The Collet–Eckmann condition for rational functions on the Riemann sphere

Aspenberg, Magnus

doi:10.1007/s00209-012-1039-3

Cited by 22 publications

(41 citation statements)

References 8 publications

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“…In the present paper, we are interested in the following simpler but related problem: is any Collet-Eckmann rational map approximated by hyperbolic rational maps? Refining Rees' results, Aspenberg [Asp2] showed that there is a set of positive Vol M d -measure of (suitable) Collet-Eckmann rational map (such maps are in the bifurcation locus). A rational map f ∈ Rat d is Collet-Eckmann if the critical set C(f ) of f is contained in J f and if there exist γ, γ 0 > 0 such that (CE(γ, γ 0 )) |(f n ) ′ (f (c))| ≥ e nγ−γ 0 , for any c ∈ C(f ) and any n ≥ 0.…”

Section: Introductionmentioning

confidence: 81%

“…This subsection is devoted to the proof of Theorem C. In order to find a set of positive measure of good Collet-Eckmann rational maps, our strategy is as follows: first, we deduce from the main result of [Asp2] the existence of a positive measure set of Collet-Eckmann rational maps satisfying all of the previous conditions with appropriate constants. Then, we follow Tsujii's presentation and improvement [Ts] of the Benedicks-Carleson theory for interval maps [BC].…”

Section: Getting a Positive Measure Set Of Good Parametersmentioning

confidence: 99%

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Collet, Eckmann and the bifurcation measure

et al. 2019

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The moduli space M d of degree d ≥ 2 rational maps can naturally be endowed with a measure µ bif detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure µ bif has positive Lebesgue measure. To do so, we establish a general sufficient condition for the conjugacy class of a rational map to belong to the support of µ bif and we exhibit a large set of Collet-Eckmann rational maps which satisfy that condition. As a consequence, we get a set of Collet-Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

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Section: Introductionmentioning

confidence: 81%

Section: Getting a Positive Measure Set Of Good Parametersmentioning

confidence: 99%

Collet, Eckmann and the bifurcation measure

et al. 2019

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“…In [2] Avila and Moreira showed: In [1] Aspenberg proved that the set of Collet-Eckmann parameters has positive Lebesgue measure in the space of coefficients of all rational maps of fixed degree d 2. Moreover, there is a conjecture that almost all parameters in this space correspond to either Collet-Eckmann or hyperbolic maps.…”

Section: Collet-eckmann Maps and The Statement Of The Principal Resultsmentioning

confidence: 99%

Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

Dudko

Yampolsky

2017

Found Comput Math

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“…We outline and develop the so called large deviation argument, first invented by Benedicks and Carleson in [4], also used in [2]. In order to state the following lemmata we need a couple of definitions.…”

Section: Large Deviationsmentioning

confidence: 99%

“…In this proof, write x = X which is now constant and does not depend on a. We consider the second iterate of x, namely T Since both T (ax) and T (ax) are positive, and moreover a > 1 and x > 0 we can easily choose a to fulfill |ξ 2 (a)| (min(T a (x))) 2 . Let γ = min log |T a (x)|.…”

Section: Proofs Of Lemma 1 and Corollaries 4 Andmentioning

confidence: 99%

Shrinking targets in parametrised families

Aspenberg

Persson

2017

Math. Proc. Camb. Phil. Soc.

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Abstract. We consider certain parametrised families of piecewise expanding maps on the interval, and estimate and sometimes calculate the Hausdor dimension of the set of parameters for which the orbit of a xed point has a certain shrinking target property. This generalises several similar results for -transformations to more general non-linear families. The proofs are based on a result by Schnellmann on typicality in parametrised families.IF Introduction vet T X M 3 M e dynmil systemF sn nlogy to hiophntine pproximtionD rill nd elni II studied the set of points x P M suh tht the orit hits shrinking trget round nother point y innitely oftenF wore preiselyD they studied the set f x P M X d@T n @xA; yA < r n iFoFg; where r n 3 H s n 3 IF sn their seD T is n expnding rtionl mp on the iemnn sphereD nd M is the orresponding tuli setF hey lulted the rusdor dimension of the set in the se r n a e n F his nd similr sets hve lter een studied in severl dierent settingsD for instne for Etrnsformtions y fugeud nd ng T nd fugeud nd vio SF yne n lso study other relted setsD suh s f y P M X d@T n @xA; yA < r n iFoFg; for xed xD s ws done for instne in VD IT nd PIF sn this pperD we will onsider fmily of mps T a D where a is prmeterF por xed x nd yD we onsider the set of prmeters a for whih the orit of x hits shrinking trget round y innitely oftenD tht isD we onsider sets of the form f a X d@T n a @xA; yA < r n iFoFg: ets of this kind were previously studied y ersson nd hmeling PPD in the se where T a re EtrnsformtionsD T X H; I 3 H;IAY x U 3 x mod I; @ > IA; nd x a I nd y a HF he motivtion for studying the set E a f X T @IA a jT @IA Hj n iFoFg me from the question how well the dynmis of T n e pproximted with sushifts of nite typeF

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The Collet–Eckmann condition for rational functions on the Riemann sphere

Cited by 22 publications

References 8 publications

Collet, Eckmann and the bifurcation measure

Collet, Eckmann and the bifurcation measure

Almost Every Real Quadratic Polynomial has a Poly-time Computable Julia Set

Shrinking targets in parametrised families

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