matter, i.e. where there are expanding directions? And this will probably inherently rule out the kind of "stability" that I'm asking for? $\endgroup$ – Julian Newman Jul 11 at 13:56

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