An exceptional set in the ergodic theory of rational maps of the Riemann sphere

1997 ◽  
Vol 17 (2) ◽  
pp. 253-267 ◽  
Author(s):  
A. G. ABERCROMBIE ◽  
R. NAIR
Keyword(s):  
Ergodic Theorem ◽  
Ergodic Theory ◽  
Hausdorff Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Rational Maps ◽  
Exceptional Set ◽  
Rational Map ◽  
Pointwise Ergodic Theorem ◽  
Conformal Measure

A rational map $T$ of degree not less than two is known to preserve a measure, called the conformal measure, equivalent to the Hausdorff measure of the same dimension as its Julia set $J$ and supported there, with respect to which it is ergodic and even exact. As a consequence of Birkhoff's pointwise ergodic theorem almost every $z$ in $J$ with respect to the conformal measure has an orbit that is asymptotically distributed on $J$ with respect to this measure. As a counterpoint to this, the following result is established in this paper. Let $\Omega(z)=\Omega_{T}(z)$ denote the closure of the set $\{T^{n}(z):n=1,2,\ldots\}$. For any expanding rational map $T$ of degree at least two we set \[ S(z_{0})=\{z\in J:z_{0}\not\in \Omega_{T}(z)\}. \] We show that for all $z_{0}$ the Hausdorff dimensions of $S(z)$ and $J$ are equal.

Geometric measures for parabolic rational maps

1992 ◽  
Vol 12 (1) ◽  
pp. 53-66 ◽  
Author(s):  
M. Denker ◽  
M. Urbański
Keyword(s):  
Hausdorff Dimension ◽  
Hausdorff Measure ◽  
Julia Set ◽  
Rational Maps ◽  
Packing Measure ◽  
Rational Map ◽  
Dimensional Hausdorff Measure ◽  
Conformal Measure

AbstractLet h denote the Hausdorff dimension of the Julia set J(T) of a parabolic rational map T. In this paper we prove that (after normalisation) the h-conformal measure on J(T) equals the h-dimensional Hausdorff measure Hh on J(T), if h ≥ 1, and equals the h-dimensional packing measure Πh on J(T), if h ≤ 1. Moreover, if h < 1, then Hh = 0 and, if h > 1, then Πh(J(T)) = ∞.

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.

ON HAUSDORFF MEASURES AND SBR MEASURES FOR PARABOLIC RATIONAL MAPS

1999 ◽  
Vol 09 (09) ◽  
pp. 1763-1769
Author(s):  
M. DENKER ◽  
S. ROHDE
Keyword(s):  
Hausdorff Dimension ◽  
Julia Set ◽  
Rational Maps ◽  
Hausdorff Measures ◽  
Rational Map ◽  
Absolutely Continuous ◽  
Conformal Measure

If J is the Julia set of a parabolic rational map having Hausdorff dimension h<1, we show that Sullivan's h-conformal measure on J is either absolutely continuous or orthogonal with respect to the Hausdorff measures defined by the function [Formula: see text], according to whether τ>τ0 or τ<τ0 for some explicitly computable τ0>0.

The geometry of conformal measures for parabolic rational maps

2000 ◽  
Vol 128 (1) ◽  
pp. 141-156 ◽  
Author(s):  
B. O. STRATMANN ◽  
M. URBAŃSKI
Keyword(s):  
Julia Set ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Uniformly Perfect ◽  
Gauge Functions ◽  
Efficient Control ◽  
Rényi Dimension ◽  
Conformal Measure ◽  
Conformal Measures

We study the h-conformal measure for parabolic rational maps, where h denotes the Hausdorff dimension of the associated Julia sets. We derive a formula which describes in a uniform way the scaling of this measure at arbitrary elements of the Julia set. Furthermore, we establish the Khintchine Limit Law for parabolic rational maps (the analogue of the ‘logarithmic law for geodesics’ in the theory of Kleinian groups) and show that this law provides some efficient control for the fluctuation of the h-conformal measure. We then show that these results lead to some refinements of the description of this measure in terms of Hausdorff and packing measures with respect to some gauge functions. Also, we derive a simple proof of the fact that the Julia set of a parabolic rational map is uniformly perfect. Finally, we obtain that the conformal measure is a regular doubling measure, we show that its Renyi dimension and its information dimension are equal to h and we compute its logarithmic index.

DYNAMICS AND BIFURCATIONS OF A FAMILY OF RATIONAL MAPS WITH PARABOLIC FIXED POINTS

2011 ◽  
Vol 21 (11) ◽  
pp. 3323-3339
Author(s):  
RIKA HAGIHARA ◽  
JANE HAWKINS
Keyword(s):  
Hausdorff Dimension ◽  
Fixed Points ◽  
Riemann Sphere ◽  
Julia Set ◽  
Critical Orbit ◽  
Rational Maps ◽  
Complex Numbers ◽  
Period 2 ◽  
The Family ◽  
Dynamical Properties

We study a family of rational maps of the Riemann sphere with the property that each map has two fixed points with multiplier -1; moreover, each map has no period 2 orbits. The family we analyze is Ra(z) = (z3 - z)/(-z2 + az + 1), where a varies over all nonzero complex numbers. We discuss many dynamical properties of Ra including bifurcations of critical orbit behavior as a varies, connectivity of the Julia set J(Ra), and we give estimates on the Hausdorff dimension of J(Ra).

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.

On the transfer operator for rational functions on the Riemann sphere

1996 ◽  
Vol 16 (2) ◽  
pp. 255-266 ◽  
Author(s):  
Manfred Denker ◽  
Feliks Przytycki ◽  
Mariusz Urbański
Keyword(s):  
Equilibrium Measure ◽  
Riemann Sphere ◽  
Julia Set ◽  
Transfer Operator ◽  
Rational Functions ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Correlation Integral ◽  
Hölder Continuous Functions ◽  
Conformal Measure

AbstractLet T be a rational function of degree ≥ 2 on the Riemann sphere. Our results are based on the lemma that the diameter of a connected component of T−n(B(x, r)), centered at any point x in its Julia set J = J(T), does not exceed Lnrp for some constants L ≥ 1 and ρ > 0. Denote the transfer operator of a Hölder-continuous function φ on J satisfying P(T,φ) > supz∈Jφ(z). We study the behavior of {: n ≥ 1} for Hölder-continuous functions ψ and show that the sequence is (uniformly) normbounded in the space of Hölder-continuous functions for sufficiently small exponent. As a consequence we obtain that the density of the equilibrium measure μ for φ with respect to the -conformal measure is Höolder-continuous. We also prove that the rate of convergence of {ψ to this density in sup-norm is . From this we deduce the corresponding decay of the correlation integral and the central limit theorem for ψ.

scholarly journals Jarník and Julia; a Diophantine analysis for parabolic rational maps for Geometrically Finite Kleinian Groups with Parabolic Elements

2002 ◽  
Vol 91 (1) ◽  
pp. 27 ◽  
Author(s):  
B. O. Stratmann ◽  
M. Urbański
Keyword(s):  
Number Theory ◽  
Multifractal Analysis ◽  
Julia Sets ◽  
Kleinian Groups ◽  
Rational Maps ◽  
Rational Map ◽  
Geometrically Finite ◽  
Conformal Measure ◽  
Further Development

In this paper we derive a Diophantine analysis for Julia sets of parabolic rational maps. We generalise two theorems of Dirichlet and Jarník in number theory to the theory of iterations of these maps. On the basis of these results, we then derive a "weak multifractal analysis" of the conformal measure naturally associated with a parabolic rational map. The results in this paper contribute to a further development of Sullivan's famous dictionary translating between the theory of Kleinian groups and the theory of rational maps.

-stability of expanding maps in non-Archimedean dynamics

2017 ◽  
Vol 39 (4) ◽  
pp. 1002-1019
Author(s):  
JUNGHUN LEE
Keyword(s):  
Characteristic Zero ◽  
Complex Dynamics ◽  
Julia Set ◽  
Rational Maps ◽  
Expanding Maps ◽  
Rational Map

The aim of this paper is to show $J$-stability of expanding rational maps over an algebraically closed, complete and non-Archimedean field of characteristic zero. More precisely, we will show that for any expanding rational map, there exists a neighborhood of it such that the dynamics on the Julia set of any rational map in the neighborhood is the same as the dynamics of the expanding rational map as a non-Archimedean analogue of a corollary of Mañé, Sad and Sullivan’s result [On the dynamics of rational maps. Ann. Sci. Éc. Norm. Supér. (4)16 (1983), 193–217] in complex dynamics.

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