A Farey tree organization of locking regions for simple circle maps

Brucks, K.; Tresser, C.

doi:10.1090/S0002-9939-96-03025-0

A Farey tree organization of locking regions for simple circle maps
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by K. M. Brucks and C. Tresser PDF

Proc. Amer. Math. Soc. 124 (1996), 637-647 Request permission

Abstract:

Let $f$ be a $C^3$ circle endomorphism of degree one with exactly two critical points and negative Schwarzian derivative. Assume that there is no real number $a$ such that $f + a$ has a unique rotation number equal to $\frac {p}{q}$. Then the same holds true for any $\frac {p’}{q’}$ such that $\frac {p}{q}$ stands above $\frac {p’}{q’}$ in the Farey tree and can be related to it by a path on the tree.

References

L. Alsedá, J. Llibre, and M. Misiurewicz, Combinatorial dynamics and entropy in dimension one, World Scientific, River Edge, NJ, 1993, .
P. M. Blecher and M. V. Jakobson, Absolutely continuous invariant measures for some maps of the circle, Statistical physics and dynamical systems (Köszeg, 1984) Progr. Phys., vol. 10, Birkhäuser Boston, Boston, MA, 1985, pp. 303–315. MR 821303
Philip L. Boyland, Bifurcations of circle maps: Arnol′d tongues, bistability and rotation intervals, Comm. Math. Phys. 106 (1986), no. 3, 353–381. MR 859816, DOI 10.1007/BF01207252
Alain Chenciner, Jean-Marc Gambaudo, and Charles Tresser, Une remarque sur les familles d’endomorphismes de degré $1$ du cercle, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 15, 771–773. MR 772091
A. Epstein, L. Keen, and C. Tresser, The set of maps $f_{a,b} : \theta \mapsto \theta + a + \frac {b}{2 \pi } \cdot \sin (2 \pi \theta )$ with any given rotation numbers is contractible, Comm. Math. Phys. (to appear).
B. Friedman and C. Tresser, Comb structure in hairy boundaries: some transition problems for circle maps, Phys. Lett. A 117 (1986), no. 1, 15–22. MR 851335, DOI 10.1016/0375-9601(86)90228-8
Jean-Marc Gambaudo and Charles Tresser, On the dynamics of quasi-contractions, Bol. Soc. Brasil. Mat. 19 (1988), no. 1, 61–114. MR 1018928, DOI 10.1007/BF02584821
L. Goldberg and C. Tresser, Rotation orbits and the farey tree, Ergodic Theory Dynamical Systems (to appear).
G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 5th ed., The Clarendon Press, Oxford University Press, New York, 1979. MR 568909
K. M. Khanin, Universal estimates for critical circle mappings, Chaos 1 (1991), no. 2, 181–186. MR 1135906, DOI 10.1063/1.165826
R. S. MacKay and C. Tresser, Transition to topological chaos for circle maps, Phys. D 19 (1986), no. 2, 206–237. MR 844701, DOI 10.1016/0167-2789(86)90020-5
H. Poincaré, Sur les courbes définies par des équations différentielles, J. Math. Pures Appl. (4) 1 (1885), 167–244, Euvres Complètes.
R. M. Siegel, C. Tresser, and G. Zettler, A decoding problem in dynamics and in number theory, Chaos 2 (1982), 473–494.
David J. Uherka, Charles Tresser, Roza Galeeva, and David K. Campbell, Errata: “Solvable models for the quasi-periodic transition to chaos” [Phys. Lett. A 170 (1992), no. 3, 189–194; MR1187926 (93h:58106)], Phys. Lett. A 177 (1993), no. 6, 461. MR 1227278, DOI 10.1016/0375-9601(93)90981-5