scholarly journals How two planets likely acquired their backward orbits

Physics Today ◽  
2021 ◽  
Vol 74 (5) ◽  
pp. 12-14
Author(s):  
Johanna L. Miller
Keyword(s):  
Backward Orbits

scholarly journals Backward orbits in the unit ball

2019 ◽  
Vol 147 (9) ◽  
pp. 3947-3954 ◽  
Author(s):  
Leandro Arosio ◽  
Lorenzo Guerini
Keyword(s):  
Unit Ball ◽  
Backward Orbits

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

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
Yi-Chiuan Chen
Keyword(s):  
Finite Number ◽  
Rotation Number ◽  
Singular Limit ◽  
Minimal System ◽  
Minimal Set ◽  
Symbolic Dynamical System ◽  
Sturmian System ◽  
Omega Limit Set ◽  
Transitive Orientation ◽  
Backward Orbits

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 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.

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

2016 ◽  
Vol 21 (4) ◽  
pp. 465-476
Author(s):  
Weihua Sun ◽  
Yongping Zhang
Keyword(s):  
Control Parameter ◽  
Julia Set ◽  
Julia Sets ◽  
The Other ◽  
Standard System ◽  
Control And Synchronization ◽  
Backward Orbits

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.

scholarly journals Backward orbits and petals of semigroups of holomorphic self-maps of the unit disc

2018 ◽  
Vol 198 (2) ◽  
pp. 411-441
Author(s):  
Filippo Bracci ◽  
Manuel D. Contreras ◽  
Santiago Díaz-Madrigal ◽  
Hervé Gaussier
Keyword(s):  
Unit Disc ◽  
Backward Orbits

scholarly journals Eventually stable rational functions

2017 ◽  
Vol 13 (09) ◽  
pp. 2299-2318 ◽  
Author(s):  
Rafe Jones ◽  
Alon Levy
Keyword(s):  
Rational Function ◽  
Function Field ◽  
Rational Functions ◽  
Global Field ◽  
Good Reduction ◽  
Integral Points ◽  
Finiteness Conditions ◽  
Backward Orbits ◽  
Function Field Case ◽  
Eventual Stability

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.

scholarly journals Integer points in backward orbits

2011 ◽  
Vol 131 (7) ◽  
pp. 1229-1239 ◽  
Author(s):  
Vijay A. Sookdeo
Keyword(s):  
Integer Points ◽  
Backward Orbits
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