divergence in lattices in semisimple lie groups and graphs of groups divergence functions of a metric space estimate the length of a path connecting two points a b at distance 3 s is a finite set of valuations of a number field k including all infinite valuations and o s is the corresponding ring of s integers

near integrable systems a two dimensional circular quantum billiard with unusual boundary conditions introduced by berry and dennis 2008 j phys a math theor 41 135203 is considered in detail it is demonstrated that most of its eigenfunctions are strongly localized and the corresponding eigenvalues are close to eigenvalues of the circular billiard with neumann boundary conditions deviations from strong localization are also discussed these results agree well with numerical calculations

single parameter scaling in one dimensional localization revisited the variance of the lyapunov exponent is calculated exactly in the one dimensional anderson model with random site energies distributed according to the cauchy distribution we find a new significant scaling parameter in the system and derive an exact analytical criterion for single parameter scaling which differs from the commonly used condition of phase randomization the results obtained are applied to the kronig penney model with the potential in the form of periodically positioned ensuremath delta functions with random strength

lipschitz constants for iterates of mean lipschitzian mappings abstract we present a sharp evaluation of the lipschitz constant for every iterate for mean lipschitzian mappings solving a problem posed by k goebel and m a japon pineda

fluctuations of the product of random matrices and generalized lyapunov exponent i present a general framework allowing to carry out explicit calculation of the moment generating function of random matrix products varpi n m nm n 1 ldots m 1 where m i s are i i d following tutubalin theor probab appl 10 1 15 27 1965 the calculation of the generating function is reduced to finding the largest eigenvalue of a certain transfer operator associated with a family of representations of the group the formalism is illustrated by considering products of random matrices from the group mathrm sl 2 mathbb r where explicit calculations are possible for concreteness i study in detail transfer matrix products for the one dimensional schrodinger equation where the random potential is a levy noise derivative of a levy process in this case i obtain a general formula for the variance of ln varpi n and for the variance of ln psi x where psi x is the wavefunction in terms of a single integral involving the fourier transform of the invariant density of the matrix product finally i discuss the continuum limit of random matrix products matrices close to the identity in particular i investigate a simple case where the spectral problem providing the generalized lyapunov exponent can be solved exactly

resonant enhancement of anderson localization analytical approach we study localization properties of the eigenstates and wave transport in a one dimensional system consisting of a set of barriers and or wells of fixed thickness and random heights the inherent peculiarity of the system resulting in the enhanced anderson localization is the presence of the resonances emerging due to the coherent interaction of the waves reflected from the interfaces between the wells and or barriers our theoretical approach allows to derive the localization length in infinite samples both out of the resonances and close to them we examine how the transport properties of finite samples can be described in terms of this length it is shown that the analytical expressions obtained by standard methods for continuous random potentials can be used in our discrete model in spite of the presence of resonances that cannot be described by conventional theories all our results are illustrated with numerical data manifesting an excellent agreement with the theory

the demiclosedness principle for mean nonexpansive mappings abstract in this paper we establish the demiclosedness principle for the class of mean nonexpansive mappings introduced in 2007 by goebel and japon pineda defined on closed convex subsets of banach spaces satisfying opial s condition we also establish the demiclosedness principle for a subclass of the mean nonexpansive maps whose domains are closed bounded convex sets in uniformly convex spaces these results extend known demiclosedness results for nonexpansive maps and lead to some fixed point results for mean nonexpansive maps which partially answer an open question posed by goebel and japon pineda

on mean nonexpansive mappings and the lifshitz constant abstract we present how the theory of polynomials can be used to describe the asymptotic behaviour of the sequence of lipschitz constants for iterates of mean nonexpansive mappings we find as consequences using the lifshitz theorem some new fixed point theorems we also prove that the fixed point set of every mean nonexpansive mapping is closed and convex provided the banach space is strictly convex

fluctuations of random matrix products and 1d dirac equation with random mass we study the fluctuations of certain random matrix products pi n m n cdots m 2m 1 of mathrm sl 2 mathbb r describing localisation properties of the one dimensional dirac equation with random mass in the continuum limit i e when matrices m n s are close to the identity matrix we obtain convenient integral representations for the variance gamma 2 lim n rightarrow infty mathrm var ln pi n n the case studied exhibits a saturation of the variance at low energy varepsilon along with a vanishing lyapunov exponent gamma 1 lim n rightarrow infty ln pi n n leading to the behaviour gamma 2 gamma 1 sim ln 1 varepsilon rightarrow infty as varepsilon rightarrow 0 our continuum description sheds new light on the kappus wegner band center anomaly

quasi convexity of hyperbolically embedded subgroups we show that any infinite order element g of a virtually cyclic hyperbolically embedded subgroup of a group g is morse that is to say any quasi geodesic connecting points in the cyclic group c generated by g stays close to c this answers a question of dahmani guirardel osin what is more we show that hyperbolically embedded subgroups are quasi convex finally we give a definition of what it means for a collection of subspaces of a metric space to be hyperbolically embedded and we show that axes of pseudo anosovs are hyperbolically embedded in teichmuller space endowed with the weil petersson metric

individual energy level distributions for one dimensional diagonal and off diagonal disorder we study the distribution of the n th energy level for two different one dimensional random potentials this distribution is shown to be related to the distribution of the distance between two consecutive nodes of the wave function we first consider the case of a white noise potential and study the distributions of energy level both in the positive and the negative part of the spectrum it is demonstrated that in the limit of a large system l the distribution of the n th energy level is given by a scaling law which is shown to be related to the extreme value statistics of a set of independent variables in the second part we consider the case of a supersymmetric random hamiltonian potential v x x 2 x we study first the case of x being a white noise with zero mean it is in particular shown that the ground state energy which behaves on average like exp l 1 3 in agreement with previous work is not a self averaging quantity in the limit l as is seen in the case of diagonal disorder then we consider the case when x has a non zero mean value

the lyapunov exponent of products of random 2 2 matrices close to the identity we study products of arbitrary random real 2 2 matrices that are close to the identity matrix using the iwasawa decomposition of sl 2 we identify a continuum regime where the mean values and the covariances of the three iwasawa parameters are simultaneously small in this regime the lyapunov exponent of the product is shown to assume a scaling form in the general case the corresponding scaling function is expressed in terms of gauss hypergeometric function a number of particular cases are also considered where the scaling function of the lyapunov exponent involves other special functions airy bessel whittaker elliptic the general solution thus obtained allows us among other things to recover in a unified framework many results known previously from exactly solvable models of one dimensional disordered systems