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Relative density ofirrational rotation numbers in families of circle diffeomorphisms

Published online by Cambridge University Press:  01 February 1998

V. AFRAIMOVICH  and
T. YOUNG
V. AFRAIMOVICH
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL, USA (e-mail: valentin%[email protected])
T. YOUNG
Affiliation:
Department of Mathematics, Northwestern University, Evanston, IL, USA (e-mail: [email protected])
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Abstract

Consider a one-parameter family of circle diffeomorphisms which unfolds a saddle-node periodic orbit at the edge of an ‘Arnold tongue’. Recently it has been shown that homoclinic orbits of the saddle-node periodic points induce a ‘transition map’ which completely describes the smooth conjugacy classes of such maps and determines the universalities of the bifurcations resulting from the disappearance of the saddle-node periodic points. We show that after the bifurcation the relative density (measure) of parameter values corresponding to irrational rotation numbers is completely determined by the transition map and give a formula for this density. It turns out that this density is always less than 1 and generically greater than 0, with the exceptional cases having infinite co-dimension.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 18 , Issue 1 , February 1998 , pp. 1 - 16
Copyright
1998 Cambridge University Press

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