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?
