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     

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

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.     

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.     

Intersections of Sets Starshaped Via Paths of Bounded Length

 Let S be a nonempty closed, simply connected set in the plane. For α > 0, let ℳ denote the family of all maximal subsets of S which are starshaped via paths of length at most α. Then ⋂{M : M in ℳ} is either starshaped via α-paths or empty. The result fails without the simple connectedness condition. However, even with a simple connectedness requirement, there is no Helly theorem for intersections of sets which are starshaped via α-paths. Received November 19, 2001; in revised form April 25, 2002 Published online November 18, 2002     

Fix-mahonian calculus III; a quadruple distribution

A four-variable distribution on permutations is derived, with two dual combinatorial interpretations. The first one includes the number of fixed points “fix”, the second the so-called “pix” statistic. This shows that the duality between derangements and desarrangements can be extended to the case of multivariable statistics. Several specializations are obtained, including the joint distribution of (des, exc), where “des” and “exc” stand for the number of descents and excedances, respectively. Authors’ addresses: Dominique Foata, Institut Lothaire, 1 rue Murner, F-67000 Strasbourg, France; Guo-Niu Han, Center for Combinatorics, LPMC, Nankai University, Tianjin 300071, P.R. China; I.R.M.A. UMR 7501, Université Louis Pasteur et CNRS, 7, rue René-Descartes, F-67084 Strasbourg, France     

Linear Weingarten Surfaces in ℝ<Superscript>3</Superscript>

 In this paper we study properties of linear Weingarten immersions and graphs related to non-existence problems and behaviour of its curvatures. The main results are obtained giving a harmonic representation of linear Weingarten surfaces and by proving optimal estimates of the height and curvatures that the immersion must satisfy, characterizing the spherical caps as the only ones achieving these bounds. Received January 25, 2001; in revised form April 4, 2002     

Short Interval Results for a Class of Integers

For any subset , we introduce the definition of -numbers, which contains the well-known k-free numbers, k-full numbers, k-full and l-free numbers (k<l–1) as special cases. In this paper we study the distribution of -numbers in short intervals. We establish the connection of this problem with the gap problem of k-free numbers and multi-dimensional divisor problems. As applications, we study the short interval distribution of k-full and l-free numbers for k=2, 3, 4, 5, 6, 7.Received June 4, 2002; in revised form January 22, 2003 Published online June 30, 2003     

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     

Cycles for Rational Maps of Good Reduction Outside a Prescribed Set

Let K be a number field and S a fixed finite set of places of K containing all the archimedean ones. Let R _S be the ring of S-integers of K. In the present paper we study the cycles in for rational maps of degree ≥2 with good reduction outside S. We say that two ordered n-tuples (P ₀, P ₁,… ,P _n−1) and (Q ₀, Q ₁,… ,Q _n−1) of points of are equivalent if there exists an automorphism A ∈ PGL₂(R _S ) such that P _i = A(Q _i ) for every index i∈{0,1,… ,n−1}. We prove that if we fix two points , then the number of inequivalent cycles for rational maps of degree ≥2 with good reduction outside S which admit P ₀, P ₁ as consecutive points is finite and depends only on S and K. We also prove that this result is in a sense best possible.     

Thermodynamic formalism of transcendental entire maps of finite singular type

We build a version of a thermodynamic formalism for maps of the form f(z) = ∑ _{j = 0} ^{p + q} a _j e ^(j−p)z where p, q > 0 and . We show in particular the existence and uniqueness of (t,α)-conformal measures and that the Hausdorff dimension HD(J _f ^r ) = h is the unique zero of the pressure function t ↦ P(t) for t > 1, where the set J _f ^r is the radial Julia set. Partially supported by NSF Grant DMS 0100078. Partially supported by Warsaw University of Technology Grant No. 504G11200023000, Polish KBN Grant No. 2PO3A03425 and Chilean FONDECYT Grant No. 11060280.     

On Dynamics Related to a Class of Numeration Systems

 In this paper, we investigate the class of numeration systems and we study the associated dynamical systems, called odometers. It is shown that these odometers are measure-theoretically isomorphic to rank one transformations on the unit interval, constructed by a cutting-stacking method. Furthermore, a symbolic coding leads to isomorphic shift systems arising from substitutions. Some skew products of the odometers by cocycles related to the sum of digits are shown to be ergodic. Received 5 March 2001; in revised form 16 August 2001     

Isomorphism and Measure Rigidity for Algebraic Actions on Zero-Dimensional Groups

We consider mixing ^d-actions on compact zero-dimensional abelian groups by automorphisms. Rigidity of invariant measures does not hold for such actions in general; we present conditions which force an invariant measure to be Haar measure on an affine subset. This is applied to isomorphism rigidity for such actions. We develop a theory of halfspace entropies which plays a similar role in the proof to that played by invariant foliations in the proof of rigidity for smooth actions.     

On Maps Preserving Zero Jordan Products

Let R be a ring, A = M _n (R) and θ: A → A a surjective additive map preserving zero Jordan products, i.e. if x,y ∈ A are such that xy + yx = 0, then θ(x)θ(y) + θ(y)θ(x) = 0. In this paper, we show that if R contains and n ≥ 4, then θ = λϕ, where λ = θ(1) is a central element of A and ϕ: A → A is a Jordan homomorphism. The third author is Corresponding author.     

Invariant Cocycles Have Abelian Ranges

 Let R be a discrete nonsingular equivalence relation on a standard probability space , and let V be an ergodic strongly asymptotically central automorphism of R. We prove that every V-invariant cocycle with values in a Polish group G takes values in an abelian subgroup of G. The hypotheses of this result are satisfied, for example, if A is a finite set, a closed, shift-invariant subset, V is the shift, μ a shift-invariant and ergodic probability measure on X, the two-sided tail-equivalence relation on X, a shift-invariant subrelation which is μ-nonsingular, and a shift-invariant cocycle. (Received 15 September 2001)     

On the Mean Value of the Product of Multiplicative Functions with Shifted Argument

In this paper, we consider the mean value of the product of multiplicative arithmetic functions with shifted argument. The investigated functions have to satisfy the following conditions: their moduli do not exceed 1; the values on the set of primes are close to 1 for one of the functions and close to a fixed complex number for the other function. Some consequences for the classical functions are given.     

Summatory Formula of the Convolution of Two Arithmetical Functions

 We study the asymptotic formula of for some arithmetical functions f and g. This generalizes the case investigated by Balakrishnan and Pétermann. Received 15 January 2001; in revised form 7 July 2001     

Kuratowski Convergence of Holomorphic Functions

The notion of Kuratowski convergence is applied to describe a kind of convergence in the context of holomorphic functions. We associate it to a convenient topology, explore its relation with the compact-open topology, thus providing a new set theoretic point of view of this classic topology, and present it in the framework of set-valued mappings.     

Rotating Charge Coupled to the Maxwell Field: Scattering Theory and Adiabatic Limit

We consider a spinning charge coupled to the Maxwell field. Through the appropriate symmetry in the initial conditions the charge remains at rest. We establish that any time-dependent finite energy solution converges to a sum of a soliton wave and an outgoing free wave. The convergence holds in global energy norm. Under a small constant external magnetic field the soliton manifold is stable in local energy seminorms and the evolution of the angular velocity is guided by an effective finite-dimensional dynamics. The proof uses a non-autonomous integral inequality method.Supported partly by the Wittgenstein 2000 Award of Peter Markowich, funded by the Austrian Science Foundation (FWF), research grants of DFG (436 RUS 113/615/0-1(R)) and RFBR (01-01-04002).On leave Department of Mechanics and Mathematics, Moscow State University, Moscow 119899, Russia. Supported partly by Max Planck Institute for the Mathematics in Sciences (Leipzig) and the Austrian Science Foundation (FWF) START Project (Y-137-TEC) of Norbert Mauser.