Synchronization of canonical measures for hyperbolic attractors. (English) Zbl 0618.58026

The author considers a \(C^ 2\) hyperbolic attractor. Under suitable conditions he proves that it is possible to change the velocity so that the measure of maximal entropy and the Sinai-Ruelle-Bowen measure for the new flow coincide.
Reviewer: A.Reinfelds (Riga)


37D99 Dynamical systems with hyperbolic behavior
28D10 One-parameter continuous families of measure-preserving transformations
Full Text: DOI


[1] Abramov, L.M.: On the entropy of a flow. Am. Math. Soc. Transl.49, 167-170 (1966) · Zbl 0185.21803
[2] Anosov, D.V.: Geodesic flows on closed Riemann manifolds with negative curvature. Proceedings of the Steklov Institute 90 (1967) · Zbl 0176.19101
[3] Anosov, D.V., Sinai, Ya.G.: Some smooth ergodic systems. Russ. Math. Surv.22, 103-167 (1967) · Zbl 0177.42002
[4] Bowen, R.: The equidistribution of closed geodesics. Am. J. Math.94, 413-423 (1972) · Zbl 0249.53033
[5] Bowen, R., Ruelle, D.: The ergodic theory of axiomA flows. Invent. Math.29, 181-202 (1975) · Zbl 0311.58010
[6] Hannay, J.H., Ozorio DeAlmeida, A.M.: Periodic orbits and a correlation function for the semiclassical density of states. J. Phys. A: Math. Gen.17, 3429-3440 (1984)
[7] Hirsch, M., Pugh, C.: Smoothness of horocycle foliations. J. Differ. Geom.10, 225-238 (1975) · Zbl 0312.58008
[8] Hirsch, M., Pugh, C., Shub, M.: Invariant manifolds. Lecture Notes in Math., Vol. 583. Berlin, Heidelberg, New York: Springer 1976 · Zbl 0355.58009
[9] Parry, W.: Bowen’s equidistribution theory and the Dirichlet density theorem. Ergodic Theory Dyn. Syst.4, 117-134 (1984) · Zbl 0567.58014
[10] Parry, W., Pollicott, M.: An analogue of the prime number theorem for closed orbits of axiomA flows. Ann. Math.118, 573-591 (1983) · Zbl 0537.58038
[11] Plante, J.F.: Anosov flows. Am. J. Math.94, 729-754 (1972) · Zbl 0257.58007
[12] Schwartzmann, S.: Asymptotic cycles. Ann. Math.66, 270-284 (1957) · Zbl 0207.22603
[13] Sinai, Ya.G.: Gibbs measures in ergodic theory. Russ. Math. Surv.166, 21-690 (1972) · Zbl 0246.28008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.