Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T23:09:07.532Z Has data issue: false hasContentIssue false
  • English
  • Français

On the construction of smooth ergodic skew-products†

Published online by Cambridge University Press:  19 September 2008

Mahesh G. Nerurkar
Mahesh G. Nerurkar
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, Ohio 43210, USA
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.

In this paper we prove results about lifting dynamical and ergodic properties of a given smooth dynamical system to its skew-product extensions by smooth cocycles. The classical small divisor argument shows that in general such results are not possible. However, using the notion of the ‘fast periodic approximation’ introduced by A. Katok, we will show that if the dynamical system admits such a ‘fast periodic approximation’ then indeed a certain qualitative behaviour which is prohibited by small divisor type conditions is now in fact generic. The techniques are also applied to show that ‘recurrent-proximal’ behaviour of solutions of linear differential equations with almost periodic coefficients is generic under suitable conditions on the coefficient matrix.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 8 , Issue 2 , June 1988 , pp. 311 - 326
Copyright
Copyright © Cambridge University Press 1988

References

REFERENCES

[1]Anosov, D. & Katok, A.. New examples in smooth ergodic theory. Trans. Moscow Math. Soc. 23 (1970), 1–35.Google Scholar
[2]Brin, M.. Topological transitivity of one class of dynamical systems and flows of frames on manifolds of negative curvature. Fund. Anal. 9 No. 1 (1975), 8–16.Google Scholar
[3]Ellis, R. & Johnson, R.. Topological dynamics and linear differential systems. J. Diff. Equations 44 No. 1 (1982), 21–39.Google Scholar
[4]Glasner, S. & Weiss, B.. On the construction of minimal skew products. Israel J. Math. 34 No. 4 (1979).Google Scholar
[5]Hawkins, J.. Smooth n valued cocycles for ergodic diffeomorphisms. Proceedings of AMS 93 No. 2 (1985), 307–311.Google Scholar
[6]Herman, M.. Construction de diffeomorphismes ergodiques (1979) preprint.Google Scholar
[7]Herman, M.. Construction d'un diffeomorphisme minimal d'entropie topologique non nulle. Ergod. Th. & Dynan. Syst. 1 (1981), 65–76.Google Scholar
[8]Herman, M.. Une Methode pour minorer les exposonts de Lyapounov. Comment. Math. Helv. 58 (1983), 453–502.Google Scholar
[9]Johnson, R.. Ergodic theory and linear differential equations. J. Diff. Equations 28 (1978), 23–34.Google Scholar
[10]Johnson, R.. On Floquet theory for almost periodic 2-dimensional linear systems. J. Diff. Equations 37 (1980).Google Scholar
[11]Johnson, R. & Sell, G.. Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential systems. J. Diff. Equations 41 (1981), 262–288.Google Scholar
[12]Katok, A.. Constructions in ergodic theory Vol. I and II. Lecture notes (to be published).Google Scholar
[13]Katok, A. & Stepin, A.. Approximations in ergodic theory. Russian Math. Surveys 22 No. 5 (1967), 77–102.Google Scholar
[14]Keynes, H. & Nerurkar, M.. Ergodicity in affine skew product toral extensions. Pacific J. Math. 123 No. 1 (1986).Google Scholar
[15]Krygin, A.. Examples of ergodic cylindrical cascades. Math. Notes U.S.S.R. 16 (1974), 1180–1186.Google Scholar
[16]Livsic, A.. Cohomology of dynamical systems. Math. U.S.S.R. Izvestija No. 6 (1972), 1278–1301.Google Scholar
[17]Millionschikov, V.. Proof of the existence of irregular systems of linear differential equation with almost periodic coefficients. Differential Equations (translated from Russian) 4 No. 3 (1968).Google Scholar
[18]Nerurkar, M.. Ergodic continuous skew product actions of amenable groups. Pacific J. Math. 119 No. 2 (1985).Google Scholar
[19]Nerurkar, M.. Recurrent-proximal linear differential systems. Proc. AMS 100 No. 4 (1987) 739.Google Scholar
[20]Sacker, R. & Sell, G.. A spectral theory for linear differential systems. J. Diff. Equations 27 (1978), 320–358.Google Scholar
[21]Vladimirov, V.. Generalized functions in Mathematical Physics. (Mir Publisher).Google Scholar
[22]Zimmer, R.. Amenable ergodic group actions and an application to Poisson boundaries of Random walks. J. Functional Analysis 27 No. 3 (1978), 350–372.Google Scholar