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 aphysical measureof $f$ 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 astable physical measureof $f$ 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.