Hostname: page-component-7f64f4797f-rcs6z Total loading time: 0 Render date: 2025-11-04T00:16:28.269Z Has data issue: false hasContentIssue false
  • English
  • Français

No round wandering domains for $C^{1}$-diffeomorphisms of tori

Published online by Cambridge University Press:  10 April 2018

SERGEI MERENKOV
SERGEI MERENKOV*
Affiliation:
Department of Mathematics, City College of New York and CUNY Graduate Center, New York, NY 10031, USA email smerenkov@ccny.cuny.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We prove that if $n\geq 2$, then there is no $C^{1}$-diffeomorphism $f$ of the $n$-torus such that $f$ is semi-conjugate to a minimal translation and its wandering domains are geometric balls. This improves a recent result of Navas, who proved it assuming $C^{n+1}$ regularity of $f$.

Information

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 11 , November 2019 , pp. 3127 - 3135
Copyright
© Cambridge University Press, 2018 

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.)

Article purchase

Temporarily unavailable

References

Bonk, M., Kleiner, B. and Merenkov, S.. Rigidity of Schottky sets. Amer. J. Math. 131(2) (2009), 409–443.Google Scholar
Bonk, M.. Uniformization of Sierpiński carpets in the plane. Invent. Math. 186 (2011), 559–665.Google Scholar
Bohl, P.. Über die Hinsichtlich der unabhängigen und abhängigen Variablen periodische Differentialgleichung erster Ordnung. Acta Math. 40(1) (1916), 321–336.Google Scholar
Denjoy, A.. Sur les courbes définies par des équations différentiales à la surface du tore. J. Math. Pures Appl. 11 (1932), 333–375.Google Scholar
Heinonen, J.. Lectures on Analysis on Metric Spaces. Springer, New York, 2001.Google Scholar
Herman, M.-R.. Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Inst. Hautes Études Sci. Publ. Math.(49) (1979), 5–233.Google Scholar
McSwiggen, P. D.. Diffeomorphisms of the torus with wandering domains. Proc. Amer. Math. Soc. 117(4) (1993), 1175–1186.Google Scholar
McSwiggen, P. D.. Diffeomorphisms of the k-torus with wandering domains. Ergod. Th. & Dynam. Sys. 15(6) (1995), 1189–1205.Google Scholar
Navas, A.. Wandering domains for diffeomorphisms of the k-torus: a remark on a theorem by Norton and Sullivan. Ann. Acad. Sci. Fenn. Math. 43 (2018), 419–424.Google Scholar
Norton, A. and Sullivan, D.. Wandering domains and invariant conformal structures for mappings of the 2-torus. Ann. Acad. Sci. Fenn. Math. 21(1) (1996), 51–68.Google Scholar
Passeggi, A. and Sambarino, M.. Examples of minimal diffeomorphisms on 𝕋2 semiconjugate to an ergodic translation. Fund. Math. 222(1) (2013), 63–97.Google Scholar