scholarly journals Reduced dynamical systems

2020 ◽  
pp. 1-15
Author(s):  
LUKA BOC THALER ◽  
UROŠ KUZMAN
Keyword(s):  
Dynamical Systems ◽  
Fixed Number ◽  
Rational Maps ◽  
Rational Map ◽  
Ergodic Properties

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$ . We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

Matings of quadratic polynomials

1992 ◽  
Vol 12 (3) ◽  
pp. 589-620 ◽  
Author(s):  
Tan Lei
Keyword(s):  
Dynamical Systems ◽  
Mandelbrot Set ◽  
Rational Maps ◽  
Rational Map ◽  
Quadratic Polynomials ◽  
Branched Coverings ◽  
Equivalence Theory

AbstractWe apply Thurston's equivalence theory between dynamical systems of post-critically finite branched coverings and rational maps, to try to construct, from a pair of polynomials, a rational map. We prove that given two post-critically finite quadratic polynomials fc: z→z2+c and fc:z→ z2+c′, one can get a rational map if and only if c, c′ are not in conjugate limbs of the Mandelbrot set.

Erratum to ‘Semi-hyperbolic fibered rational maps and rational semigroups’ (Ergodic Theory and Dynamical Systems 26 (2006), 893–922)

2008 ◽  
Vol 28 (3) ◽  
pp. 1043-1045 ◽  
Author(s):  
HIROKI SUMI
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theory ◽  
Rational Maps

AbstractWe give a correction to the assumption of Theorems 1.12 and 2.6 in the paper [H. Sumi. Semi-hyperbolic fibered rational maps and rational semigroups. Ergod. Th. & Dynam. Sys.26 (2006), 893–922].

Ergodic Properties of Lamperti Operators

1978 ◽  
Vol 30 (6) ◽  
pp. 1206-1214 ◽  
Author(s):  
Charn-Huen Kan
Keyword(s):  
Fixed Number ◽  
Ergodic Properties

We shall assume throughout this paper, unless otherwise specified, that p is a fixed number, 1 < p < ∞.It is well known that to prove the poin.twise ergodic convergence of a contraction T on an Lp-space it is enough to prove a Dominated Ergodic Estimate (DEE) for T (see e.g. [11]).

scholarly journals THE SCENERY FLOW FOR HYPERBOLIC JULIA SETS

2002 ◽  
Vol 85 (2) ◽  
pp. 467-492 ◽  
Author(s):  
TIM BEDFORD ◽  
ALBERT M. FISHER ◽  
MARIUSZ URBAŃSKI
Keyword(s):  
Julia Set ◽  
Density Theorem ◽  
Finite Union ◽  
Maximal Entropy ◽  
Rational Maps ◽  
Rational Map ◽  
Open Set ◽  
Almost All ◽  
Scenery Flow ◽  
Measure Of Maximal Entropy

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.

Density of hyperbolicity for rational maps with Cantor Julia sets

2011 ◽  
Vol 32 (5) ◽  
pp. 1711-1726 ◽  
Author(s):  
WENJUAN PENG ◽  
YONGCHENG YIN ◽  
YU ZHAI
Keyword(s):  
Julia Set ◽  
Julia Sets ◽  
Rational Maps ◽  
Rational Map

AbstractIn this paper, taking advantage of quasi-conformal surgery, we prove that each non-hyperbolic rational map with a Cantor Julia set can be approximated by hyperbolic rational maps with Cantor Julia sets of the same degree.

scholarly journals RATIONAL MAPS ADMITTING MEROMORPHIC INVARIANT LINE FIELDS

2009 ◽  
Vol 80 (3) ◽  
pp. 454-461 ◽  
Author(s):  
XIAOGUANG WANG
Keyword(s):  
Chebyshev Polynomial ◽  
Rational Maps ◽  
Invariant Line ◽  
Rational Map ◽  
Line Field

AbstractIt is shown that a rational map of degree at least 2 admits a meromorphic invariant line field if and only if it is conformally conjugate to either an integral Lattès map, a power map, or a Chebyshev polynomial.

Transitive maps which are not ergodic with respect to Lebesgue measure

1996 ◽  
Vol 16 (4) ◽  
pp. 833-848 ◽  
Author(s):  
Sebastian Van Strien
Keyword(s):  
Lebesgue Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Iteration Scheme ◽  
Rational Maps ◽  
Positive Lebesgue Measure ◽  
Rational Map ◽  
Interval Maps ◽  
Open Set ◽  
Totally Disconnected

AbstractIn this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

scholarly journals Logical entropy of quantum dynamical systems

Open Physics ◽  
2016 ◽  
Vol 14 (1) ◽  
pp. 1-5 ◽  
Author(s):  
Abolfazl Ebrahimzadeh
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Quantum Logic ◽  
Quantum Dynamical ◽  
Quantum Dynamical System ◽  
Ergodic Properties ◽  
Quantum Dynamical Systems ◽  
Logical Entropy

AbstractThis paper introduces the concepts of logical entropy and conditional logical entropy of hnite partitions on a quantum logic. Some of their ergodic properties are presented. Also logical entropy of a quantum dynamical system is dehned and ergodic properties of dynamical systems on a quantum logic are investigated. Finally, the version of Kolmogorov-Sinai theorem is proved.

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