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

#### Abstract

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 $R_n(f)$.

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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