Hyperbolic Rational Maps Research Articles

Julia Sets of Hyperbolic Rational Maps have Positive Fourier Dimension

Let \(f:\widehat{{\mathbb {C}}}\rightarrow \widehat{{\mathbb {C}}}\) be a hyperbolic rational map of degree \(d \ge 2\), and let \(J \subset {\mathbb {C}}\) be its Julia set. We prove that J always has positive Fourier dimension. The case where J is included in a circle follows from a recent work of Sahlsten and Stevens (Fourier transform and expanding maps on Cantor sets preprint. arXiv:2009.01703, 2020). In the case where J is not included in a circle, we prove that a large family of probability measures supported on J exhibit polynomial Fourier decay: our result applies in particular to the measure of maximal entropy and to the conformal measure.

On geometrically finite degenerations II: convergence and divergence

In this paper, we study quasi post-critically finite degenerations for rational maps. We construct limits for such degenerations as geometrically finite rational maps on a finite tree of Riemann spheres. We prove the boundedness for such degenerations of hyperbolic rational maps with Sierpinski carpet Julia set and give criteria for the convergence for quasi-Blaschke products Q B d \mathcal {QB}_d , making progress towards the analogues of Thurston’s compactness theorem for acylindrical 3 3 -manifold and the double limit theorem for quasi-Fuchsian groups in complex dynamics. In the appendix, we apply such convergence results to show the existence of certain polynomial matings.

Statistics of multipliers for hyperbolic rational maps

<p style='text-indent:20px;'>In this article, we consider a counting problem for orbits of hyperbolic rational maps on the Riemann sphere, where constraints are placed on the multipliers of orbits. Using arguments from work of Dolgopyat, we consider varying and potentially shrinking intervals, and obtain a result which resembles a local central limit theorem for the logarithm of the absolute value of the multiplier and an equidistribution theorem for the holonomies.</p>

On hyperbolic rational maps with finitely connected Fatou sets

In this paper, we study hyperbolic rational maps with finitely connected Fatou sets. We construct models of post-critically finite hyperbolic tree mapping schemes for such maps, generalizing post-critically finite rational maps in the case of connected Julia set. We show they are general limits of rational maps as we quasiconformally stretch the dynamics. Conversely, we use quasiconformal surgery to show that any post-critically finite hyperbolic tree mapping scheme arises as such a limit. We construct abundant examples thanks to the flexibilities of the models, and use them to construct a sequence of rational maps of a fixed degree with infinitely many non-monomial rescaling limits.

Quantitative equidistribution of angles of multipliers

We study angles of multipliers of repelling cycles for hyperbolic rational maps in $\mathbb {C}(z)$. For a fixed $K \gg 1$, we show that nearly every interval of length $2\pi /K$ in $(-\pi ,\pi ]$ contains a multiplier angle with the property that the abs

Quasisymmetric uniformization and Hausdorff dimensions of Cantor circle Julia sets

For Cantor circle Julia sets of hyperbolic rational maps, we prove that they are quasisymmetrically equivalent to standard Cantor circles (i.e., connected components are round circles). This gives a quasisymmetric uniformization of all Cantor circle Julia sets of hyperbolic rational maps. By analyzing the combinatorial information of the rational maps whose Julia sets are Cantor circles, we give a computational formula of the number of the Cantor circle hyperbolic components in the moduli space of rational maps for any fixed degree. We calculate the Hausdorff dimensions of the Julia sets which are Cantor circles, and prove that for any Cantor circle hyperbolic component $\mathcal{H}$ in the space of rational maps, the infimum of the Hausdorff dimensions of the Julia sets of the maps in $\mathcal{H}$ is equal to the conformal dimension of the Julia set of any representative $f_0\in\mathcal{H}$, and that the supremum of the Hausdorff dimensions is equal to $2$.

On uniformly disconnected Julia sets

It is well-known that the Julia set of a hyperbolic rational map is quasisymmetrically equivalent to the standard Cantor set. Using the uniformization theorem of David and Semmes, this result comes down to the fact that such a Julia set is both uniformly perfect and uniformly disconnected. We study the analogous question for Julia sets of UQR maps in \(\mathbb {S}^n\), for \(n\ge 2\). Introducing hyperbolic UQR maps, we show that the Julia set of such a map is uniformly disconnected if it is totally disconnected. Moreover, we show that if E is a compact, uniformly perfect and uniformly disconnected set in \(\mathbb {S}^n\), then it is the Julia set of a hyperbolic UQR map \(f:\mathbb {S}^N \rightarrow \mathbb {S}^N\) where \(N=n\) if \(n=2\) and \(N=n+1\) otherwise.

On the cycles of components of disconnected Julia sets

For any integers $$d\ge 3$$ and $$n\ge 1$$ , we construct a hyperbolic rational map of degree d such that it has n cycles of the connected components of its Julia set except single points and Jordan curves.

Collet, Eckmann and the bifurcation measure

The moduli space $$\mathcal {M}_d$$ of degree $$d\ge 2$$ rational maps can naturally be endowed with a measure $$\mu _{\text{ bif }}$$ detecting maximal bifurcations, called the bifurcation measure. We prove that the support of the bifurcation measure $$\mu _{\text{ bif }}$$ 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 $$\mu _{\text{ bif }}$$ and we exhibit a large set of Collet–Eckmann rational maps which satisfy this condition. As a consequence, we get a set of Collet–Eckmann rational maps of positive Lebesgue measure which are approximated by hyperbolic rational maps.

Semi-hyperbolic rational maps and size of Fatou components

Recently Merenkov and Sabitova introduced the notion of a homogeneous planar set. Using this notion they proved a result for Sierpi${\'n}$ski carpet Julia sets of hyperbolic rational maps that relates the diameters of the peripheral circles to the Hausdorff dimension of the Julia set. We extend this theorem to Julia sets (not necessarily Sierpi${\'n}$ski carpets) of semi-hyperbolic rational maps, and prove a stronger version of the theorem that was conjectured by Merenkov and Sabitova.

Prime number theorems and holonomies for hyperbolic rational maps

We discuss analogues of the prime number theorem for a hyperbolic rational map f of degree at least two on the Riemann sphere. More precisely, we provide counting estimates for the number of primitive periodic orbits of f ordered by their multiplier, and also obtain equidistribution of the associated holonomies; both estimates have power savings error terms. Our counting and equidistribution results will follow from a study of dynamical zeta functions that have been twisted by characters of $S^1$. We will show that these zeta functions are non-vanishing on a half plane $\Re(s) > \delta - \epsilon$, where $\delta$ is the Hausdorff dimension of the Julia set of f.

From rubber bands to rational maps: a research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.

From Apollonian packings to homogeneous sets

We extend fundamental results concerning Apollonian packings, which constitute a major object of study in number theory, to certain homogeneous sets that arise naturally in complex dynamics and geometric group theory. In particular, we give an analogue of D. W. Boyd's theorem (relating the curvature distribution function of an Apollonian packing to its exponent and the Hausdorff dimension of the residual set) for Sierpi\'nski carpets that are Julia sets of hyperbolic rational maps.

A family of rational maps with buried Julia components

It is known that the disconnected Julia set of any polynomial map does not contain buried Julia components. But such Julia components may arise for rational maps. The first example is due to Curtis T. McMullen who provided a family of rational maps for which the Julia sets are Cantor of Jordan curves. However, all known examples of buried Julia components, up to now, are points or Jordan curves and comes from rational maps of degree at least five. This paper introduces a family of hyperbolic rational maps with disconnected Julia set whose exchanging dynamics of postcritically separating Julia components is encoded by a weighted dynamical tree. Each of these Julia sets presents buried Julia components of several types: points, Jordan curves, but also Julia components which are neither points nor Jordan curves. Moreover, this family contains some rational maps of degree three with explicit formula that answers a question McMullen raised.

Quasisymmetrically inequivalent hyperbolic Julia sets

We give explicit examples of pairs of Julia sets of hyperbolic rational maps which are homeomorphic but not quasisymmetrically homeomorphic.

A characterization of hyperbolic potentials of rational maps

Consider a rational map f of degree at least 2 acting on its Julia set J(f), a Hölder continuous potential φ: J(f) → ℝ and the pressure P(f,φ). In the case where $$\mathop {\sup }\limits_{J(f)} \phi < P(f,\phi ),$$ the uniqueness and stochastic properties of the corresponding equilibrium states have been extensively studied. In this paper we characterize those potentials φ for which this property is satisfied for some iterate of f, in terms of the expanding properties of the corresponding equilibrium states. A direct consequence of this result is that for a non-uniformly hyperbolic rational map every Hölder continuous potential has a unique equilibrium state and that this measure is exponentially mixing.

On Large Deviations for Some Dynamical Systems and for Gibbs States at Zero Temperature

In this article we analyze two issues related with large deviations in dynamical systems: 1. We show that the level-2 large deviation principle established by Comman and Rivera-Letelier[1], is satisfied by maps with a specification property and, with some additional condition, also by those with the almost property product, which is weaker than specification. The earlier mentioned authors proved that their principle, which is a generalization of previous results by Kifer, is verified by a class of hyperbolic rational maps. 2. In a previous article[5] we have considered a family of Gibbs states {μq } q≥1, which had an accumulation point μ∞ (zero temperature limit since the interpretation of q as the inverse of the temperature). We proved that μ∞ is a maximizing measure for more general systems than symbolics. In a recent article by Lopes and Mengue[4] were considered similar families of states and proved, for symbolic dynamics, that an accumulation point of the family was maximizing. Also they established a large deviation principle. In this note we show how to use the results of our previous work to describe a large deviation process in a more general context than[4].

Density of hyperbolicity for rational maps with Cantor Julia sets

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.

Julia and John revisited

We show that Fatou components of a semi-hyperbolic rational map are John domains and that the converse does not hold. This generalizes a famous result of Carleson, Jones and Yoccoz. We show that a connected Julia set is locally connected for a large class of non-uniformly hyperbolic rational maps. This class is more general than semi-hyperbolicity and includes Collet-Eckmann and Topological Collet-Eckmann maps and maps verifying a summability condition (as considered by Graczyk and Smirnov).

Nice Inducing Schemes and the Thermodynamics of Rational Maps

We study the thermodynamic formalism of a complex rational map f of degree at least two, viewed as a dynamical system acting on the Riemann sphere. More precisely, for a real parameter t we study the existence of equilibrium states of f for the potential \({-t {\rm ln} \left|f^{\prime} \right|}\) , and the analytic dependence on t of the corresponding pressure function. We give a fairly complete description of the thermodynamic formalism for a large class of rational maps, including well known classes of non-uniformly hyperbolic rational maps, such as (topological) Collet-Eckmann maps, and much beyond. In fact, our results apply to all non-renormalizable polynomials without indifferent periodic points, to infinitely renormalizable quadratic polynomials with a priori bounds, and all quadratic polynomials with real coefficients. As an application, for these maps we describe the dimension spectrum for Lyapunov exponents, and for pointwise dimensions of the measure of maximal entropy, and obtain some level-1 large deviations results. For polynomials as above, we conclude that the integral means spectrum of the basin of attraction of infinity is real analytic at each parameter in \({\mathbb{R}}\) , with at most two exceptions.