Some remarks on the geometry of circle maps with a break point



We study circle homeomorphisms $f\in
C^{2}(\mathbb{S}^1\backslash\{{x_b}\})$ whose rotation number $\rho_f$ is irrational,
with a single break point $x_b$ at which $f'$ has a jump discontinuity. We prove that the behavior
of the ratios of the lengths of any two adjacent intervals of the dynamical partition depends
on the size of break and on the continued fraction decomposition of $\rho_f$.
We also prove a result analogous to Yoccoz's lemma on asymptotic behaviour of the lengths of the
intervals of trajectories of renormalization transformation $R_n(f)$.


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