Ergodic invariant probability measures and entire functions

Fundamenta Mathematicae

We prove that if A is the basin of immediate attraction to a periodic attracting or parabolic point for a rational map f on the Riemann sphere, if A is completely invariant (i.e. f 01 (A) = A), and if is an arbitrary f-invariant measure with positive Lyapunov exponents on @A, then-almost every point q 2 @A is accessible along a curve from A. In fact we prove the accessability of every "good" q i.e. such q for which "small neighbourhoods arrive at large scale" under iteration of f. This generalizes Douady-Eremenko-Levin-Petersen theorem on the accessability of periodic sources. We prove a general "tree" version of this theorem. This allows to deduce that on the limit set of a geometric coding tree (in particular on the whole Julia set), if diameters of the edges converge to 0 uniformly with the number of generation converging to 1, every finvariant probability ergodic measure with positive Lyapunov exponent is the image through coding with the help of the tree, of an invariant measure on the full one-sided shift space.

Mathematische Annalen, 2010

Let f be a transcendental entire function for which the set of critical and asymptotic values is bounded. The Denjoy-Carleman-Ahlfors theorem implies that if the set of all z for which |f (z)| > R has N components for some R > 0, then the order of f is at least N/2. More precisely, we have log log M (r, f) ≥ 1 2 N log r − O(1), where M (r, f) denotes the maximum modulus of f. We show that if f does not grow much faster than this, then the escaping set and the Julia set of f have positive Lebesgue measure. However, as soon as the order of f exceeds N/2, this need not be true. The proof requires a sharpened form of an estimate of Tsuji related to the Denjoy-Carleman-Ahlfors theorem.

Indagationes Mathematicae, 2003

We consider maps in the tangent family for which the asymptotic values are eventually mapped onto poles. For such functions the Julia set .I@) = a,. We prove that for almost all z E J(f) the limit set W(Z) is the post-singular set andf is non-ergodic on /cf). We also prove that for suchf does not exist af-invariant measure absolutely continuous with respect to the Lebesgue measure finite on compact subsets of C.

Proceedings of the American Mathematical Society, 2008

If the preimage of a four-point set under a meromorphic function belongs to the real line, then the image of the real line is contained in a circle in the Riemann sphere. We include an application of this result to holomorphic dynamics: if the Julia set of a rational function is contained in a smooth curve, then it is contained in a circle.

Arxiv preprint math/9402215, 1994

In this paper we shall show that there exists ℓ 0 such that for each even integer ℓ ≥ ℓ 0 there exists c 1 ∈ R for which the Julia set of z → z ℓ + c 1 has positive Lebesgue measure. This solves an old problem. Editor's note: In 1997, it was shown by Xavier Buff that there was a serious flaw leaving a gap in the proof. Currently (1999), the question of positive measure Julia sets remains open. * A mistake which was pointed out to us by J.C. Yoccoz has been corrected in this version † e-mail: strien at fwi.uva.nl. ‡ supported by the NWO, KBN-GR91. S. van Strien and T. Nowicki 6 Quasiconformal rigidity of the return maps; renormalization and the proof of Theorem A 25 7 The random walk argument 42 8 A nested sequence of discs 43 9 An induced mapping with Markov properties 58 10 An asymptotic expression for the real induced map 64 10.

2016

In a recent paper, one of the authors — along with coauthors — extended the famous theorem of Beurling to the context of subspaces that are invariant under the class of subalgebras of H∞ of the form IH∞, where I is the inner function z. In recent times, several researchers have replaced z by an arbitrary inner function I and this has proved important and fruitful in applications such as to interpolation problems of the Pick–Nevanlinna type. Keeping in mind the great deal of interest in such problems, in this paper, we provide analogues of the above mentioned IH∞ related extension of Beurling’s theorem in the setting of the Banach space BMOA, in the context of uniform algebras, on compact abelian groups with ordered duals and the Lebesgue space on the real line. We also provide a significant simplification of the proof of Beurling’s theorem in the setting of uniform algebras and a new proof of Helson’s generalization of Beurling’s theorem in the context of compact abelian groups with...

The Journal of Analysis, 2022

In this paper we investigate the primeness of a class of entire functions and discuss the dynamics of a periodic member f of this class with respect to a transcendental entire function g that permutes with f. In particular we show that the Julia sets of f and g are identical.

Contemporary Mathematics, 2011

Two dynamical systems u t (t, •) = (Lu)(t, •), L = A, B, u(0, •) = f , which can serve as toy models for infinite systems of interacting particles in continuum are studied. Here u(t, •) and f are holomorphic functions in some K ⊂ C, and A, B are linear operators in a certain Banach space E of such functions. It is proven that both A and B generate C 0 semigroups and hence the above Cauchy problems have solutions in E. In some particular cases, ergodicity and reversibility are proven.

Fundamenta Mathematicae, 2005

We prove that for Ω being an immediate basin of attraction to an attracting fixed point for a rational mapping of the Riemann sphere, and for an ergodic invariant measure µ on the boundary Fr Ω, with positive Lyapunov exponent, there is an invariant subset of Fr Ω which is an expanding repeller of Hausdorff dimension arbitrarily close to the Hausdorff dimension of µ. We also prove generalizations and a geometric coding tree abstract version. The paper is a continuation of a paper in Fund. Math. 145 (1994) by the author and Anna Zdunik, where the density of periodic orbits in Fr Ω was proved. 1. Introduction. Let Ω be a simply connected domain in C and f be a holomorphic map defined on a neighbourhood W of Fr Ω to C. Assume f (W ∩ Ω) ⊂ Ω, f (Fr Ω) ⊂ Fr Ω and Fr Ω repells to the side of Ω, that is, ∞ n=0 f −n (W ∩ Ω) = Fr Ω. An important special case is where Ω is an immediate basin of attraction of an attracting fixed point for a rational function. This covers also the case of a component of the immediate basin of attraction to a periodic attracting orbit, as one can consider an iterate of f mapping the component to itself. Distances and derivatives are considered in the Riemann spherical metric on C. Let R : D → Ω be a Riemann mapping from the unit disc onto Ω and let g be a holomorphic extension of R −1 • f • R to a neighbourhood of the unit circle ∂D. It exists and it is expanding on ∂D (see [P2, Section 7]). We prove the following Theorem A. Let ν be an ergodic g-invariant probability measure on ∂D such that for ν-a.e. ζ ∈ ∂D the radial limit R(ζ) := lim rր1 R(rζ) exists. Assume that the measure µ := R * (ν) has positive Lyapunov exponent χ µ (f).

In this paper we study the ergodic theory of a class of symbolic dynamical systems (Ω, T, µ) where T : Ω → Ω the left shift transformation on Ω = ∞ 0 {0, 1} and µ is a σ-finite T -invariant measure having the property that one can find a real number d so that µ(τ d ) = ∞ but µ(τ d−ǫ ) < ∞ for all ǫ > 0, where τ is the first passage time function in the reference state 1. In particular we shall consider invariant measures µ arising from a potential V which is uniformly continuous but not of summable variation. If d > 0 then µ can be normalized to give the unique non-atomic equilibrium probability measure of V for which we compute the (asymptotically) exact mixing rate, of order n −d . We also establish the weak-Bernoulli property and a polynomial cluster property (decay of correlations) for observables of polynomial variation. If instead d ≤ 0 then µ is an infinite measure with scaling rate of order n d . Moreover, the analytic properties of the weighted dynamical zeta function and those of the Fourier transform of correlation functions are shown to be related to one another via the spectral properties of an operator-valued power series which naturally arises from a standard inducing procedure. A detailed control of the singular behaviour of these functions in the vicinity of their non-polar singularity at z = 1 is achieved through an approximation scheme which uses generating functions of a suitable renewal process. In the perspective of differentiable dynamics, these are statements about the unique absolutely continuous invariant measure of a class of piecewise smooth interval maps with an indifferent fixed point.