16-78 Yi-Chiuan Chen

Transitions and Anti-integrable Limits for Multi-hole Sturmian Systems and Denjoy Counterexamples (587K, PDF) Sep 22, 16

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Abstract. For a Denjoy homeomorphism $f$ of the circle $S$, we call a pair of distinct points of the $\omega$-limit set $\omega (f)$ whose forward and backward orbits converge together a {\it gap}, and call an orbit of gaps a {\it hole}. In this paper, we generalise the Sturmian system of Morse and Hedlund and show that the dynamics of any Denjoy minimal set of finite number of holes is conjugate to a generalised Sturmian system. Moreover, for any Denjoy homeomorphism $f$ having a finite number of holes and for any transitive orientation-preserving homeomorphism $f_1$ of the circle with the same rotation number $ ho (f_1)$ as $ ho(f)$, we construct a family $f_\epsilon$ of Denjoy homeomorphisms of rotation number $ ho (f)$ containing $f$ such that $(\omega (f_\epsilon), f_\epsilon)$ is conjugate to $(\omega (f), f)$ for $0<\epsilon< ilde{\epsilon}<1$ but the number of holes changes at $\epsilon= ilde{\epsilon}$, that $(\omega (f_\epsilon), f_\epsilon)$ is conjugate to $(\omega (f_{ ilde{\epsilon}}), f_{ ilde{\epsilon}})$ for $ ilde{\epsilon}\le\epsilon<1$ but $\lim_{\epsilon earrow 1}f_\epsilon(t)=f_1(t)$ for any $t\in S$, and that $f_\epsilon$ has a singular limit when $\epsilon\searrow 0$. We show this singular limit is an anti-integrable limit in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The anti-integrable limit can be degenerate or non-degenerate. All transitions can be precisely described in terms of the generalised Sturmian systems.

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