Elliptic Functions With Critical Points Eventually Mapped Onto Infinity

We consider the class of elliptic functions whose critical points in the Julia set are eventually mapped onto ∞. This paper is a continuation of our previous papers, namely [11] and [12]. We study the geometry and ergodic properties of this class of elliptic functions. In particular, we obtain a lower bound on the Hausdorff dimension of the Julia set that is bigger than the estimate proved in [11]. Let h be the Hausdorff dimension of the Julia set of f. We construct an atomless h-conformal measure m and prove the existence of a (unique up to a multiplicative constant) σ-finite f-invariant measure μ equivalent to m. The measure μ is ergodic and conservative.     

Trees and hairs for some hyperbolic entire maps of finite order

Let f be an entire transcendental map of finite order, such that all the singularities of f ⁻¹ are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from . This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps. Research supported by Polish KBN Grant No 2 P03A 034 25.     

Group invariant Colombeau generalized functions

Colombeau generalized functions invariant under smooth (additive) one-parameter groups are characterized. This characterization is applied to generalized functions invariant under orthogonal groups of arbitrary signature, such as groups of rotations or the Lorentz group. Further, a one-dimensional Colombeau generalized function with two (real) periods is shown to be a generalized constant, when the ratio of the periods is an algebraic nonrational number. Finally, a nonstandard Colombeau generalized function invariant under standard translations is shown to be constant. Supported by research grants M949 and Y237 of the Austrian Science Foundation (FWF). Author’s address: Institut für Grundlagen der Bauingenieurwissenschaften, Technikerstra?e 13, 6020 Innsbruck, Austria     

An explicit bound for uniform perfectness of the Julia sets of rational maps

A compact set C in the Riemann sphere is called uniformly perfect if there is a uniform upper bound on the moduli of annuli which separate C. Julia sets of rational maps of degree at least two are uniformly perfect. Proofs have been given independently by Ma né and da Rocha and by Hinkkanen, but no explicit bounds are given. In this note, we shall provide such an explicit bound and, as a result, give another proof of uniform perfectness of Julia sets of rational maps of degree at least two. As an application, we provide a lower estimate of the Hausdorff dimension of the Julia sets. We also give a concrete bound for the family of quadratic polynomials in terms of the parameter c. Received: 7 June 1999; in final form: 9 November 1999 / Published online: 17 May 2001     

Parabolic Implosion and Julia-Lavaurs Sets in the Exponential Family

We deal with all the maps from the exponential family f _ε(z) = (e ⁻¹ + ε)exp(z), with ε ≥ 0. Let h _ε = HD(J _r,ε) be the Hausdorff dimension of the radial Julia sets J _r,ε. Observing the phenomenon of parabolic implosion, it is shown that the function ε ↦ h _ε is not continuous from the right. The research of the first author was supported in part by the NSF Grant DMS 0100078.     

Geometric rigidity of transcendental meromorphic functions

The classes of dynamically and geometrically tame functions meromorphic outside a small set are introduced. The Julia sets of geometrically tame functions are proven to be either geometrical circle (in ) or to have Hausdorff dimension strictly larger than 1. Vast classes of dynamically and geometrically tame functions are identified. The research of both authors was supported in part by the Polish KBN Grant No 2 PO3A 034 25, the Warsaw University of Technology Grant no 504G 11200043000 and by the NSF/PAN grant INT-0306004. The research of the second author was supported in part by the NSF Grant DMS 0400481.     

Smooth singular flows in dimension 2 with the minimal self-joining property

It is proved that some velocity changes in flows on the torus determined by quasi-periodic Hamiltonians on : where α₁/α₂ is an irrational number with bounded partial quotients, lead to singular flows on with an ergodic component having a minimal set of self-joinings. Authors’ address: K. Frączek and M. Lemańczyk, Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, ul. Chopina 12/18, 87-100 Toruń, Poland Research partially supported by KBN grant 1 P03A and by Marie Curie “Transfer of Knowledge” program, project MTKD-CT-2005-030042 (TODEQ) 03826.     

Ergodic Optimization for Renewal Type Shifts

We consider a class of countable Markov shifts and a locally H?lder potential φ. We prove that the existence of φ-optimal measures is closely related to the behaviour of the pressure function t → P(tφ). Using a Theorem by Sarig it is possible to prove that there exists a critical value t _c ∈ (0, ∞] such that for t < t _c the pressure is analytic and for t > t _c is linear. We prove that if t _c is finite, then there are no φ-optimal measures, and if it is infinite, then φ-optimal measures do exist. The author was partially supported by FCT/POCTI/FEDER and the grant SFRH/BPD/21927/2005.     

Diophantine Analysis of Conformal Iterated Function Systems

 A formula for the Hausdorff dimension of a subset of the limit set of a conformal infinite iterated function system is derived and expressed as a unique zero of the topological pressure of corresponding strongly H?lder family of functions. Received August 10, 2001; in revised form February 23, 2002     

Spectral Properties of Interval Exchange Maps

 In [7], Nogueira and Rudolph proved that for irreducible permutations not of rotation class almost every (a.e.) interval exchange transformation (i.e.t.) is topological weak mixing. It is conjectured that the claim holds if topological weak mixing is replaced by weak mixing. Here we study the behaviour of eigenfunctions of i.e.t. Our analysis gives alternative proofs of results due to Katok and Stepin [4] and Veech [10]: for certain permutations a.e. i.e.t. is weak mixing and for irreducible permutations a.e. i.e.t. is totally ergodic. (Received 1 February 2001)     

Global Existence and Exponential Stability of Smooth Solutions to a Full Hydrodynamic Model to Semiconductors

 We study a full hydrodynamic semiconductor model in multi-space dimension. The global existence of smooth solutions is established and the exponential stability of the solutions as is investigated. Received November 14, 2000; in revised form March 25, 2002 Published online August 5, 2002     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)     

Ergodic Properties and Julia Sets of Weierstrass Elliptic Functions

 We discuss properties of the Julia and Fatou sets of Weierstrass elliptic ℘ functions arising from real lattices. We give sufficient conditions for the Julia sets to be the whole sphere and for the maps to be ergodic, exact, and conservative. We also give examples for which the Julia set is not the whole sphere. Received September 4, 2001; in revised form March 26, 2002     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Large time behavior in Wasserstein spaces and relative entropy for bipolar drift-diffusion-Poisson models

We prove asymptotic stability results for nonlinear bipolar drift-diffusion-Poisson Systems arising in semiconductor device modeling and plasma physics in one space dimension. In particular, we prove that, under certain structural assumptions on the external potentials and on the doping profile, all solutions match for large times with respect to all q-Wasserstein distances. We also prove exponential convergence to stationary solutions in relative entropy via the so called entropy dissipation (or Bakry-émery) method. Authors’ addresses: Marco Di Francesco, Sezione di Matematica per L’Ingegneria, Dipartimento di Matematica Pura ed Applicata, Università di L'Aquila, Piazzale E. Pontieri, 2, Monteluco di Roio, 67040 L’Aquila, Italy; Marcus Wunsch, Fakult?t für Mathematik, Universit?t Wien, Nordbergstra?e 15, A-1090 Wien, Austria     

Some results on supersolvability of finite groups

We get new characterizations of finite supersolvable groups based on the structure of the generalized Fitting subgroup. Then we extend our results to the saturated formations containing the class of all supersolvable groups. Authors’ addresses: M. Asaad, Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt; Piroska Cs?rg?, Department of Algebra and Number Theory, E?tv?s University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary     

On Cocycles with Values in the Group SU(2)

 In this paper we introduce the notion of degree for C ¹-cocycles over irrational rotations on the circle with values in the group SU(2). It is shown that if a C ¹-cocycle over an irrational rotation by α has nonzero degree, then the skew product

Zeros of Generalized Holomorphic Functions

Let f be a generalized holomorphic function on a connected open set . It is proved that f equals zero if and only if there exists a smooth curve and a set A of positive (one-dimensional) measure such that f takes zero value on A. Also, a holomorphic generalized function different from zero on the disc, which takes zero values on a dense G _δ-set of the disc, is constructed. The generalized zero set of a holomorphic function is introduced and studied in an analogous way.