Equilibrium states and weighted uniform distribution of closed orbits.

*(English)*Zbl 0667.58056
Dynamical systems, Proc. Spec. Year, College Park/Maryland, Lect. Notes Math. 1342, 617-625 (1988).

[For the entire collection see Zbl 0653.00011.]

This talk is concerned with a theorem proved by Bowen in 1973 and some generalisations via a zeta function. The theorem of Bowen asserts that closed orbits for Axiom A flows are uniformly distributed within a basic set with respect to the associated measure of maximal entropy. This paper addresses the question as to what one can say about closed orbits in relation to other equilibrium states. The main result of this paper has already been observed by J. H. Hannay and A. M. Ozorio de Almeida [J. Phys. A 17, 3429-3440 (1984)], who present what might be called a physicists proof. The present author now gives a clearer understanding of the result.

This talk is concerned with a theorem proved by Bowen in 1973 and some generalisations via a zeta function. The theorem of Bowen asserts that closed orbits for Axiom A flows are uniformly distributed within a basic set with respect to the associated measure of maximal entropy. This paper addresses the question as to what one can say about closed orbits in relation to other equilibrium states. The main result of this paper has already been observed by J. H. Hannay and A. M. Ozorio de Almeida [J. Phys. A 17, 3429-3440 (1984)], who present what might be called a physicists proof. The present author now gives a clearer understanding of the result.

Reviewer: Ding Tongren

##### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |

37A99 | Ergodic theory |

37D99 | Dynamical systems with hyperbolic behavior |