Multidimensional continued fractions and a Minkowski function

The Minkowski question mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the M?nkemeyer continued fraction algorithm with an appropriate tent map. Author’s address: Department of Mathematics, University of Udine, via delle Scienze 208, 33100 Udine, Italy     

Prime Fano threefolds and integrable systems

For a general K3 surface S of genus g, with 2 ≤ g ≤ 10, we prove that the intermediate Jacobians of the family of prime Fano threefolds of genus g containing S as a hyperplane section, form generically an algebraic completely integrable Hamiltonian system. The first author is partially supported by grant MI1503/2005 of the Bulgarian Foundation for Scientific Research.     

Valleys and the Maximum Local Time for Random Walk in Random Environment

Let ξ (n, x) be the local time at x for a recurrent one-dimensional random walk in random environment after n steps, and consider the maximum ξ^*(n) = max _x ξ(n, x). It is known that lim sup is a positive constant a.s. We prove that lim inf is a positive constant a.s. this answers a question of P. Révész [5]. The proof is based on an analysis of the valleys in the environment, defined as the potential wells of record depth. In particular, we show that almost surely, at any time n large enough, the random walker has spent almost all of its lifetime in the two deepest valleys of the environment it has encountered. We also prove a uniform exponential tail bound for the ratio of the expected total occupation time of a valley and the expected local time at its bottom.     

Algebraic independence of reciprocal sums of binary recurrences

Algebraic independence of the numbers , where{R _n } _n 0 is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler's method.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

The approximate decomposition of exponential order of slow-fast motions in multifrequency systems

This paper deals with multifrequency slow-fast systems. It is shown that, under a suitable change of coordinates, the system can be reduced to a simple form such that slow motions are described by autonomous equations except for exponential error of perturbations. Hence, the fast and slow motions are decoupled. The Newton rapid iteration is used. In addition, for a perturbation, only the smallness condition is needed.     

Lyapunov exponents,bifurcation currents and laminations in bifurcation loci

Bifurcation loci in the moduli space of degree d rational maps are shaped by the hypersurfaces defined by the existence of a cycle of period n and multiplier 0 or e ^iθ. Using potential-theoretic arguments, we establish two equidistribution properties for these hypersurfaces with respect to the bifurcation current. To this purpose we first establish approximation formulas for the Lyapunov function. In degree d = 2, this allows us to build holomorphic motions and show that the bifurcation locus has a lamination structure in the regions where an attracting basin exists.     

On integer and fractional parts of powers of Salem numbers

Let N be a positive rational integer and let P be the set of powers of a Salem number of degree d. We prove that for any α∈P the fractional parts of the numbers , when n runs through the set of positive rational integers, are dense in the unit interval if and only if N≦ 2d − 4. We also show that for any α∈P the integer parts of the numbers αⁿ are divisible by N for infinitely many n if and only if N≦ 2d − 3. Received: 27 April 2005     

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.     

Analytic-non-integrability of an integrable analytic Hamiltonian system

We introduce the polynomial Hamiltonian and we prove that the associated Hamiltonian system is Liouville-C^∞-integrable, but fails to be real-analytically integrable in any neighbourhood of an equilibrium point. The proof only uses power series expansions, and is elementary.     

Annealed deviations of random walk in random scenery

Let (Zn)_n∈N be a d-dimensional random walk in random scenery, i.e., with (Sk)_k∈N0 a random walk in Zd and (Y(z))_z∈Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and some finite exponential moments. We identify the speed and the rate of the logarithmic decay of for various choices of sequences n(bn) in [1,∞). Depending on n(bn) and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work [A. Asselah, F. Castell, Large deviations for Brownian motion in a random scenery, Probab. Theory Related Fields 126 (2003) 497-527] by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen [X. Chen, Exponential asymptotics and law of the iterated logarithm for intersection local times of random walks, Ann. Probab. 32 (4) 2004].     

Spanning triangulations in graphs

We prove that every graph of sufficiently large order n and minimum degree at least 2n/3 contains a triangulation as a spanning subgraph. This is best possible: for all integers n, there are graphs of order n and minimum degree ?2n/3? ? 1 without a spanning triangulation. © 2005 Wiley Periodicals, Inc. J Graph Theory     

Invariant measures of Hamiltonian systems with prescribed asymptotic Maslov index

We study the properties of the asymptotic Maslov index of invariant measures for time-periodic Hamiltonian systems on the cotangent bundle of a compact manifold M. We show that if M has finite fundamental group and the Hamiltonian satisfies some general growth assumptions on the momenta, then the asymptotic Maslov indices of periodic orbits are dense in the half line [0,+∞). Furthermore, if the Hamiltonian is the Fenchel dual of an electromagnetic Lagrangian, then every non-negative number r is the limit of the asymptotic Maslov indices of a sequence of periodic orbits which converges narrowly to an invariant measure with asymptotic Maslov index r. We discuss the existence of minimal ergodic invariant measures with prescribed asymptotic Maslov index by the analogue of Mather’s theory of the beta function, the asymptotic Maslov index playing the role of the rotation vector. Dedicated to Vladimir Igorevich Arnold     

On the number of halving planes

LetS ³ be ann-set in general position. A plane containing three of the points is called a halving plane if it dissectsS into two parts of equal cardinality. It is proved that the number of halving planes is at mostO(n ^2.998).As a main tool, for every setY ofn points in the plane a setN of sizeO(n ⁴) is constructed such that the points ofN are distributed almost evenly in the triangles determined byY.Research supported partly by the Hungarian National Foundation for Scientific Research grant No. 1812     

Multifractal analysis of divergence points of deformed measure theoretical Birkhoff averages. IV: Divergence points and packing dimension

During the past 10 years multifractal analysis has received an enormous interest. For a sequence n(φn) of functions on a metric space X, multifractal analysis refers to the study of the Hausdorff and/or packing dimension of the level sets(1) of the limit function limnφn. However, recently a more general notion of multifractal analysis, focusing not only on points x for which the limit limnφn(x) exists, has emerged and attracted considerable interest. Namely, for a sequence n(xn) in a metric space X, we let A(xn) denote the set of accumulation points of the sequence n(xn). The problem of computing that the Hausdorff dimension of the set of points x for which the set of accumulation points of the sequence (φnn(x)) equals a given set C, i.e. computing the Hausdorff dimension of the set(2){x∈X|A(φn(x))=C} has recently attracted considerable interest and a number of interesting results have been obtained. However, almost nothing is known about the packing dimension of sets of this type except for a few special cases investigated in [I.S. Baek, L. Olsen, N. Snigireva, Divergence points of self-similar measures and packing dimension, Adv. Math. 214 (2007) 267–287]. The purpose of this paper is to compute the packing dimension of those sets for a very general class of maps φn, including many examples that have been studied previously, cf. Theorem 3.1 and Corollary 3.2. Surprisingly, in many cases, the packing dimension and the Hausdorff dimension of the sets in (2) do not coincide. This is in sharp contrast to well-known results in multifractal analysis saying that the Hausdorff and packing dimensions of the sets in (1) coincide.     

On the index of composition of integers from various sets

Given an integer n ≥ 2, let λ(n) := (log n)/(log γ(n)), where γ(n) = Π _p|n p, stand for the index of composition of n, with λ(1) = 1. We study the distribution function of (λ(n) – 1) log n as n runs through particular sets of integers, such as the shifted primes, the values of a given irreducible cubic polynomial and the shifted powerful numbers. Research supported in part by a grant from NSERC. Research supported by the Applied Number Theory Research Group of the Hungarian Academy of Science and by a grant from OTKA. Professor M.V. Subbarao passed away on February 15, 2006. Received: 3 March 2006 Revised: 28 October 2006     

Moments, moderate and large deviations for a branching process in a random environment

Let (Z_n) be a supercritical branching process in a random environment ξ, and W be the limit of the normalized population size Z_n/E[Z_n|ξ]. We show large and moderate deviation principles for the sequence logZ_n (with appropriate normalization). For the proof, we calculate the critical value for the existence of harmonic moments of W, and show an equivalence for all the moments of Z_n. Central limit theorems on W−W_n and logZ_n are also established.     

Counting Closed Orbits of Hyperbolic Diffeomorphisms

In this paper we count closed orbits of a hyperbolic diffeomorphism restricted to a basic set. In fact we shall be dealing with two types of extensions of a hyperbolic diffeomorphism: a finite group extension (also called G-extensions) and also an automorphism extension (also called (G, ρ)-extensions). In particular we present Chebotarev type theorems and prime orbit theorem for such diffeomorphisms. These counting results are in the form of an asymptotic formula derived in complete analogy with the number theoretic result. The main difficulty is extending the domain of analyticity of the zeta and L-functions and this is overcome by resorting to symbolic dynamics. Unlike the case of a flow, the proof of this extension result does not rely on the properties of the Ruelle–Perron–Frobenius operator. Also the counting results do not use any tauberian theorems. Received: January 7, 2005.