Abstract
We investigate the family of double standard maps of the circle onto itself, given by \({f_{a,b}(x)=2x+a+(b/{\pi})\sin(2\pi x)}\) (mod 1), where the parameters a,b are real and 0 ≤ b ≤ 1. Similarly to the well known family of (Arnold) standard maps of the circle, \({A_{a,b}(x)=x+a+(b/(2\pi))\sin(2\pi x)}\) (mod 1), any such map has at most one attracting periodic orbit and the set of parameters (a,b) for which such orbit exists is divided into tongues. However, unlike the classical Arnold tongues that begin at the level b = 0, for double standard maps the tongues begin at higher levels, depending on the tongue. Moreover, the order of the tongues is different. For the standard maps it is governed by the continued fraction expansions of rational numbers; for the double standard maps it is governed by their binary expansions. We investigate closer two families of tongues with different behavior.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnold, V.I.: Small denominators,. IMappings of the Circumference onto Itself. Amer. Math. Soc. Translations 46, 213–284 (1965)
Bergweiler, W.: Iteration of meromorphic functions. Bull. Amer. Math. Soc. 29, 151–188 (1993)
Blokh, A., Cleveland, C.,Misiurewicz, M.: Expanding polymodials. In: Modern Dynamical Systems and Applications eds. M. Brin, B. Hasselblatt, Ya. Pesin, Cambridge:Cambridge University Press, 2004, pp.253-270
Blokh, A., Cleveland, C.,Misiurewicz, M.: Julia sets of expanding polymodials. Ergodic Theory Dynam. Systems 25, 1691–1718 (2005)
Blokh, A., Misiurewicz, M.: Branched derivatives. Nonlinearity 18, 703–715 (2005)
Epstein, A., Keen, L., Tresser, C.: The set of maps \({F_{a,b}:x\mapsto x+a+\frac{b}{2\pi}\sin(2\pi x)}\) with any given rotation interval is contractible. Commun. Math. Phys. 173, 313–333 (1995)
Jonker, L.B.: The scaling ofArnold’s tongues for differentiable homeomorphisms of the circle. Commun. Math. Phys. 129, 1–25 (1990)
Levin, G., Świątek, G.: Universality of critical circle covers. Commun. Math. Phys. 228, 371–399 (2002)
de Melo, W., van Strien, S.: One-dimensional dynamics. Berlin: Springer-Verlag, 1993
Newhouse, S.E., Palis, J., Takens, F.: Bifurcations and stability of families of diffeomorphisms. Publ. Math. IHES 57, 5–71 (1983)
Shub, M., Sullivan, D.: Expanding endomorphisms of the circle revisited. Ergodic Theory Dynam Systems 5, 285–289 (1985)
Wenzel, W., Biham, O., Jayaprakash, C.: Periodic orbits in the dissipative standard map. Phys. Rev. A 43, 6550–6557 (1991)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by G. Gallavotti.
The first author was partially supported by NSF grant DMS 0456526.
The second author was supported by FCT Grant SFRH/BD/18631/2004.
CMUP is supported by FCT through POCTI and POSI of Quadro Comunitário de apoio III (2000-2006) with FEDER and national funding.
Rights and permissions
About this article
Cite this article
Misiurewicz, M., Rodrigues, A. Double Standard Maps. Commun. Math. Phys. 273, 37–65 (2007). https://doi.org/10.1007/s00220-007-0223-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-007-0223-5