on some integrals over the u n unitary group and their large n limit the integral over the u n unitary group i du exp tr au bu is re examined various approaches and extensions are first reviewed the second half of the paper deals with more recent developments relation with integrable toda lattice hierarchy diagrammatic expansion and combinatorics and what they teach us on the large n limit of log i

the stability of partial difference systems with retarded arguments abstract for certain partial difference systems with retarded arguments sufficient conditions for the existence of decaying solutions are given the stability of these systems is discussed also

integration over matrix spaces with unique invariant measures we present a method to calculate integrals over monomials of matrix elements with invariant measures in terms of wick contractions the method gives exact results for monomials of low order for higher order monomials it leads to an error of order 1 n where n is the dimension of the matrix and where is independent of the degree of the monomial we give a lower bound on the integer and show how can be increased systematically the method is particularly suited for symbolic computer calculation explicit results are given for o n u n and for the circular orthogonal ensemble

iterated networks and the spectra of renormalizable electromechanical systems beginning with anlrc network with impedance functionz a sequence of iterated networksnk with impedance functionsz k k 1 2 3 is introduced the asymptotic comportment ofz k and the spectra ofnk are analyzed in terms of the julia set ofz an example is given of an iterated network associated with a cascade of period doubling bifurcations

a family of almost periodic schrodinger operators bellissard et al 1 introduced a one dimensional almost periodic discrete schrodinger operator which is defined by a parameter we allow this parameter to become complex and develop a geometric formalism to control the operator the support of the spectrum of this operator is the julia set of the mapping x x2 we prove that the almost periodicity holds over wide regions of the complex plane even though hermiticity fails

orthogonal and l q extremal polynomials on inverse images of polynomial mappings abstract let t be a polynomial of degree n and let k be a compact set with c first it is shown if zero is a best approximation to f from p n on k with respect to the lq norm q 1 then zero is also a best approximation to f t on t 1 k with respect to the l q t norm where t arises from by the transformation t in particular t is the equilibrium measure on t 1 k if is the equilibrium measure on k for q i e the sup norm a corresponding result is presented in this way polynomials minimal on several intervals on lemniscates on equipotential lines of compact sets etc are obtained special attention is given to lq minimal polynomials on julia sets next based on asymptotic results of widom we show that the minimum deviation of polynomials orthogonal with respect to a positive measure on t 1 k behaves asymptotically periodic and that the orthogonal polynomials have an asymptotically periodic behaviour too some open problems are also given

orthogonal polynomials and cubic transformations abstract starting from a sequence pn of orthogonal polynomials with respect to a quasi definite or regular linear functional we construct a new sequence qn of orthogonal polynomials such that q3n x pn x3 the basic tool is the three term recurrence relation which such a family satisfies as an application we present some examples concerning chebyshev and hermite polynomials

almost periodic jacobi matrices associated with julia sets for polynomials let be the jacobi matrix associated with polynomialt z of degreen 2 the spectrum of is the julia set associated witht z which in many cases is a cantor set let 1 denote the result of omitting the first row and column ofj then it is shown that the spectrum of 1 may be purely discrete

on a sequence of rational functions abstract we study a sequence of rational functions considered by bessis mehta and moussa 1982 and we show that a conjecture given in this paper is not correct moreover we give a relation between values of these rational functions and a finite automaton

meblin transforms associated with julia sets and physical applications we introduce the mellin transform of the balanced invariant measure associated to the julia set generated by a rational transformation we show that its analytic continuation is a meromorphic function the poles of which are on a semi infinite periodic lattice this allows one to have an understanding of the behavior of the measure near a repulsive fixed point trace identities corresponding to the fact that the analytically continued mellin transform vanishes at negative integers are derived for the polynomial case the quadratic map is first analyzed in detail and the analytic properties of the inverse of the green s function are exhibited of interest is the appearance of a dense set of spikes at dyadic points when the julia set is disconnected these results are used to study the residues of the mellin transform a certain number of physically interesting consequences are derived for the spectral dimensionality of quantum mechanical systems the excitation spectrum of which displays unusual oscillations the appearance of complex critical indices for thermodynamical systems is also discussed in the conclusion

geometry and combinatorics of julia sets of real quadratic maps for real a correspondence is made between the julia setb forz z 2 in the hyperbolic case and the set of chains with the aid of cremer s theorem it is shown how a number of features ofb can be understood in terms of chains the structure ofb is determined by certain equivalence classes of chains fixed by orders of visitation of certain real cycles and the bifurcation history of a given cycle can be conveniently computed via the combinatorics of chains the functional equations obeyed by attractive cycles are investigated and their relation to chains is given the first cascade of period doubling bifurcations is described from the point of view of the associated julia sets and chains certain julia sets associated with the feigenbaum function and some theorems of lanford are discussed

orthogonality properties of iterated polynomial mappings we consider a measure defined on a complex contour and its associated orthogonal polynomials the action of a polynomial transformation on the measure and the transformation laws of the corresponding orthogonal polynomials are given iterating the transformation provides an invariant measure whose support is the julia set corresponding to the polynomial transformation it appears that up to a constant the iterated polynomials generated by the initial mapping form a subset of the invariant set of orthogonal polynomials which fulfill a three term recursion relation an algorithm is given to compute the coefficients of this recursion relation which can be interpreted as a linear extension of the iterative procedure

the stability of certain partial difference equations abstract this paper is concerned with the linear delay partial difference equation where and are positive integers a i j b i j p i j are real sequences defined on i j sufficient conditions for this equation to be stable are derived stability of certain nonlinear partial difference equations is studied also

on the invariant sets of a family of quadratic maps the julia setb for the mappingz z 2 is considered where is a complex parameter for 2 a new upper bound for the hausdorff dimension is given and the monic polynomials orthogonal with respect to the equilibrium measure onb are introduced a method for calculating all of the polynomials is provided and certain identities which obtain among coefficients of the three term recurrence relations are given a unifying theme is the relationship betweenb and chains which is explored for 1 4 2 and for with 1 4 with the aid of the bottcher equation thenb is shown to be a holder continuous curve for 1 4

block orthogonal systems for symmetric p forms abstract to study polynomials orthogonal with respect to the logarithmic equilibrium measure on the julia set of a nonlinear polynomial p bessis and his coworkers introduced a symmetric bilinear form the symmetric p form it is a non hermitian variant of a genuine inner product the hermitian p form the main reason for doing so was the existence of a three term recurrence relation between the polynomials of successive degree of an orthogonal system however this is only true if the moment matrix of the symmetric p form is strongly regular in this paper we investigate what happens if this condition is not met we distinguish between the case where an orthogonal system still exists and the one where it does not and illustrate them by z z 2 and p z z 2 1 respectively

level repulsion in constrained gaussian random matrix ensembles introducing sets of constraints we define new classes of random matrix ensembles the constrained gaussian unitary cgue and the deformed gaussian unitary dgue ensembles the latter interpolate between the gue and the cgue we derive a sufficient condition for gue type level repulsion to persist in the presence of constraints for special classes of constraints we extend this approach to the orthogonal and to the symplectic ensembles a generalized fourier theorem relates the spectral properties of the constraining ensembles with those of the constrained ones we find that in the dgues level repulsion always prevails at a sufficiently short distance and may be lifted only in the limit of strictly enforced constraints

spectral fluctuation properties of constrained unitary ensembles of gaussian distributed random matrices we investigate the spectral fluctuation properties of constrained ensembles of random matrices defined by the condition that a number nq of matrix elements vanish identically that condition is imposed in unitarily invariant form in the limit of large matrix dimension we show that as long as nq is smaller than a critical value at which the quadratic level repulsion of the gaussian unitary ensemble of random matrices may be destroyed all spectral fluctuation measures have the same form as for the gaussian unitary ensemble

review of the k body embedded ensembles of gaussian random matrices the embedded ensembles were introduced by mon and french 1975 ann phys ny 95 90 as physically more plausible stochastic models of many body systems governed by one and two body interactions than provided by standard random matrix theory we review several approaches aimed at determining the spectral density the spectral fluctuation properties and the ergodic properties of these ensembles moments methods numerical simulations the replica trick the eigenvector decomposition of the matrix of second moments and supersymmetry the binary correlation approximation and the study of correlations between matrix elements

inverse images of polynomial mappings and polynomials orthogonal on them let j be a polynomial with complex coefficients first we study the inverse images of the real and imaginary axes under a polynomial mapping j in detail then for an arbitrary polynomial and a sequence pn of orthogonal polynomials the orthogonality behaviour of the sequence of polynomials pn j n n is investigated in particular necessary and sufficient conditions are given such that pn j n n is a subsequence of polynomials orthogonal with respect to a positive measure supported on a compact subset of the real line

stability criteria for a class of linear delay partial difference equations this paper is concerned with the linear delay partial difference equation a i j u i 1 j u i j 1 b i j u i j p i j u i j 0 where a i j b i j p i j i j n0 are real sequences sufficient conditions for this equation to be stable are derived some conditions for this equation to be unstable are obtained also

orthogonal polynomials on the unit circle via a polynomial mapping on the real line let f n n 0 be a sequence of monic orthogonal polynomials on the unit circle opuc with respect to a symmetric and finite positive borel measure email protected on 0 email protected and let 1 a 0 a 1 a 2 be the associated sequence of verblunsky coefficients in this paper we study the sequence email protected n n 0 of monic opuc whose sequence of verblunsky coefficients is 1 b 1 b 2 b n 1 a 0 b n 1 b 2 b 1 a 1 b 1 b 2 b n 1 a 2 b n 1 b 2 b 1 a 3 where b 1 b 2 b n 1 are n 1 fixed real numbers such that b email protected 1 1 for all j 1 2 n 1 so that email protected n n 0 is also orthogonal with respect to a symmetric and finite positive borel measure email protected on the unit circle we show that the sequences of monic orthogonal polynomials on the real line oprl corresponding to f n n 0 and email protected n n 0 by szego s transformation are related by some polynomial mapping giving rise to a one to one correspondence between the monic opuc email protected n n 0 on the unit circle and a pair of monic oprl on a subset of the interval 1 1 in particular we prove email protected q email protected n 1 q email protected q n q email protected q n q q n q supported on a subset of the union of 2n intervals contained in 0 email protected such that any two of these intervals have at most one common point and where up to an affine change in the variable z n 1 and email protected n are algebraic polynomials in email protected of degrees n 1 and n respectively defined only in terms of a 0 b 1 b n 1 this measure induces a measure on the unit circle supported on the union of 2n arcs pairwise symmetric with respect to the real axis the restriction to symmetric measures or real verblunsky coefficients is needed in order that szego s transformation may be applicable

spectra of certain jacobi operators we discuss the spectra and in particular the essential spectra of some bounded self adjoint jacobi operators associated with orthogonal polynomial sequences obtained via polynomial mappings