Quantitative approximation of invariant measures by periodic ones

Question

It is well known that many strongly chaotic dynamical systems have the property that periodic measures are (weak-star) dense in the space of all invariant probability measures. Is there some knowledge on how accurate on can approximate a given good measure by a periodic one in terms of length of the periodic orbit supporting it?

I mean, is there a known statement of the following kind: Let $M$ be a compact manifold, $f:M\to M$ is continuous, periodic measures are dense in the space of invariant probability measures. Let $f$ has a physical measure $\mu$ (i.e. for Lebesgue-almost every point $x$ one has $\frac{1}{n}\sum_{i=0}^{n-1} \delta_{f^i(x)} \to \mu$ ) that is not a finite sum of $\delta$-measures. Then under some additional condition on $f$ (for example, specification property) there exists an accuracy function $\gamma:\mathbb{N} \to \mathbb{R}_{\geq 0}$, $\gamma(n)\to 0, n\to \infty$ such that for every natural $N$ there exists a natural $n>N$ and an $n$-periodic orbit $p$ such that $$ \rho\left( \frac{1}{n}\sum_{i=0}^{n-1} \delta_{f^i(p)} , \mu \right) < \gamma(n) , $$ where $\rho$ is some metrisation of the weak-* topology.

If there is no general statement like this, are there examples (say, piecewise-expanding maps of the interval) where something like this is known to hold?

Answer 1

For a result of a somewhat similar type, check the paper "Rate of approximation of minimizing measures" by Bressaud and Quas, 2007, especially Theorem 4. In their case the dynamics is a subshift of finite type. They consider an arbitrary invariant measure, and look for periodic orbits in an $\varepsilon$-neighborhood of the support of the measure. True, it's not the type of approximation you want (weak-star), but it may be something to start with. They prove that $\varepsilon$ can be took superpolynomially small as a function of the period of the orbit, i.e., for any $K>0$ and $n_0 \ge 1$ it's possible to find a periodic orbit of period $n \ge n_0$ in the $n^{-K}$-neighborhood of $\mathrm{supp}(\mu)$. They also show that these bounds are sharp, so for example exponential approximation is not always possible.