Components of degree two hyperbolic rational maps

Rees, Mary

doi:10.1007/bf01231191

Cited by 49 publications

(47 citation statements)

References 3 publications

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“…It turnes out M od 2 is isomorphic to C 2 . The problem of multipliers as coordinates for hyperbolic (and some neutral) degree 2 rational maps is settled in [31]. Theorem 6 allows us to deal with not necessary hyperbolic maps.…”

Section: Comment 10mentioning

confidence: 99%

“…V takes values 1, 2, or 6. The spaces Rat 2 and M od 2 have been studied intensively, see [31], [32], [33], [25]. Global coordinates in M od 2 are introduced in [25].…”

Section: Comment 10mentioning

confidence: 99%

“…This theorem [4], [29], [3] is a cornerstone of the DouadyHubbard theory of the Mandelbrot set. It has been generalized to components of hyperbolic polynomials [24] and degree two rational maps [31].…”

Section: Introductionmentioning

confidence: 99%

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Multipliers of periodic orbits in spaces of rational maps

Levin

2010

Ergod. Th. Dynam. Sys.

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Given a polynomial or a rational function f we include it in a space of maps. We introduce local coordinates in this space, which are essentially the set of critical values of the map. Then we consider an arbitrary periodic orbit of f with multiplier ρ = 1 as a function of the local coordinates, and establish a simple connection between the dynamical plane of f and the function ρ in the space associated to f . The proof is based on the theory of quasiconformal deformations of rational maps. As a corollary, we show that multipliers of non-repelling periodic orbits are also local coordinates in the space.

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Section: Comment 10mentioning

confidence: 99%

“…V takes values 1, 2, or 6. The spaces Rat 2 and M od 2 have been studied intensively, see [31], [32], [33], [25]. Global coordinates in M od 2 are introduced in [25].…”

Section: Comment 10mentioning

confidence: 99%

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Multipliers of periodic orbits in spaces of rational maps

Levin

2010

Ergod. Th. Dynam. Sys.

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“…Much work has been done on the dynamics of quadratic rational maps. Milnor [13,14] and Rees [15] worked on the moduli space of quadratic rational maps quite generally. Specific slices of the moduli space have also been studied by Hawkins [7] and Milnor [13], with the latter work including parabolic ones.…”

Section: Theorem 11 ([1]) Let R Be a Rational Map Of Degree D Whermentioning

confidence: 99%

Quadratic rational maps lacking period 2 orbits

Hagihara

2009

Proc. Amer. Math. Soc.

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“…Rees has carried out an in-depth study of the parameter space of quadratic rational maps (see, for example, [10,11]). The main results of this paper give a classification of rational maps belonging to certain type D hyperbolic components (components containing rational maps whose two critical points belong to the attracting basins of two disjoint (super)attracting periodic orbits) in the parameter space of bicritical rational maps.…”

Section: T Sharlandmentioning

confidence: 99%

Thurston equivalence for rational maps with clusters

Sharland¹

2012

Ergod. Th. Dynam. Sys.

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Abstract. We investigate rational maps with period-one and period-two cluster cycles. Given the definition of a cluster, we show that, in the case where the degree is d and the cluster is fixed, the Thurston class of a rational map is fixed by the combinatorial rotation number ρ and the critical displacement δ of the cluster cycle. The same result will also be proved in the case where the rational map is quadratic and has a period-two cluster cycle, and we will also show that the statement is no longer true in the higher-degree case. IntroductionThe emergence of complex dynamics as a popular subject for mathematical research came about as a result of the rediscovery of the early twentieth-century works of Fatou [5,6] and Julia [7]. After a comparatively quiet period, the subject was given new life in the 1980s. Perhaps the most notable contributions were supplied by Douady and Hubbard, whose 'Orsay lecture notes' [1, 2] provide a number of enlightening and amazing results about the behaviour of such systems. Since then, the study of complex dynamical systems has grown enormously and is now a very fruitful area for research.In dynamical systems, one often wants to be able to understand the systems one works with up to some form of equivalence. The study of complex dynamics on the Riemann sphere is no different, and we are fortunate that there is a powerful criterion, due to Thurston, that tells us whether or not two rational maps on the sphere are equivalent. However, despite its simplicity and power, the criterion suffers from the fact that it can, in practice, be very difficult to check. In this paper, we investigate a specific class of maps, those bicritical rational maps with periodic cluster cycles of period one or period two (the latter only in the quadratic case), and show that in these cases the application of the Thurston criterion is simple and allows a classification of such maps. This work fits into the branch of complex dynamics which attempts to describe rational maps combinatorially (or perhaps symbolically), following a programme initiated by Douady, Hubbard and Thurston. Perhaps the most remarkable feature of the results in this paper is that the rational maps in question are classified purely by the behaviour within their clusters. It is hoped that the techniques in this paper can be refined to tackle a larger class of rational maps, perhaps

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Components of degree two hyperbolic rational maps

Cited by 49 publications

References 3 publications

Multipliers of periodic orbits in spaces of rational maps

Multipliers of periodic orbits in spaces of rational maps

Quadratic rational maps lacking period 2 orbits

Thurston equivalence for rational maps with clusters

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