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

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

DOI: https://doi.org/10.1090/S0002-9939-96-03025-0

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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.

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