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I've recently been looking at dynamical systems recently, more precisely expanding maps. I know that for a dynamical system $T:X \to X$ such that $T$ is continuous on $X$ with $X$ being a compact metric space. A measure defined by

$\mu_{n} = \frac{1}{n}\sum_{j=0}^{n-1}T_{*}^{j}\delta$

converges to an invariant measure $\mu$ for $T$

For the main part of the proof $|\int_{X}f \circ Td\mu-\int_{X}fd\mu| \leq lim_{k \to \infty} \frac{2\|f\|_{\infty}}{n_{k}} = 0$. I want to know if there is some way to find an explicit expression for $n_{k}$ or alternatively some kind of bound for the sequence.

Essentially what I'm trying to do is find some constant $C$ and a sequence $x_{n}$ which tends to $0$ such that $|\mu_{n}(A) - \mu(A)| \leq Cx_{n} \forall A \in \mathcal{B}(X)$ where $\mathcal{B}(X)$ is the borel algebra.

I've been told that this limit can be bounded by some exponentially decreasing sequence for expanding maps but I've not managed to find a source for this yet.

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    In general the sequence $\mu_n$ does not converge (quite easy to give examples), but it has always a converging subsequence. You should look for "exponential decay of correlations" – 2017-02-16
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    Is there any way to find an explicit formula for $n_{k}$ or find a sequence such that $\frac{1}{n_{k}} \leq x_{k}$ such that $x_{k}$ tends to $0$ for the case when $T$ is an expanding map? – 2017-02-16
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    The problem is that you are asking something that is close to research, in case you really want/need an explicit sequence. That's why I pointed out to the expression, without telling us more you need to look for it yourself. – 2017-02-16
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    However, in case you only aim is to prove the part "converges to an invariant measure $\mu$ for $T$", then simply look for the Krylov-Bogolubov theorem. The proof is a simple application of the contraction mapping principle, once you put a distance on the space of probability measures (on a compact metric space). – 2017-02-16

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