Abstract
Continuous flows, whose orbits are the unstable manifolds of certain Axiom A attractors, are shown to be uniquely ergodic. The approach used is symbolic dynamics. Equicontinuity (and lack of it) for these flows is also discussed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R. Bellman,Introduction to Matrix Analysis, 2nd Edition, McGraw-Hill, 1970.
R. Bowen,Markov partitions for Axiom A diffeomorphisms, Amer. J. Math.92 (1970), 725–747.
R. Bowen,Markov partitions and minimal sets for Axiom A diffeomorphisms, Amer. J. Math.92 (1970), 907–918.
R. Bowen,Periodic points and measures for Axiom A diffeomorphisms, Trans. Amer. Math. Soc.154 (1971), 377–397.
R. Bowen,Equilibrium states and ergodic theory of Anosov diffeomorphisms, lecture notes.
J. Franks,Anosov diffeomorphisms, Proc. Symp. Pure Math.14, Amer. Math. Soc., Providence, R. I., 1970, 61–93.
H. Furstenberg,Strict ergodicity and transformation of the torus, Amer. J. Math.83 (1961), 573–601.
H. Furstenberg,The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics, Springer-Verlag Lecture Notes in Math. 318, 95–114.
M. Hirsch and C. Pugh,Stable manifolds and hyperbolic sets, Proc. Symp. Pure Math.14, Amer. Math. Soc., Providence, R. I., 1970, 133–163.
N. Kryloff and N. Bogoliuboff,La théorie générale de la mesure dans son application à l’étude des systèmes dynamiques non linéaires. Ann. of Math.38 (1937), 65–113.
B. Marcus,Reparametrizations of uniquely ergodic flows, to appear in J. Differential Equations.
S. Newhouse,On codimension one Anosov diffeomorphisms, Amer. J. Math.92 (1970), 761–770.
V. V. Nemytzkii and V. V. Stepanov,Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960, Ch. V.2.
D. Ruelle and D. Sullivan,Currents, Flows, and Diffeomorphisms, preprint.
Ya. G. Sinai,Markov partitions and C-diffeomorphisms, Functional Anal. Appl.2 (1968), 64–89.
S. Smale,Differentiable dynamical systems, Bull. Amer. Math. Soc.73 (1967), 747–817.
P. Walters,Introductory Lectures on Ergodic Theory, lecture notes, University of Maryland.
H. Whitney,Regular families of curves, Ann. of Math.34 (1933), 269.
R. F. Williams,One dimensional non-wandering sets, Topology6 (1967), 473–487.
R. F. Williams,Classification of one dimensional attractors, Proc. Symp. Pure Math.14, Amer. Math. Soc., Providence, R. I., 1970, 341–361.
R. F. Williams, written communication.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Marcus, B. Unique ergodicity of some flows related to axiom a diffeomorphisms. Israel J. Math. 21, 111–132 (1975). https://doi.org/10.1007/BF02760790
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02760790