Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-28T02:22:43.179Z Has data issue: false hasContentIssue false
  • English
  • Français

Invariant measures for some one-dimensional attractors

Published online by Cambridge University Press:  19 September 2008

M. V. Jacobson
M. V. Jacobson
Affiliation:
Moscow, USSR
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider certain non-invertible maps of the square which are extensions of the quadratic maps of the interval and their small perturbations. We show that several maps of the type possess attractors which are not hyperbolic but have invariant measures similar to Bowen-Ruelle measures for hyperbolic attractors.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 3-4 , December 1982 , pp. 317 - 337
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Alexeyev, V. M., Quasi-random dynamical systems, I: Quasi-random diffeomorphisms. Math, of USSR-Sbornik 5 No. 1 (1968), 73–128.Google Scholar
[2]Adler, R. L., F-expansions revisited. Lecture Notes in Math. No. 318. Springer: Berlin, 1973, pp. 1–5.Google Scholar
[3]Anosov, D. V. & Sinai, Ya. G.. Some smooth ergodic systems. Russian Math. Surveys 22 No. 5 (1967), 102–167.CrossRefGoogle Scholar
[4]Bowen, R., On axiom A diffeomorphisms. Conference Board of the Mathematical Sciences. Regional Conference Series in Mathematics 35 (1978).Google Scholar
[5]Blanchard, F.. K-flots et théorème de renouvellement. Z. Wahrscheinlichkeitstheorie verw. Gebiete 36 (1976), 345–358.CrossRefGoogle Scholar
[6]Gurevich, B. M.. Some existence conditions for K-decompositions for special flows. Tr. Mosc. Mat. Ob. 17 (1967), 89–116. (In Russian).Google Scholar
[7]Guckenheimer, J., Oster, G. & Ipaktchi, A.. The dynamics of density dependent population models, Theor. Pop. Biol. 19 (1976).Google Scholar
[8]Jacobson, M. V.. Absolutely continuous invariant measures for one-parameter families of one-dimensioal maps. Commun. Math. Phys. 81 (1981), 39–88.CrossRefGoogle Scholar
[9]Jacobson, M. V.. Topological and metric properties of one-dimensional endomorphisms. Sou. Math. Dokl. 19 (1978), 1452–1455.Google Scholar
[10]Ledrappier, F.. Some properties of absolutely continuous measures on an interval. Ergod. Th. & Dynam. Sys. 1, No. 3 (1981).CrossRefGoogle Scholar
[11]Ratner, M.. Bernoulli flows over maps of the interval. Israel Math. J. 31 (1978), 298–314.CrossRefGoogle Scholar
[12]Rohlin, V. A.. Exact endomorphisms of a Lebesgue space. Izv. AN SSR, Ser. Mat. 25 (1961), 499–530. (In Russian).Google Scholar
[13]Walters, P.. Invariant measures and equilibrium states for some mappings which expand distances. Trans. Amer. Math. Soc. 236 (1978), 121–153.CrossRefGoogle Scholar
[14]Williams, R.. One-dimensional non-wandering sets. Topology 6 (1967), 473–487.CrossRefGoogle Scholar