Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-16T16:15:07.615Z Has data issue: false hasContentIssue false
  • English
  • Français

Cyclicity in families of circle maps

Published online by Cambridge University Press:  03 June 2013

O. KOZLOVSKI
O. KOZLOVSKI*
Affiliation:
Mathematics Institute, University of Warwick, UK email [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we will study families of circle maps of the form $x\mapsto x+ 2\pi r+ af(x)({\rm mod} \hspace{0.334em} 2\pi )$ and investigate how many periodic trajectories maps from this family can have for a ‘typical’ function $f$ provided the parameter $a$ is small.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 33 , Issue 5 , October 2013 , pp. 1502 - 1518
Copyright
© Cambridge University Press, 2013 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Arnol’d, V. I. and Il’yashenko, Yu. S.. Ordinary Differential Equations (Current Problems in Mathematics. Fundamental Directions, 1). Akad. Nauk SSSR Vsesoyuz. Inst. Nauchn. i Tekhn. Inform. Moscow, Itogi Nauki i Tekhniki, Moscow, 1985, pp. 7–149; 244.Google Scholar
Benyamini, Y. and Lindenstrauss, J.. Geometric Nonlinear Functional Analysis. American Mathematical Society, Providence, RI, 2000.Google Scholar
Christensen, J. P. R.. On sets of Haar measure zero in abelian Polish groups. Proceedings of the International Symposium on Partial Differential Equations and the Geometry of Normed Linear Spaces (Jerusalem, 1972), Israel J. Math. 13 (1972), 255–260.Google Scholar
Hunt, B. R., Sauer, T. and Yorke, J. A.. Prevalence: a translation-invariant ‘almost every’ on infinite-dimensional spaces. Bull. Amer. Math. Soc. 27 (2) (1992), 217–239.Google Scholar
Hurwitz, A.. Über die Fourierschen Konstanten integrierbarer Funktionen. Math. Ann. 57 (4) (1903), 425–446.Google Scholar
Ilyashenko, Yu.. Centennial history of Hilbert’s 16th problem. Bull. Amer. Math. Soc. (N.S.) 39 (3) (2002), 301–354.CrossRefGoogle Scholar
Preiss, D., Lindenstrauss, J. and Tiser, J.. Fréchet Differentiability of Lipschitz Functions and Porous Sets in Banach Spaces. Princeton University Press, Princeton, NJ, 2012.Google Scholar
Kozlovski, O. S.. Getting rid of the negative Schwarzian derivative condition. Ann. Math. (2) 152 (3) (2000), 743–762.CrossRefGoogle Scholar
Ott, W. and Yorke, J. A.. Prevalence. Bull. Amer. Math. Soc. 42 (3) (2005), 263–290.Google Scholar
Pólya, G. and Szegő, G.. Problems and Theorems in Analysis: Theory of Functions, Zeros, Polynomials II. Springer, Berlin, 1976.Google Scholar
Yakobson, M. V.. The number of periodic trajectories for analytic diffeomorphisms of a circle. Funktsional. Anal. i Prilozhen. 19 (1) (1985), 91–92.Google Scholar