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Some large deviation results for dynamical systems. (English) Zbl 0721.58030
Let X be a compact metric space with a continuous mapping $$f:X\to X$$, and let m be a Borel probability measure on X which is not necessarily f- invariant. The large deviations considered in this paper concern
(A) for a continuous real function $$\phi$$ on X with $$n^{-1}S_ n\phi =n^{-1}\sum^{n-1}_{i=0}\phi \circ f^ i$$ converging to a constant $${\bar \phi}$$ the way in which $$m\{$$ $$x\in X:| n^{-1}S_ n\phi (x)-{\bar \phi}| >\delta \}$$ tends to 0 if $$n\to \infty;$$
(B) for an attractor $$\Lambda$$ of f with an attracted neighbourhood U and special interesting f-invariant measures $$\mu$$ on $$\Lambda$$ (Sinai-Bowen- Ruelle measures or equilibrium states) the behaviour of $$m\{$$ $$x\in U:| n^{-1}S_ n\phi (x)-\int \phi d\mu | >\delta \}$$ if $$n\to \infty$$ (here it is assumed that X is a manifold, f is differentiable and $$\Lambda$$ satisfies certain hyperbolicity conditions);
(C) for f as in (B) and $$\Lambda$$ a compact invariant set with a neighbourhood U satisfying $$\Lambda =\cap^{\infty}_{i=-\infty}f^ i(U)$$ the behaviour of $$m\{$$ $$x\in X:$$ $$f^ i(x)\in U$$ for $$0\leq i<n\}$$ if $$n\to \infty;$$
(D) new proofs for known results in the case that X is a shift space [see R. Ellis: Entropy, large deviations, and statistical mechanics, Springer (1985; Zbl 0566.60097); M. Donker and S. R. S. Varadhan: Commun. Pure Appl. Math. 36, 183-212 (1983; Zbl 0512.60068)].
The main theorem from which the further results are derived states that for any continuous function $$\phi$$ and any $$c\in {\mathbb{R}}$$ three inequalities hold, provided f has finite topological entropy:
(1) $$\liminf_{n\to \infty} n^{-1}\log m\{x\in X:$$ $$n^{-1}S_ n\phi (x)>c\} \geq \sup \{h_{\nu}(f)-h_ m(f,\nu):$$
$$\nu$$ f-invariant ergodic probability measure satisfying $$\int \phi d\nu >c\}$$
(2) $$\limsup_{n\to \infty}n^{-1}\log m\{x\in X:$$ $$n^{-1}S_ n\phi (x)\geq c\} \leq \sup \{h_{\nu}(f)-\int \xi d\nu:$$
$$\nu$$ f-invariant probability measure satisfying $$\int \phi d\nu \geq c\}.$$
Here $$h_{\nu}(f)$$ denotes the entropy with respect to $$\nu$$, $$h_ m(f,\nu)$$ is the $$\nu$$-essential supremum of the function $x \mapsto - \lim_{\epsilon \to 0} \limsup_{n\to \infty} n^{-1}\log m\{y\in X:\;d(f^ i(x),f^ i(y))<\epsilon \text{ for } 0\leq i<n\}$ (x$$\in X)$$, and $$\xi$$ is any real continuous function on X for which there are positive C, $$\epsilon$$ such that for any $$x\in X$$, $$n\geq 0$$ $m\{y\in X:\;d(f^ i(x),f^ i(y))<\epsilon \text{ for } 0<i\leq n\} < Ce^{-S_ n\phi (x)}.$ The third inequality provides another lower bound for the left hand side of (1) under the condition that some kind of shadowing is possible in X.
Reviewer: H.G.Bothe (Berlin)

##### MSC:
 37A99 Ergodic theory 60F10 Large deviations 82B05 Classical equilibrium statistical mechanics (general) 54C70 Entropy in general topology
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