Some large deviation results for dynamical systems.

*(English)*Zbl 0721.58030Let 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.

(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 |