Mikhail Lyubich

• Stony Brook University

We formulate and prove $\textit{a priori}$ bounds for the renormalization of H\'enon-like maps (under certain regularity assumptions). This provides a certain uniform control on the small-scale geometry of the dynamics, and ensures pre-compactness of the renormalization sequence. In a sequel to this paper, a priori bounds are used in the proof of t...

In this paper, we bring together four different branches of antiholomorphic dynamics: of global anti-rational maps, reflection groups, Schwarz reflections in quadrature domains, and antiholomorphic correspondences. We establish the first general realization theorems for bi-degree d d : d d correspondences on the Riemann sphere (for d ≥ 2 d\geq 2 )...

In this paper, we study matings of (anti-)polynomials and Fuchsian, reflection groups as Schwarz reflections, B-involutions or as (anti-)holomorphic correspondences, as well as their parameter spaces. We prove the existence of matings of generic (anti-)polynomials, such as periodically repelling, or geometrically finite (anti-)polynomials, with cir...

We prove that any degree $d$ rational map having a parabolic fixed point of multiplier $1$ with a fully invariant and simply connected immediate basin of attraction is mateable with the Hecke group $H_{d+1}$, with the mating realized by an algebraic correspondence. This solves the parabolic version of the Bullett-Freiberger Conjecture from 2003 on...

The goal of this survey is to present intimate interactions between four branches of conformal dynamics: iterations of anti-rational maps, actions of Kleinian reflection groups, dynamics generated by Schwarz reflections in quadrature domains, and algebraic correspondences. We start with several examples of Schwarz reflections as well as algebraic c...

We prove {\em a priori} bounds for Feigenbaum quadratic polynomials, i.e., infinitely renormalizable polynomials $f_c: z\mapsto z^2+c$ of bounded type. It implies local connectivity of the corresponding Julia sets $J(f_c)$ and MLC (local connectivity of the Mandelbrot set $\Mandel$) at the corresponding parameters $c$. It also yields the scaling Un...

We explore geometric properties of the Mandelbrot set \({{\mathcal {M}}}\), and the corresponding Julia sets \({{\mathfrak {J}}}_c\), near the main cardioid. Namely, we establish that: (a) \({{\mathcal {M}}}\) is locally connected at certain infinitely renormalizable parameters c of bounded satellite type, providing first examples of this kind; (b)...

In this paper, we study the dynamics of a general class of antiholomorphic correspondences; i.e., multi-valued maps with antiholomorphic local branches, on the Riemann sphere. Such correspondences are closely related to a class of single-valued antiholomorphic maps in one complex variable; namely, Schwarz reflection maps of simply connected quadrat...

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group H H whose limit set is a generalized Apollonian gasket Λ H \Lambda _H . We design a surgery that relates H H to a rational map g g whose Julia set J g \mathcal {J}...

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two vari...

We prove uniform ``pseudo-Siegel'' a priori bounds for Siegel disks of bounded type that give a uniform control of oscillations of their boundaries in all scales. As a consequence, we construct the Mother Hedgehog controlling the postcritical set for any quadratic polynomial with a neutral periodic point and show that this hedgehog has a star-like...

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in [14], [15]. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns thes...

In this paper we continue to explore infinitely renormalizable H\'enon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-H\"older. Another formulation of this phenomenon is that the...

We provide a David extension result for circle homeomorphisms conjugating two dynamical systems such that parabolic periodic points go to parabolic periodic points, but hyperbolic points can go to parabolics as well. We use this result, in particular, to produce matings of anti-polynomials and necklace reflection groups, show conformal removability...

In this paper, we explore the spectral measures of the Laplacian on Schreier graphs for several self-similar groups (the Grigorchuk, Lamplighter, and Hanoi groups) from the dynamical and algebro-geometric viewpoints. For these graphs, classical Schur renormalization transformations act on appropriate spectral parameters as rational maps in two vari...

According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group $H$ whose limit set is an Apollonian-like gasket $\Lambda_H$. We design a surgery that relates $H$ to a rational map $g$ whose Julia set $\mathcal{J}_g$ is (non-qu...

We prove that all smooth Fibonacci maps with quadratic critical point are quasisymmetrically conjugate. The proof is based upon an idea of asymptotically conformal extension, which provides a link between smooth and holomorphic dynamics.

In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in \cite{LLMM1,LLMM2}. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that tu...

We continue our study of the family $\mathcal{S}$ of Schwarz reflection maps with respect to a cardioid and a circle which was started in [LLMM1]. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic...

In this paper, we initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to a deltoid, and Schwarz reflections with respect to a cardioid and a family of circumscribing circles. We describe the dyn...

In this paper, we initiate the exploration of a new class of anti-holomorphic dynamical systems generated by Schwarz reflection maps associated with quadrature domains. More precisely, we study Schwarz reflection with respect to a deltoid, and Schwarz reflections with respect to a cardioid and a family of circumscribing circles. We describe the dyn...

We consider the unstable manifold of a pacman renormalization operator constructed in [DLS]. It comprises rescaled limits of quadratic polynomials. Every such limit admits a maximal extension to a sigma-proper branched covering of the complex plane. Using methods and ideas from transcendental dynamics, we show that certain maps on the unstable mani...

We give a description of the group of all quasisymmetric self-maps of the Julia set of f(z) = z²−1 that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is qua...

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to...

In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of "Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. We show that Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to...

We give a description of the group of all quasisymmetric self-maps of the Julia set of $f(z)=z^2-1$ that have orientation preserving homeomorphic extensions to the whole plane. More precisely, we prove that this group is the uniform closure of the group generated by the Thompson group of the unit circle and an inversion. Moreover, this result is qu...

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher...

In this paper we study the dynamics of germs of holomorphic diffeomorphisms of $(\mathbb{C}^{n},0)$ with a fixed point at the origin with exactly one neutral eigenvalue. We prove that the map on any local center manifold of $0$ is quasiconformally conjugate to a holomorphic map and use this to transport results from one complex dimension to higher...

We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of P\'erez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of $(\mathbb{C},0)...

We prove the existence of hedgehogs for germs of complex analytic diffeomorphisms of $(\mathbb{C}^{2},0)$ with a semi-neutral fixed point at the origin, using topological techniques. This approach also provides an alternative proof of a theorem of P\'erez-Marco on the existence of hedgehogs for germs of univalent holomorphic maps of $(\mathbb{C},0)...

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.

Fatou components for rational endomorphisms of the Riemann sphere are fully classified and play an important role in our view of one-dimensional dynamics. In higher dimensions, the situation is less satisfactory. In this work we give a nearly complete classification of invariant Fatou components for moderately dissipative Hénon maps. Namely, we pro...

We prove a version of the classical $\lambda$-lemma for holomorphic families of Riemann surfaces. We then use it to show that critical loci for complex H\'{e}non maps that are small perturbations of quadratic polynomials with Cantor Julia sets are all quasiconformally equivalent.

John Milnor, best known for his work in differential topology, K-theory, and dynamical systems, is one of only three mathematicians to have won the Fields medal, the Abel prize, and the Wolf prize, and is the only one to have received all three of the Leroy P. Steele prizes. In honor of his eightieth birthday, this book gathers together surveys and...

This survey overviews selected themes in real and complex dynamics that have been strongly influenced by Milnor’s work over the past 40 years.

We study stability and bifurcations in holomorphic families of polynomial automorphisms of C^2. We say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there. We first show that this defines a meaningful notion of stability, which parallels in many ways the classical notion of J-stability in one-dim...

Let $\mathcal{K}$ be a knot, G be the knot group, K be its commutator subgroup, and x be a distinguished meridian. Let Σ be a finite abelian group. The dynamical system introduced by Silver and Williams in [Augmented group systems and n-knots, Math. Ann.296 (1993) 585–593; Augmented group systems and shifts of finite type, Israel J. Math.95 (1996)...

We show that repelling periodic points are landing points of periodic rays for exponential maps whose singular value has bounded orbit. For polynomials with connected Julia sets, this is a celebrated theorem by Douady, for which we present a new proof. In both cases we also show that points in hyperbolic sets are accessible by at least one and at m...

1. FIRST 30 YEARS The field of one-dimensional dynamics, real and complex, emerged from obscurity in the 1970s and has been intensely explored ever since. It combines the depth and complexity of chaotic phenomena with a chance to fully understand it in probabilistic terms: to describe the dynamics of typical orbits for typical maps. It also reveale...

We show that given a one-parameter family Fb of strongly dissipative infinitely renormalizable Hénon-like maps, parametrized by a quantity called the ‘average Jacobian’ b, the set of all parameters b such that Fb has a Cantor set with unbounded geometry has full Lebesgue measure.

We study highly dissipative Hénon maps Fc,b: (x,y) ® (c-x2-by, x)F_{c,b}: (x,y) \mapsto (c-x^2-by, x) with zero entropy. They form a region Π in the parameter plane bounded on the left by the curve W of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in Π, but there exist infinitely many different topological types of su...

In a classical work of the 1950's, Lee and Yang proved that for fixed nonnegative temperature, the zeros of the partition functions of a ferromagnetic Ising model always lie on the unit circle in the complex magnetic field. Zeros of the partition function in the complex temperature were then considered by Fisher, when the magnetic field is set to z...

Period doubling cascades are observed at transition to chaos in many models used in the sciences and in physical experiments. These period doubling cascades are very well understood in one-dimensional dynamics. In particular, the microscopic geometrical properties of the attractors do not depend on the actual system, they are universal. Moreover, t...

In a classical work of the 1950's, Lee and Yang proved that the zeros of the partition functions of a ferromagnetic Ising models always lie on the unit circle. Distribution of these zeros is physically important as it controls phase transitions in the model. We study this distribution for the Migdal-Kadanoff Diamond Hierarchical Lattice (DHL). In t...

We prove exponential contraction of renormalization along hybrid classes of infinitely renormalizable unimodal maps (with arbitrary combinatorics), in any even degree $d$. We then conclude that orbits of renormalization are asymptotic to the full renormalization horseshoe, which we construct. Our argument for exponential contraction is based on a p...

We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable H\'enon-like maps, parametrised by a quantity called the `average Jacobian' $b$, the set of all parameters $b$ such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure. Comment: 29 pages, 2 figures

The most prominent component of the interior of the Mandelbrot set M is the component bounded by the main cardioid. There are infinitely many secondary hyperbolic components of intM attached to it. In turn, infinitely many hyperbolic components are attached to each of the secondary components, etc. Let us take the union of all hyperbolic components...

We study the affine orbifold laminations that were constructed by Lyubich and Minsky (J. Differential Geom. 47(1) (1997), pp. 17-94). An important question left open in the original construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regul...

We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$. For complex parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic or infinitely renormalizable. For real parameters, we prove that for Lebesgue almost every $c$, the map $f_c$ is either hyperbolic, or Collet-Eckmann, or infinitely ren...

We study highly dissipative H\'enon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $\Pi$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $\Pi$, but there exist infinitely many different topological types of such maps (ev...

In this paper we prove {\it a priori bounds} for infinitely renormalizable quadratic polynomials satisfying a ``molecule condition''. Roughly speaking, this condition ensures that the renormalization combinatorics stay away from the satellite types. These {\it a priori bounds} imply local connectivity of the corresponding Julia sets and the Mandelb...

Holomorphic dynamics is a thriving field which has experienced tremendous progress over the last 25 years, involving mathematicians such as Douady, Hubbard, McMullen, Milnor, Sullivan, Thurston, and Yoccoz. Holomorphic dynamics is gifted with the tools of conformal and hyperbolic geometry that allow a deep penetration into its nature, with many fur...

A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two external rays landing at some tip of a satellite copy of $M$ attached to the main cardioid. In this paper we consider infinitely renormalizable quadratic polynomials satisfying the decoration condition, which means that the combinatorics of the renormalization operators involved...

In this paper geometric properties of infinitely renormalizable real Hénon-like maps F in \(\mathbb{R} ^2\) are studied. It is shown that the appropriately defined renormalizations R n F converge exponentially to the one-dimensional renormalization fixed point. The convergence to one-dimensional systems is at a super-exponen- tial rate controlled b...

We prove that any unicritical polynomial $f_c:z\mapsto z^d+c$ which is at most finitely renormalizable and has only repelling periodic points is combinatorially rigid. It implies that the connectedness locus (the ``Multibrot set'') is locally connected at the corresponding parameter values. It generalizes Yoccoz's Theorem for quadratics to the high...

On a Riemann surface $S$ of finite type containing a family of $N$ disjoint disks $D_i$ (``islands''), we consider several natural conformal invariants measuring the distance from the islands to $\di S$ and separation between different islands. In a near degenerate situation we establish a relation between them called the Quasi-Additivity Law. We t...

We prove that the Julia set $J(f)$ of at most finitely renormalizable unicritical polynomial $f:z\mapsto z^d+c$ with all periodic points repelling is locally connected. (For $d=2$ it was proved by Yoccoz around 1990.) It follows from a priori bounds in a modified Principle Nest of puzzle pieces. The proof of a priori bounds makes use of new analyti...

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 Poincar\'e critical exponent $\de_\crit$ is equal...

We give examples of infinitely renormalizable quadratic polynomials $F_c: z\maps to z^2+c$ with stationary combinatorics whose Julia sets have Hausdorff dimension arbitrar y close to 1. The combinatorics of the renormalization involved is close to the Chebyshev one . The argument is based upon a new tool, a ``Recursive Quadratic Estimate'' for the...

In this paper we prove that in any non-trivial real analytic family of quasiquadratic maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimension-one analytic submanifolds, hybrid clas...

The goal of this technical note is to show that the geometry of generalized parabolic towers cannot be essentially bounded. It fills a gap in author's paper "Combinatorics, geomerty and attractors of quasi-quadratic maps", Annals of Math., 1992.

this paper we will assume that the families are equipped, unless otherwise is explicitly stated

The framework of affine and hyperbolic laminations provides a unifyingfoundation for many aspects of conformal dynamics and hyperbolic geometry. Thecentral objects of this approach are an affine Riemann surface lamination A and theassociated hyperbolic 3-lamination H endowed with an action of a discrete group of isomorphisms.

In this paper we prove that in any non-trivial real analytic family of unimodal maps, almost any map is either regular (i.e., it has an attracting cycle) or stochastic (i.e., it has an absolutely continuous invariant measure). To this end we show that the space of analytic maps is foliated by codimensionone analytic submanifolds, hybrid classes". T...

1042 NOTICES OF THE AMS VOLUME 47, NUMBER 9 I n the last quarter of the twentieth century the real quadratic family f c : R → R, f c : x → x 2 + c (c ∈ R) was recognized as a very interesting and repre-sentative model of chaotic dynamics. It contains regular and stochastic maps intertwined in an in-tricate manner. It also has remarkable universal-i...

We prove the Feigenbaum-Coullet-Tresser conjecture on the hyperbolicity of the renormalization transformation of bounded type. This gives the first computer-free proof of the original Feigenbaum observation of the universal parameter scaling laws. We use the Hyperbolicity Theorem to prove Milnor's conjectures on self-similarity and "hairiness" of t...

We prove the Regulat or Stochastic Conjecture for the real quadratic family which asserts that almost every real quadratic map Pc, c in [-2, 1/4], has either an attracting cycle or an absolutely continuous invariant measure.

We prove uniform hyperbolicity of the renormalization operator for all possible real combinatorial types. We derive from it that the set of innitely renormalizable parameter values in the real quadratic family P c : x 7! x 2 + c has zero measure. This yields the statement in the title (where \ regular" means to have an attracting cycle and \stochas...

. We prove complex bounds for infinitely renormalizable real quadratic maps with essentially bounded combinatorics. This is the last missing ingredient in the problem of complex bounds for all infinitely renormalizable real quadratics. One of the corollaries is that the Julia set of any real quadratic map z 7! z 2 + c, c 2 [Gamma2; 1=4], is locally...

This is a continuation of notes on dynamics of quadratic polynomials. In this part we transfer the our prior geometric result to the parameter plane. To any parameter value c in the Mandelbrot set (which lies outside of the main cardioid and little Mandelbrot sets attached to it) we associate a ``principal nest of parapuzzle pieces'' and show that...

This is a continuation of the series of notes on the dynamics of quadratic polynomials. We show the following Rigidity Theorem: Any combinatorial class contains at most one quadratic polynomial satisfying the secondary limbs condition with a-priori bounds. As a corollary, such maps are combinatorially and topologically rigid, and as a consequence,...

We extend Sullivan's complex a priori bounds to real quadratic polynomials with essentially bounded combinatorics. Combined with the previous results of the first author, this yields complex bounds for all real quadratics. Local connectivity of the corresponding Julia sets follows.

This work studies combinatorics and geometry of the Yoccoz puzzle for quadratic polynomials. It is proven that the moduli of the ``principal nest'' of annuli grow at linear rate. As a corollary we obtain complex a priori bounds and local connectivity of the Julia set for many infinitely renormalizable quadratics.

We suggest a way to associate to a rational map of the Riemann sphere a three dimensional object called a hyperbolic orbifold 3-lamination. The relation of this object to the map is analogous to the relation of a hyperbolic 3-manifold to a Kleinian group. In order to construct the 3-lamination we analyze the natural extension of a rational map and...

We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to u...

Let J be a Cantor repeller of a conformal map f. Provided f is polynomial-like or ℝ-symmetric, we prove that the harmonic measure on J is equivalent to the measure of maximal entropy if and only if f is conformally equivalent to a polynomial. We also show that this is not true for general Cantor repellers: there is a nonpolynomial algebraic functio...

We will describe recent developments in several intimately related problems of complex and real one-dimensional dynamics: rigidity of polynomials and local connectivity of the Mandelbrot set, measure of Julia sets, and attractors of quasi-quadratic maps. A combinatorial basis for this study is provided by the Yoccoz puzzle. The main problem is to u...

This paper concerns the dynamics of polynomial automorphisms of ${\bf C}^2$. One can associate to such an automorphism two currents $\mu^\pm$ and the equilibrium measure $\mu=\mu^+\wedge\mu^-$. In this paper we study some geometric and dynamical properties of these objects. First, we characterize $\mu$ as the unique measure of maximal entropy. Then...

According to Sullivan, a space ${\cal E}$ of unimodal maps with the same combinatorics (modulo smooth conjugacy) should be treated as an infinitely-dimensional Teichm\"{u}ller space. This is a basic idea in Sullivan's approach to the Renormalization Conjecture. One of its principle ingredients is to supply ${\cal E}$ with the Teichm\"{u}ller metric...

A while ago MLC (the conjecture that the Mandelbrot set is locally connected) was proven for quasi-hyperbolic points by Douady and Hubbard, and for boundaries of hyperbolic components by Yoccoz. More recently Yoccoz proved MLC for all at most finitely renormalizable parameter values. One of our goals is to prove MLC for some infinitely renormalizab...

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan. It turns out that the situation can be understood c...

This paper deals with the dynamics of a simple family of holomorphic diffeomorphisms of $\C^2$: the polynomial automorphisms. This family of maps has been studied by a number of authors. We refer to [BLS] for a general introduction to this class of dynamical systems. An interesting object from the point of view of potential theory is the equilibriu...

The Milnor problem on one-dimensional attractors is solved for S-unimodal maps with a non-degenerate critical point c. It provides us with a complete understanding of the possible limit behavior for Lebesgue almost every point. This theorem follows from a geometric study of the critical set $\omega(c)$ of a "non-renormalizable" map. It is proven th...

Contents: 1. Quasiconformal Surgery and Deformations: Ben Bielefeld, Questions in quasiconformal surgery; Curt McMullen, Rational maps and Teichm\"uller space; John Milnor, Thurston's algorithm without critical finiteness; Mary Rees, A possible approach to a complex renormalization problem. 2. Geometry of Julia Sets: Lennart Carleson, Geometry of J...

The paper is concerned with the dynamics of an entire transcendental function whose inverse has only finitely many singularities. It is rpoven that there are no escaping orbits on the Fatou set. Under some extra assumptions the set of escaping orbits has zero Lebesgue measure. If a function depends analytically on parameters then a periodic point a...

Let J be a Cantor repeller of a rational map f. We prove that the harmonic measure on J is absolutely continuous with respect to the measure of maximal entropy m if and only if f is conformally equivalent to a polynomial. The same is true in the case of any hyperbolic rational map.

This paper will study topological, geometrical and measure-theoretical properties of the real Fibonacci map. Our goal was to figure out if this type of recurrence really gives any pathological examples and to compare it with the infinitely renormalizable patterns of recurrence studied by Sullivan. It turns out that the situation can be understood c...

In this paper we study measurable dynamics for the widest reasonable class of smooth one dimensional maps. Three principle decompositions are described in this class : decomposition of the global measure-theoretical attractor into primitive ones, ergodic decomposition and Hopf decomposition. For maps with negative Schwarzian derivative this was don...

The goal of this note is to prove the following theorem: Let $p_a(z) = z^2+a$ be a quadratic polynomial which has no irrational indifferent periodic points, and is not infinitely renormalizable. Then the Lebesgue measure of the Julia set $J(p_a)$ is equal to zero. As part of the proof we discuss a property of the critical point to be {\it persisten...