on the distribution of a random variable occurring in 1d disordered systems the authors consider the random variable z 1 x1 x1x2 x1x2x3 where the xi are independent identically distributed variables they derive some asymptotic properties of the distribution of z which are related e g to the low temperature behaviour of the random field ising chain for a special class of distributions of the xi exact solutions are presented they also study the cases where the distribution function of z exhibits a power law fall off modulated by a periodic critical amplitude

hierarchical pinning models quadratic maps and quenched disorder we consider a hierarchical model of polymer pinning in presence of quenched disorder introduced by derrida et al j stat phys 66 1189 1213 1992 which can be re interpreted as an infinite dimensional dynamical system with random initial condition the disorder it is defined through a recurrence relation for the law of a random variable r n n 1 2 which in absence of disorder i e when the initial condition is degenerate reduces to a particular case of the well known logistic map the large n limit of the sequence of random variables 2 n log r n a non random quantity which is naturally interpreted as a free energy plays a central role in our analysis the model depends on a parameter 0 1 related to the geometry of the hierarchical lattice and has a phase transition in the sense that the free energy is positive if the expectation of r 0 is larger than a certain threshold value and it is zero otherwise it was conjectured in derrida et al j stat phys 66 1189 1213 1992 that disorder is relevant respectively irrelevant or marginally relevant if 1 2 1 2 we find the correct scaling form for weak disorder of the critical point shift

periodic oscillations of coefficients of power series that satisfy functional equations abstract it is shown that the coefficients an of the power series f z n 1anzn which satisfies the functional equation f z z f z 2 z 3 display periodic oscillations a n o n n u log n as n where o 1 5 1 2 2 and u x is a positive nonconstant continuous function which is periodic with period log 4 o similar results are obtained for a wide class of power series that satisfy similar functional equations power series of these types are of interest in combinatorics and computer science since they often represent generating functions for example the nth coefficient of the power series satisfying enumerates 2 3 trees with n leaves

boundedness in nonlinear oscillations at resonance abstract in this paper we are concerned with the boundedness of all the solutions for the semilinear duffing equation x n 2 x x p t with the lazer leach condition where p t is a smooth 2 periodic function and the perturbation x is bounded

oscillatory critical amplitudes in hierarchical models we study the oscillatory critical amplitudes of theq states potts model on a diamond hierarchical lattice we consider an example of the generic case finite critical index as well as the degenerate case essential singularity in both cases we compare the magnitude of the oscillations with geometrical characteristics of the julia set of zeroes of the partition function

log periodic oscillations for biased diffusion on random lattice abstract random walks with a fixed bias direction on randomly diluted cubic lattices far above the percolation threshold exhibit log periodic oscillations in the effective exponent versus time a scaling argument accounts for the numerical results in the limit of large biases and small dilution and shows the importance of the interplay of these two ingredients in the generation of the log periodicity these results show that log periodicity is the dominant effect compared to previous predictions of and reports on anomalous diffusion

semilinear duffing equations crossing resonance points abstract in this paper using a generalized form of the poincare birkhoff theorem and a fixed point theorem we prove under weaker conditions two theorems for the equation x g x p t p t p t 2 of which one shows the existence of a harmonic solution the other that the equation may have an infinite number of harmonic solutions in the resonance case this is an enhancement of the results already obtained

marginal relevance of disorder for pinning models the effect of disorder on pinning and wetting models has attracted much attention in theoretical physics in particular it has been predicted on the basis of the harris criterion that disorder is relevant annealed and quenched model have different critical points and critical exponents if the return probability exponent alpha a positive number that characterizes the model is larger than 1 2 weak disorder has been predicted to be irrelevant i e coinciding critical points and exponents if alpha 1 2 recent mathematical work has put these predictions on firm grounds in renormalization group terms the case alpha 1 2 is a marginal case and there is no agreement in the literature as to whether one should expect disorder relevance or irrelevance at marginality the question is particularly intriguing also because the case alpha 1 2 includes the classical models of two dimensional wetting of a rough substrate of pinning of directed polymers on a defect line in dimension 3 1 or 1 1 and of pinning of an heteropolymer by a point potential in three dimensional space here we prove disorder relevance both for the general alpha 1 2 pinning model and for the hierarchical version of the model proposed by b derrida v hakim and j vannimenus jsp 1992 in the sense that we prove a shift of the quenched critical point with respect to the annealed one in both cases we work with gaussian disorder and we show that the shift is at least of order exp 1 beta 4 for beta small if beta is the standard deviation of the disorder

multiplicity of periodic solutions of semilinear duffing s equation at resonance in this paper we prove an infinity of periodic solutions to the periodically forced nonlinear duffing equation under new conditions

boundedness in a piecewise linear oscillator and a variant of the small twist theorem consider the differential equation ddot x n 2 x h l x p t where n 1 2 dots is an integer p is a 2 pi periodic function and h l is the piecewise linear function h l x begin cases l text if x geq 1 lx text if x leq 1 l text if x leq 1 end cases a classical result of lazer and leach implies that this equation has a 2 pi periodic solution if and only if begin equation label ll hat p n 2l over pi end equation where hat p n 1 over 2 pi int 0 2 pi p t e int dt in this paper i prove that if p is of class c 5 then the condition ref ll is also necessary and sufficient for the boundedness of all the solutions of the equation the proof of this theorem motivates a new variant of moser s small twist theorem this variant guarantees the existence of invariant curves for certain mappings of the cylinder which have a twist that may depend on the angle 1991 mathematics subject classification 34c11 58f35

singular behaviour of certain infinite products of random 2 2 matrices the authors consider an infinite product of random matrices which appears in several physical problems in particular the ising chain in a random field the random matrices depend analytically on a parameter epsilon in such a way that for 0 they all commute under certain conditions they find that the lyapunov index of this product behaves approximately as c 2 for to 0 their approach is based on the decomposition of an exact integral equation for this problem into two reduced equations they give an expression for the exponent alpha in terms of the probability distribution of the matrices and for the proportionality constant c in terms of the solutions of the reduced integral equations in cases where exact results are available agreement is obtained the physical consequences for disordered one dimensional systems are pointed out

spatial log periodic oscillations of first passage observables in fractals for transport processes in geometrically restricted domains the mean first passage time mfpt admits a general scaling dependence on space parameters for diffusion anomalous diffusion and diffusion in disordered or fractal media for transport in self similar fractal structures we obtain an expression for the source target distance dependence of the mfpt that exhibits both the leading power law behavior depending on the hausdorff and spectral dimension of the fractal as well as small log periodic oscillations that are a clear and definitive signal of the underlying fractal structure we also present refined numerical results for the sierpinski gasket that confirm this oscillatory behavior

fractal structure of zeros in hierarchical models we consider an ising and aq state potts model on a diamond hierarchical lattice we give pictures of the distribution of zeros of the partition function in the complex plane of temperatures for several choices ofq these zeros are just the julia set corresponding to the renormalization group transformation

distribution of repetitions of ancestors in genealogical trees we calculate the probability distribution of repetitions of ancestors in a genealogical tree for simple neutral models of a closed population with sexual reproduction and non overlapping generations each ancestor at generation g in the past has a weight w which is up to a normalization the number of times this ancestor appears in the genealogical tree of an individual at present the distribution pg w of these weights reaches a stationary shape p1 w for large g i e for a large number of generations back in the past for small w p1 w is a power law p1 w w with a non trivial exponent which can be computed exactly using a standard procedure of the renormalization group approach some extensions of the model are discussed and the eect of these variants on the shape of p1 w are analysed c 2000 elsevier science b v all rights reserved

log periodic critical amplitudes a perturbative approach log periodic amplitudes appear in the critical behavior of a large class of systems in particular when a discrete scale invariance is present here we show how to compute these critical amplitudes perturbatively when they originate from a renormalization map which is close to a monomial in this case the log periodic amplitudes of the subdominant corrections to the leading critical behavior can also be calculated

periodic and unbounded motions in asymmetric oscillators at resonance abstract we consider the existence of periodic and unbounded solutions for the asymmetric oscillator x a x b x g x p t where x max x 0 x max x 0 a and b are two positive constants p t is a 2 periodic smooth function and g x satisfies lim x x 1 g x 0 we have proved previously that the boundedness of all the solutions and the existence of unbounded solutions have a close relation to the interaction of some well defined functions p and h in this paper we consider some critical cases and obtain some new sufficient conditions for the existence of 2 periodic and unbounded solutions in particular the obtained periodic solutions are new and cannot be deduced from landesman lazer s and fabry fonda s existence conditions in contrast with many existing results the function g x may be unbounded or oscillatory without asymptotic limits

convergence to the critical attractor of dissipative maps log periodic oscillations fractality and nonextensivity for a family of logisticlike maps we investigate the rate of convergence to the critical attractor when an ensemble of initial conditions is uniformly spread over the entire phase space we found that the phase space volume occupied by the ensemble w t depicts a power law decay with log periodic oscillations reflecting the multifractal character of the critical attractor we explore the parametric dependence of the power law exponent and the amplitude of the log periodic oscillations with the attractor s fractal dimension governed by the inflection of the map near its extremal point further we investigate the temporal evolution of w t for the circle map whose critical attractor is dense in this case we found w t to exhibit a rich pattern with a slow logarithmic decay of the lower bounds these results are discussed in the context of nonextensive tsallis entropies

behavior of lacunary series at the natural boundary abstract we develop a local theory of lacunary dirichlet series of the form k 1 c k exp z g k re z 0 as z approaches the boundary i r under the assumption g and further assumptions on c k these series occur in many applications in fourier analysis infinite order differential operators number theory and holomorphic dynamics among others for relatively general series with c k 1 the case we primarily focus on we obtain blow up rates in measure along the imaginary line and asymptotic information at z 0 when sufficient analyticity information on g exists we obtain borel summable expansions at points on the boundary giving exact local description borel summability of the expansions provides property preserving extensions beyond the barrier the singular behavior has remarkable universality and self similarity features if g k k b c k 1 b n or b n 1 n n n behavior near the boundary is roughly of the standard form re z b q x where q x 1 q if x p q q and zero otherwise the botcher map at infinity of polynomial iterations of the form x n 1 p x n 0 p turns out to have uniformly convergent fourier expansions in terms of simple lacunary series for the quadratic map p x x x 2 0 1 and the julia set is the graph of this fourier expansion in the main cardioid of the mandelbrot set

kac moody lie algebras and soliton equations ii lax equations associated with a1 1 the soliton equations associated with sl 2 eigenvalue problems polynomial in the eigenvalue parameter are given a unified treatment they are shown to be generated by a single family of commuting hamiltonians on a subalgebra of the loop algebra of sl 2 the conserved densities and fluxes of the usual anks hierarchy are identified with conserved densities and fluxes for the polynomial eigenvalue problems the hamiltonian structures of the soliton equations associated with the polynomial eigenvalue problems are given a unified treatment

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

oscillatory critical amplitudes in hierarchical models and the harris function of branching processes oscillatory critical amplitudes have been repeatedly observed in hierarchical models and in the cases that have been taken into consideration these oscillations are so small to be hardly detectable hierarchical models are tightly related to iteration of maps and in fact very similar phenomena have been repeatedly reported in many fields of mathematics like combinatorial evaluations and discrete branching processes it is precisely in the context of branching processes with bounded off spring that t harris in 1948 first set forth the possibility that the logarithm of the moment generating function of the rescaled population size in the super critical regime does not grow near infinity as a power but it has an oscillatory prefactor the harris function these oscillations have been observed numerically only much later and while the origin is clearly tied to the discrete character of the iteration the amplitude size is not so well understood the purpose of this note is to reconsider the issue for hierarchical models and in what is arguably the most elementary setting the pinning model that actually just boils down to iteration of polynomial maps and notably quadratic maps in this note we show that the oscillatory critical amplitude for pinning models and the harris function coincide moreover we make explicit the link between these oscillatory functions and the geometry of the julia set of the map making thus rigorous and quantitative some ideas set forth in derrida et al commun math phys 94 115 132 1984

a small twist theorem and boundedness of solutions for semilinear duffing equations at resonance abstract we obtain a new variant of moser s small twist theorem and apply this new version to investigate the boundedness of solutions for the following semilinear duffing equation x n 2 x g x p t where p is a 2 periodic smooth function and lim x x 1 g x 0 we obtain some sharp sufficient conditions for the boundedness of all solutions to the above equation at resonance unlike many existing results in the literature where the function g is required to be a bounded function with asymptotic limits our main results here allow g be unbounded or oscillatory without asymptotic limits

boundedness of solutions for semilinear duffing equations in this paper we are concerned with the boundedness of all the solutions and the existence of quasi periodic solutions for some semilinear duffing equationx g x e t wheree t is a smooth 2 periodic function and0 g x x k