We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $\de_\crit$ is equal to the hyperbolic dimension $\HD_\hyp(J(f))$. Moreover, if $\area J(f)=0$ then $\HD_\hyp (J(f))=\HD(J(f))$. In the stationary case, the last statement can be reversed: if $\area J(f)> 0$ then $\HD_\hyp (J(f))< 2$. We also give a new construction of conformal measures on $J(f)$ that implies that they exist for any $\de\in [\de_\crit, \infty)$, and analyze their scaling and dissipativity/conservativity properties.
We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The corresponding set of parameters has positive Hausdorff dimension.
We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.
Let $d(c)$ denote the Hausdorff dimension of the Julia set $J_c$ of the polynomial $f_c(z)=z^2+c$. We will investigate behavior of the function $d(c)$ when real parameter $c$ tends to a parabolic parameter.
We introduce new variants of the notion of geometric pressure for rational functions on the Riemann sphere, including non-hyperbolic functions, in the hope some of them occur useful to achieve a fast approximation from below of the hyperbolic Hausdorff dimension of Julia set.
We prove that for every polynomial of one complex variable of degree at least 2 and Julia set not being totally disconnected nor a circle, nor interval, Hausdorff dimension of this Julia set is larger than 1. Till now this was known only in the connected Julia set case. We give also an example of a polynomial with non-connected but not totally disconnected Julia set and such that all its components comprising of more than single points are analytic arcs, thus resolving a question by Christopher Bishop, who asked whether every such component must have Hausdorff dimension larger than 1.
We show that if $β>1$ is a rational number and the Julia set $J$ of the holomorphic correspondence $z^β+c$ is a locally eventually onto hyperbolic repeller, then the Hausdorff dimension of $J$ is bounded from above by the zero of the associated pressure function. As a consequence, we conclude that the Julia set of the correspondence has zero Lebesgue measure for parameters close to zero, whenever $q^2<p$ and $β=p/q$ in lowest terms.
A closed interval and circle are the only smooth Julia sets in polynomial dynamics. D. Ruelle proved that the Hausdorff dimension of unicritical Julia sets close to the circle depends analytically on the parameter. Near the tip of the Mandelbrot set M, the Hausdorff dimension is generally discontinuous. Answering a question of J-C. Yoccoz in the conformal setting, we observe that the Hausdorff dimension of quadratic Julia sets depends continuously on $c$ and find explicit bounds at the tip of M for most real parameters in the the sense of 1-dimensional Lebesgue measure.
We calculate rigorous bounds on the Hausdorff dimension of the attractor at the accumulation of the period-doubling cascade for families of maps with quadratic, cubic, and quartic critical point. To do this, we express the attractors as the limit sets of appropriate Iterated Function Systems constructed using rigorous bounds on the corresponding renormalisation fixed point functions. We use interval arithmetic with rigorous directed rounding modes to show that the respective dimensions lie in subintervals of the intervals (0.5370,0.5392), (0.6040,0.6091), and (0.6395,0.6474).
Let $d(δ)$ denote the Hausdorff dimension of the Julia set of the polynomial $f_δ(z)=z^2-2+δ$. In this paper we will study the directional derivative of the function $d$ along directions landing at the parameter $0$, which corresponds to $-2$ in the case of family $p_c(z)=z^2+c$. We will consider all directions, except the one $δ\in\mathbb{R}^+$, which is inside the Mandelbrot set. We will prove asymptotic formula for the directional derivative of $d$. Moreover, we will see that the derivative is negative for all directions in the closed left half-plane. Computer calculations show that it is n…
Let $d(\varepsilon)$ and $\mathcal D(δ)$ denote the Hausdorff dimension of the Julia sets of the polynomials $p_\varepsilon(z)=z^2+1/4+\varepsilon$ and $f_δ(z)=(1+δ)z+z^2$ respectively. In this paper we will study the directional derivative of the functions $d(\varepsilon)$ and $\mathcal D(δ)$ along directions landing at the parameter $0$, which corresponds to $1/4$ in the case of family $z^2+c$. We will consider all directions, except the one $\varepsilon\in\mathbb{R}^+$ (or two imaginary directions in the $δ$ parametrization) which is outside the Mandelbrot set and is related to the paraboli…
We prove the existence of rational maps whose Julia sets are Sierpiński carpets having positive area. Such rational maps can be constructed such that they either contain a Cremer fixed point, a Siegel disk or are infinitely renormalizable. We also construct some Sierpiński carpet Julia sets with zero area but with Hausdorff dimension two. Moreover, for any given number $s\in(1,2)$, we prove the existence of Sierpiński carpet Julia sets having Hausdorff dimension exactly $s$.
We show that for each $d\in (0,2]$ there exists a meromorphic function $f$ such that the inverse function of $f$ has three singularities and the Julia set of $f$ has Hausdorff dimension $d$.
We prove the existence of Cantor Julia sets with Hausdorff dimension two. In particular, such examples can be found in cubic polynomials. The proof is based on the characterization of the parameter spaces and dynamical planes of cubic polynomials by Branner-Hubbard, and the parabolic bifurcation theory developed by Shishikura in 1990s.
Any finite union of disjoint, mutually exterior Jordan curves in the complex plane can be approximated arbitrarily well in the Hausdorff topology by polynomial Julia sets. Furthermore, the proof is constructive.
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…
A generalized family of transcendental (non-polynomial entire) functions is constructed, where the Hausdorff dimension and the packing dimension of the Julia sets are equal to one. Further, there exist multiply connected wandering domains, the dynamics can be completed described, and for any $s\in(0,+\infty]$, there is a function taken from this family with the order of growth $s$. Baker proved that the Hausdorff dimension of the transcendental function is no less than one in 1975, the minimum value was obtained via an elegant construction by Bishop in 2018. The order of growth is zero in Bish…
Hereditarily non uniformly perfect (HNUP) sets were introduced by Stankewitz, Sugawa, and Sumi in \cite{SSS} who gave several examples of such sets based on Cantor set-like constructions using nested intervals. We exhibit a class of examples in non-autonomous iteration where one considers compositions of polynomials from a sequence which is in general allowed to vary. In particular, we give a sharp criterion for when Julia sets from our class will be HNUP and we show that the maximum possible Hausdorff dimension of $1$ for these Julia sets can be attained. The proof of the latter considers the…
A tree, embedded into plane, is a dessin d'enfant and its Belyi function is a polynomial --- Shabat polynomial. The Zapponi form of this polynomial is unique, so we can correspond to an embedded tree the Julia set of its Shabat-Zapponi polynomial. In this purely experimental work we study relations between the form of a tree and properties (form, connectedness, Hausdorff dimension) of its Julia set.
A point $z$ in the Julia set of a polynomial $p$ is called biaccessible if two dynamic rays land at $z$; a point $z$ in the Mandelbrot set is called biaccessible if two parameter rays land at $z$. In both cases, we say that the external angles of these two rays are biaccessible as well. In this paper we give upper and lower bounds for the Hausdorff dimension of biaccessible external angles of quadratic polynomials, both in the dynamical and parameter space. In particular, explicitly describe those quadratic polynomials where this dimension equals 1 (if and only if the Julia set is an interval)…
Devaney and Krych showed that for $0<λ<1/e$ the Julia set of $λe^z$ consists of pairwise disjoint curves, called hairs, which connect finite points, called the endpoints of the hairs, with $\infty$. McMullen showed that the Julia set has Hausdorff dimension $2$ and Karpińska showed that the set of hairs without endpoints has Hausdorff dimension $1$. We study for which gauge functions the Hausdorff measure of the set of hairs without endpoints is finite.
Given a rational map of the Riemann sphere and a subset $A$ of its Julia set, we study the $A$-exceptional set, that is, the set of points whose orbit does not accumulate at $A$. We prove that if the topological entropy of $A$ is less than the topological entropy of the full system then the $A$-exceptional set has full topological entropy. Furthermore, if the Hausdorff dimension of $A$ is smaller than the dynamical dimension of the system then the Hausdorff dimension of the $A$-exceptional set is larger than or equal to the dynamical dimension, with equality in the particular case when the dyn…
We provide an expanded and clarified proof of the famous result of Bowen and Ruelle giving an asymptotic formula for the Hausdorff dimension of quasi-circles corresponding to the Julia sets of $f(z)=z^2+c$ for small $c$. The proof does not contain new material but has been rewritten to make it more accessible to MSc or PhD students with an interest in dimension theory.
We show that the Hausdorff dimension of the set of points of bounded orbit in the Julia set of a meromorphic map with a simply connected direct tract and a certain restriction on the singular values is strictly greater than one. This result is obtained by proving new results related to Wiman-Valiron theory.
We study the possible Hausdorff limits of the Julia sets and filled Julia sets of subsequences of the sequence of dual Chebyshev polynomials of a non-polar compact set K in C and compare such limits to K. Moreover, we prove that the measures of maximal entropy for the sequence of dual Chebyshev polynomials of K converges weak* to the equilibrium measure on K.
For any $ δ>0$ we construct an entire function $f$ with three singular values whose Julia set has Hausdorff dimension at most $1=δ$. Stallard proved that the dimension must be strictly larger than 1 whenever $f$ has a bounded singular set, but no examples with finite singular set and dimension strictly less than 2 were previously known.
We construct a family of transcendental entire functions whose Julia sets have packing dimension in $(1,2)$. These are the first examples where the computed packing dimension is not $1$ or $2$. Our construction will allow us further show that the set of packing dimensions attained is dense in the interval $(1,2)$, and that the Hausdorff dimension of the Julia sets can be made arbitrarily close to the corresponding packing dimension.
