Transitions and anti-integrable limits for multi-hole Sturmian systems and Denjoy counterexamples

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 gap, and call an orbit of gaps a hole. In this paper, we generalize 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 generalized 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 $\rho (\,f_1)$ as $\rho (\,f)$, we construct a family $f_\varepsilon$ of Denjoy homeomorphisms of rotation number $\rho (\,f)$ containing $f$ such that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f), f)$ for $0<\varepsilon <\tilde{\varepsilon }<1$, but the number of holes changes at $\varepsilon =\tilde{\varepsilon }$, that $(\omega (\,f_\varepsilon ), f_\varepsilon )$ is conjugate to $(\omega (\,f_{\tilde{\varepsilon }}), f_{\tilde{\varepsilon }})$ for $\tilde{\varepsilon }\leqslant \varepsilon <1$ but $\lim _{\varepsilon \nearrow 1}f_\varepsilon (t)=f_1(t)$ for any $t\in S$, and that $f_\varepsilon$ has a singular limit when $\varepsilon \searrow 0$. We show this singular limit is an anti-integrable limit (AI-limit) in the sense of Aubry. That is, the Denjoy minimal system reduces to a symbolic dynamical system. The AI-limit can be degenerate or nondegenerate. All transitions can be precisely described in terms of the generalized Sturmian systems.

Eventually stable rational functions

For a field [Formula: see text], rational function [Formula: see text] of degree at least two, and [Formula: see text], we study the polynomials in [Formula: see text] whose roots are given by the solutions in [Formula: see text] to [Formula: see text], where [Formula: see text] denotes the [Formula: see text]th iterate of [Formula: see text]. When the number of irreducible factors of these polynomials stabilizes as [Formula: see text] grows, the pair [Formula: see text] is called eventually stable over [Formula: see text]. We conjecture that [Formula: see text] is eventually stable over [Formula: see text] when [Formula: see text] is any global field and [Formula: see text] is any point not periodic under [Formula: see text] (an additional non-isotriviality hypothesis is necessary in the function field case). We prove the conjecture when [Formula: see text] has a discrete valuation for which (1) [Formula: see text] has good reduction and (2) [Formula: see text] acts bijectively on all finite residue extensions. As a corollary, we prove for these maps a conjecture of Sookdeo on the finiteness of [Formula: see text]-integral points in backward orbits. We also give several characterizations of eventual stability in terms of natural finiteness conditions, and survey previous work on the phenomenon.

Control and synchronization of Julia sets of the complex dissipative standard system

The fractal behaviors of the complex dissipative standard system are discussed in this paper. By using the boundedness of the forward and backward orbits, Julia set of the system is introduced and visualization of Julia set is also given. Then a controller is designed to achieve Julia set shrinking or expanding with the changing of the control parameter. And synchronization of two different Julia sets is discussed by adding a coupling item, which makes one Julia set change to be the other. The simulations illustrate the efficacy of these methods.