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Preprints Archive: Abstract of IC2010011 (2010)

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Periodic orbits of circle homeomorphisms with a break point

by Akhtam Dzhalilov and Abdumajid Begmatov

Document info: Pages 18, Figures 3.

Let $f_{\theta}(x)=F_{0}(x)+\theta {\rm\; (mod 1)},$ $x\in S^{1}$, $\theta\in [0,1]$ be a family of preserving orientation circle homeomorphisms with a single break point $x_{b}$, i.e. with a jump in the first derivative $F_{0}$ at the point $x=x_{b}$. Suppose that $ F^\prime_{0}(x)$ is absolutely continuous on $[x_{b},x_{b}+1]$ and $F^{\prime \prime}_{0}(x)\in L_{\alpha}([0,1])$ for some $\alpha>1$. Consider $f_{\theta}$ with rational rotation number $\rho_{\theta}=\frac{p}{q}$ of rank $n$, i.e. $\frac{p}{q}=[k_{1},k_{2},...,k_{n}]$. We prove that for sufficiently large $n$, the homeomorphism $f_{\theta}$ has either a unique periodic orbit of period $q$ or two periodic orbits of period $q$. Also the renormalization behaviour of $f_{\theta}$ with rational rotation number $\rho_{\theta}=\frac{p}{q}$ is studied.

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