quasi multiplicativity of typical cocycles we show that typical in the sense of bonatti and viana ergod theory dyn syst 24 5 1295 1330 2004 and avila and viana port math 64 311 376 2007 holder and fiber bunched text gl d mathbb r valued cocycles over subshifts of finite type are uniformly quasi multiplicative with respect to all singular value potentials we prove the continuity of the singular value pressure and its corresponding necessarily unique equilibrium state for such cocycles and apply this result to repellers moreover we show that the pointwise lyapunov spectrum is closed and convex and establish partial multifractal analysis on the level sets of pointwise lyapunov exponents for such cocycles

analyticity of smooth eigenfunctions and spectral analysis of the gauss map we provide a sufficient condition of analyticity of infinitely differentiable eigenfunctions of operators of the form uf x a x y f b x y dy acting on functions f u user1 v to mathbb c evolution operators of one dimensional dynamical systems and markov processes have this form we estimate from below the region of analyticity of the eigenfunctions and apply these results for studying the spectral properties of the frobenius perron operator of the continuous fraction gauss map we prove that any infinitely differentiable eigenfunction f of this frobenius perron operator corresponding to a non zero eigenvalue admits a unique analytic extension to the set mathbb c backslash infty 1 analyzing the spectrum of the frobenius perron operator in spaces of smooth functions we extend significantly the domain of validity of the mayer and ropstorff asymptotic formula for the decay of correlations of the gauss map

on the essential spectrum of the transfer operator for expanding markov maps the essential spectrum of the transfer operator for expanding markov maps of the interval is studied in detail to this end we construct explicityly an infinite set of eigenfunctions which allows us to prove that the essential spectrum inc k is a disk whose radius is related to the free energy of the liapunov exponent

spectral decomposition of the renyi map the authors construct a generalized spectral decomposition of the frobenius perron operator of the general beta adic renyi map using a general iterative operator method applicable in principle to any mixing dynamical system they also explicitly define appropriate rigged hilbert spaces which provide mathematical meaning to the formally obtained spectral decomposition the explicit construction of the eigenvalues and eigenvectors allows one to show that the essential spectral radius of the frobenius perron operator decreases as the smoothness of the domain increases the reason for the change of the spectrum from the unit disk to isolated eigenvalues is the existence of coherent states of the frobenius perron operator which are not infinitely differentiable

wegner estimate for gaussian random magnetic fields for the schrodinger operator on l2 r2 with the magnetic field which is a sample path of a stationary gaussian random field a wegner type estimate applicable for the proof of the anderson localization is proven by referring a recent method by erd s and hasler and the theory of the malliavin calculus

local geometry and dynamical behavior on folded basic sets we study new phenomena associated with the dynamics of higher dimensional non invertible hyperbolic maps f on basic sets of saddle type the dynamics in this case presents important differences from the case of diffeomorphisms or expanding maps we show that the stable dimension i e the hausdorff dimension of the intersection between local stable manifolds and the basic set and the unstable dimension similar definition give a lot of information about the dynamical ergodic properties of endomorphisms on folded basic sets we prove a geometric flattening phenomenon associated to the stable dimension i e we show that if the stable dimension is zero at a point then the fractal must be contained in a submanifold and f is expanding on we characterize folded attractors and folded repellers as those basic sets with full unstable dimension respectively with full stable dimension we classify possible dynamical behaviors and establish when is the system f 1 sided or 2 sided bernoulli for certain equilibrium measures on folded basic sets for a class of perturbation maps

a p q oscillator realization of two parameter quantum algebras it is noted that the study of a quantum algebra sup q 2 with two independent deformation parameters p q leads to a p q oscillator realization for it the analysis is extended to the p q analogues of su 1 1 osp 2 1 and the centreless virasoro algebra the standard single parameter q deformations are obtained in the limit p q

unusual solutions to the yang baxter equation the complete list of non symmetric eight vertex constant solutions to the yang baxter equation is obtained and new solutions dependent on more than one variable are presented the corresponding spin hamiltonians are derived by sutherland s method 1970

existence of an exact solution for a one phase stefan problem with nonlinear thermal coefficients from tirskii s method abstract the mathematical analysis of a one phase lame clapeyron stefan problem with nonlinear thermal coefficients following g a tirskii two exact solutions of stefan s nonlinear problem sov phys dokl 4 1959 288 292 is obtained two related cases are considered one of them has a temperature condition on the fixed face x 0 and the other one has a flux condition of the type q 0 t q 0 0 we obtain in both cases sufficient conditions for data in order to have the existence of an explicit solution of a similarity type which is given by using a double fixed point

computer assisted bounds for the rate of decay of correlations the rate of decay of correlations quantitatively describes the rate at which a chaotic system mixes the state space we present a new rigorous method to estimate a bound for this rate of mixing the technique may be implemented on a computer and is applicable to both multidimensional expanding and hyperbolic systems the bounds produced are significantly less conservative than current rigorous bounds in some situations it is possible to approximate resonant eigenfunctions and to strengthen our bound to an estimate of the decay rate order of convergence results are stated

spectral analysis and an area preserving extension of a piecewise linear intermittent map we investigate the spectral properties of a one dimensional piecewise linear intermittent map which has not only a marginal fixed point but also a singular structure suppressing injections of the orbits into neighborhoods of the marginal fixed point we explicitly derive generalized eigenvalues and eigenfunctions of the frobenius perron operator of the map for classes of observables and piecewise constant initial densities and it is found that the frobenius perron operator has two simple real eigenvalues 1 and ensuremath lambda d ensuremath 1 0 and a continuous spectrum on the real line 0 1 from these spectral properties we also found that this system exhibits a power law decay of correlations this analytical result is found to be in a good agreement with numerical simulations moreover the system can be extended to an area preserving invertible map defined on the unit square this extended system is similar to the baker transformation but does not satisfy hyperbolicity a relation between this area preserving map and a billiard system is also discussed

an analogue of the prime number theorem for closed orbits of axiom a flows for an axiom a flow restricted to a basic set we extend the zeta function to an open set containing w s h where h is the topological entropy this enables us to give an asymptotic formula for the number of closed orbits by adapting the wiener ikehara proof of the prime number theorem

q analogues of the parabose and parafermi oscillators and representations of quantum algebras deformations of the one dimensional parabose oscillators are presented they are shown to provide fock representations of the quantum sp q 2 algebra and ospq 1 2 superalgebra the representations of suq 2 are used to define the q analogues of the parafermi oscillators

microlocalization percolation and anderson localization for the magnetic schrodinger operator with a random potential abstract we study the spectral properties of the magnetic schrodinger operator with a random potential using results from microlocal analysis and percolation we show that away from the landau levels the spectrum is almost surely pure point with at least exponentially decaying eigenfunctions moreover it is shown that the rate of decay is proportional to b

simple deterministic dynamical systems with fractal diffusion coefficients we analyze a simple model of deterministic diffusion the model consists of a one dimensional array of scatterers with moving point particles the particles move from one scatterer to the next according to a piecewise linear expanding deterministic map on unit intervals the microscopic chaotic scattering process of the map can be changed by a control parameter the macroscopic diffusion coefficient for the moving particles is well defined and depends upon the control parameter we calculate the diffusion coefficent and the largest eigenmodes of the system by using markov partitions and by solving the eigenvalue problems of respective topological transition matrices for different boundary conditions we find that the largest eigenmodes of the map match the ones of the simple phenomenological diffusion equation our main result is that the diffusion coefficient exhibits a fractal structure as a function of the control parameter we provide qualitative and quantitative arguments to explain features of this fractal structure

periodic orbit expansions for classical smooth flows the authors derive a generalized selberg type zeta function for a smooth deterministic flow which relates the spectrum of an evolution operator to the periodic orbits of the flow this relation is a classical analogue of the quantum trace formulae and selberg type zeta functions

expanding direction of the period doubling operator we prove that the period doubling operator has an expanding direction at the fixed point we use the induced operator a perron frobenius type operator to study the linearization of the period doubling operator at its fixed point we then use a sequence of linear operators with finite ranks to study this induced operator the proof is constructive one can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point

on thermodynamics of rational mapsi negative spectrum we study the pressure spectrum p t of the maximal measure for arbitrary rational maps we also consider its modified version which is defined by means of the variational principle with respect to non atomic invariant measures it is shown that for negative values of t the modified spectrum has all major features of the hyperbolic case analyticity the existence of a spectral gap for the corresponding transfer operator rigidity properties etc the spectrum p t can be computed in terms of their legendre transforms are the hausdorff and the box counting dimension spectra of the maximal measure respectively this work is closely related to a paper 32 by d ruelle

quantum deformations of su 2 explicit representations of two deformations of su 2 one in the cartan basis and one in the cartesian basis are constructed the cartan deformation subsumes previously known cases while the cartesian deformation appears to be new in the latter case explicit reversible mappings in terms of representations of su 2 and a casimir for the algebra are constructed

the algebra of harmonic functions for a matrix valued transfer operator abstract we analyze matrix valued transfer operators we prove that the fixed points of transfer operators form a finite dimensional c algebra for matrix weights satisfying a low pass condition we identify the minimal projections in this algebra as correlations of scaling functions i e limits of cascade algorithms

the resonance spectrum of the cusp map in the space of analytic functions we prove that the frobenius perron operator u of the cusp map f 1 1 1 1 f x 1 2 x which is an approximation of the poincare section of the lorenz attractor has no analytic eigenfunctions corresponding to eigenvalues different from 0 and 1 we also prove that for any q 0 1 the spectrum of u in the hardy space in the disk z c z q 1 q is the union of the segment 0 1 and some finite or countably infinite set of isolated eigenvalues of finite multiplicity

landau hamiltonians with random potentials localization and the density of states we prove the existence of localized states at the edges of the bands for the two dimensional landau hamiltonian with a random potential of arbitrary disorder provided that the magnetic field is sufficiently large the corresponding eigenfunctions decay exponentially with the magnetic field and distance we also prove that the integrated density of states is lipschitz continuous away from the landau energies the proof relies on a wegner estimate for the finite area magnetic hamiltonians with random potentials and exponential decay estimates for the finitearea green s functions the proof of the decay estimates for the green s functions uses fundamental results from two dimensional bond percolation theory

preperiodic points and unlikely intersections in this article we combine complex analytic and arithmetic tools to study the preperiodic points of one dimensional complex dynamical systems we show that for any fixed a b c and any integer d 2 the set of c c for which both a and b are preperiodic for z c is infinite if and only if a b this provides an affirmative answer to a question of zannier which itself arose from questions of masser concerning simultaneous torsion sections on families of elliptic curves using similar techniques we prove that if rational functions c z have infinitely many preperiodic points in common then all of their preperiodic points coincide and in particular the maps must have the same julia set this generalizes a theorem of mimar who established the same result assuming that and are defined over q the main arithmetic ingredient in the proofs is an adelic equidistribution theorem for preperiodic points over number fields and function fields with non archimedean berkovich spaces playing an essential role

low lying eigenvalues of randomly curved quantum waveguides abstract we consider the negative dirichlet laplacian on an infinite waveguide embedded in r 2 and finite segments thereof the waveguide is a perturbation of a periodic strip in terms of a sequence of independent identically distributed random variables which influence the curvature we derive explicit lower bounds on the first eigenvalue of finite segments of the randomly curved waveguide in the small coupling i e weak disorder regime this allows us to estimate the probability of low lying eigenvalues a tool which is relevant in the context of anderson localization for random schrodinger operators

drinfel d realization of quantum affine superalgebra using the super reshetikhin semenov tian shansky method and gauss decomposition we obtain drinfel d s currents realization of the quantum affine superalgebra the hopf structure for drinfel d s currents is also given

on the distribution of zeros of a ruelle zeta function we study the limit distribution of zeros of a ruelle function for the dynamical systemz z2 c whenc is real andc 2 0 and apply the results to the correlation functions of this dynamical system

orthonormal expansions of invariant densities for expanding maps abstract we give a novel way of constructing the density function for the absolutely continuous invariant measure of piecewise expanding c markov maps this is a classical problem with one of the standard approaches being ulam s method problems in modern mathematics interscience new york 1960 of phase space discretisation our method hinges instead on the expansion of the density function with respect to an l 2 orthonormal basis and the computation of the expansion coefficients in terms of the periodic orbits of the expanding map the efficiency of the method and its extension to c k expanding maps are also discussed

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

unwrapping eigenfunctions to discover the geometry of almost invariant sets in hyperbolic maps the numerical approximation of perron frobenius operators allows efficient determination of the physical invariant measure of chaotic dynamical systems as a fixed point of the operator eigenfunctions of the perron frobenius operator corresponding to large subunit eigenvalues have been shown to describe almost invariant dynamics in one dimensional expanding maps we extend these ideas to hyperbolic maps in higher dimensions while the eigendistributions of the operator are relatively uninformative applying a new procedure called unwrapping to regularised versions of the eigendistributions clearly reveals the geometric structures associated with almost invariant dynamics this unwrapping procedure is applied to a uniformly hyperbolic map of the unit square to discover this map s dominant underlying dynamical structure and to the standard map to pinpoint clusters of period 6 orbits

multiparameter dependent solutions of the yang baxter equation a general method is developed for constructing representations of the temperley lieb algebra which in turn yield solutions of the yang baxter equation several classes of multiparameter dependent r matrices are obtained using this method and the connection of the method with quantum groups is discussed

the differential zeta function for axiom a attractors closed orbit of least period x t an important problem is to construct the meromorphic extensions for these zeta functions a detailed knowledge of the domain of s gives information on the long term behaviour of the flow for example this is a standard approach to deriving asymptotic formulae for closed orbits cf 9 for the case of geodesic flows on surfaces of constant negative curvature the zeta function is known to have a meromorphic extension to the entire complex plane unfortunately an obstruction to a general theory along these lines is the existence of examples of axiom a flows for which d s does not have a meromorphic extension to c in a recent article william parry proposed a modified definition of a zeta function for axiom a attractors he defined the differential zeta function by ns s h i e s t l where the least period x x of a closed orbit is replaced by the expansion xu t along the unstable manifold around the orbit 8 parry also defined the useful notion of a synchronized flow i e one for which the measure of maximal entropy and the srb measure coincide the following remark is taken from page 273 of his article when the flow is synchronized the zeta functions coincide this is the case for example with a geodesic flow on a surface of constant negative curvature one wonders whether synchronization has something to do with the very desirable features of the zeta function of such a geodesic flow in this article we shall give a solution to parry s question which shows that in many cases du s has a meromorphic extension to the entire complex plane our main result can be stated as follows

zeta functions and transfer operators for piecewise monotone transformations given a piecewise monotone transformationt of the interval and a piecewise continuous complex weight functiong of bounded variation we prove that the ruelle zeta function z of t g extends meromorphically to z 1 where lim g tn 1 g tg 1 n and thatz is a pole of if and only ifz 1 is an eigenvalue of the corresponding transfer operator l we do not assume that l leaves a reference measure invariant

wegner estimate and anderson localization for random magnetic fields we consider a two dimensional magnetic schrodinger operator with a spatially stationary random magnetic field we assume that the magnetic field has a positive lower bound and that it has fourier modes on arbitrarily short scales we prove the wegner estimate at arbitrary energy i e we show that the averaged density of states is finite throughout the whole spectrum we also prove anderson localization at the bottom of the spectrum

dynamical determinants via dynamical conjugacies for postcritically finite polynomials we give an analogue of levin sodin yuditskii s study of the dynamical ruelle determinants of hyperbolic rational maps in the case of subhyperbolic quadratic polynomials our main tool is to reduce to an expanding situation we do so by applying a dynamical change of coordinates on the domains of a markov partition constructed from the landing ray at the postcritical repelling orbit we express the dynamical determinants d beta z exp sum k geqslant 1 tfrac z k k sum w in text fix f k tfrac 1 f c k w beta tfrac 1 1 tfrac 1 f c k w beta in mathbb z as the product of an entire determinant with an explicit expression involving the postcritical repelling orbit thus explaining the poles in d z

q analogues of clifford and weyl algebras spinor and oscillator representations of quantum enveloping algebras we introduceq analogues of clifford and weyl algebras using these we construct spinor and oscillator representations of quantum enveloping algebras of typean 1 bn cn dn andan 1 1 also we discuss the irreducibility and the unitarity of these representations

new braid group representations of the b2 b3 and b4 types their associated link polynomials and quantum r matrices new braid group representations of the b2 b3 and b4 types are obtained by solving the defining relations of artin s braid group bn directly they give their associated link polynomials the authors discuss a procedure baxterization which allows them to construct their corresponding quantum r matrices

non equilibrium critical behaviour on fractal lattices a simple one component surface reaction model exhibiting a critical point has been studied on deterministic fractal lattices with fractal dimension 1 df 2 results from time dependent simulations show a change in the value of the critical exponents with df two of the exponents interpolate very nicely between the values in one and two dimensions

spectral characterization of anomalous diffusion of a periodic piecewise linear intermittent map abstract for a piecewise linear version of the periodic map with anomalous diffusion the evolution of statistical averages of a class of observables with respect to piecewise constant initial densities is investigated and generalized eigenfunctions of the frobenius perron fp operator are explicitly derived the evolution of the averages is controlled by real eigenvalues as well as continuous spectra terminating at the unit circle appropriate scaling limits are shown to give a normal diffusion if the reduced map is in the stationary regime with normal fluctuations a levy flight if the reduced map is in the stationary regime with levy type fluctuations and a transport of ballistic type if the reduced map is in the non stationary regime

linear response and consequences for differentiability of statistical quantities and multifractal analysis in this article we initially prove the differentiability of the topological pressure equilibrium states and their densities with respect to smooth expanding dynamical systems and any smooth potential this is done by proving the regularity of the dominant eigenvalue of the transfer operator with respect to dynamics and potential from that we obtain strong consequences on the regularity of the dynamical system statistical properties that apply in more general contexts indeed we prove that the average and variance obtained from the central limit theorem vary c r 1 with respect to the c r expanding dynamics and c r potential and also there is a large deviations principle exhibiting a c r 1 rate with respect to the dynamics and the potential an application for multifractal analysis is given we also obtained asymptotic formulas for the derivatives of the topological pressure and other thermodynamical quantities

transfer operators acting on zygmund functions we obtain a formula for the essential spectral radius pess of transfer type operators associated with families of c1 6 diffeomorphisms of the line and zygmund or h6lder weights acting on banach spaces of zygmund respectively h6lder functions in the uniformly contracting case the essential spectral radius is strictly smaller than the spectral radius when the weights are positive

hilbert spaces built on a similarity and on dynamical renormalization we develop a hilbert space framework for a number of general multiscale problems from dynamics the aim is to identify a spectral theory for a class of systems based on iterations of a non invertible endomorphism we are motivated by the more familiar approach to wavelet theory which starts with the two to one endomorphism r z z2 in the one torus t a wavelet filter and an associated transfer operator this leads to a scaling function and a corresponding closed subspace v0 in the hilbert space l2 r using the dyadic scaling on the line r one has a nested family of closed subspaces vn n z with trivial intersection and with dense union in l2 r more generally we achieve the same outcome but in different hilbert spaces for a class of non linear problems in fact we see that the geometry of scales of subspaces in hilbert space is ubiquitous in the analysis of multiscale problems e g martingales complex iteration dynamical systems graph iterated function systems of affine type and subshifts in

optimal volume wegner estimate for random magnetic laplacians on z2 we consider a two dimensional magnetic schrodinger operator on a square lattice with a spatially stationary random magnetic field we prove a wegner estimate with optimal volume dependence the wegner estimate holds around the spectral edges and it implies holder continuity of the integrated density of states in this region the proof is based on the wegner estimate obtained in erd s and hasler wegner estimate for random magnetic laplacians on z2 ann henri poincare 12 1719 1731 2012 10 1007 s00023 012 0177 9

a diamagnetic inequality for semigroup differences the diamagnetic inequality for the magnetic schrodinger semigroup is extended to the difference of the semigroups of magnetic schrodinger operators with neumann and dirichlet boundary conditions on arbitrary open domains and rather general magnetic vector potentials a and potentials v in particular this bound renders moot all the technical issues in the recent proofs of the independence of the boundary conditions for the integrated density of states for magnetic schrodinger operators independence of the boundary conditions for the free case that is for vanishing potentials and vector potentials immediately implies independence of the boundary conditions of the integrated density of states for a large class of magnetic schrodinger operators

graded solutions of the yang baxter relation and link polynomials from a family of graded solvable models the authors derive representations of the braid group associated with the lie superalgebra gl m mod n and give explicitly a general form of the markov traces on the representations the braid operators thus obtained satisfy the hecke algebra the authors construct composite solvable models and obtain link polynomials from the braid operators for the composite models

consistent multiparameter quantisation of gl n the author describes a manifold of quantum group structures on the vector space of the universal enveloping algebra of gl n and on its dual the space of polynomials in n2 variables the dimension of the manifold is n2 n 2 2

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

diffusion effusion and chaotic scattering an exactly solvable liouvillian dynamics we study diffusion and chaotic scattering in a chain of baker maps coupled together which forms an area preserving mapping of an infinitely extended strip onto itself this exactly solvable mapping sustains chaotic behaviors and diffusion processes the relationship between the diffusion coefficient the lyapunov exponent and the entropy per unit time is derived the long lived classical resonances of the liouville evolution operator are proved to converge toward the eigenvalues of the phenomenological diffusion equation in this sense there is a quasi isomorphism between the resonance spectrum of the liouville evolution and the eigenvalue spectrum of the phenomenological diffusion equation furthermore we show that a fractal repeller is associated to each non equilibrium state in the isolated and finite multibaker chain the nonequilibrium states are all unstable with respect to the equilibrium validating a weak form of the second principle of thermodynamics for the present dynamical system consequences of nonequilibrium fractals on classical measurements are discussed we then describe the open multibaker chain as a scattering system fractal properties of chaotic scattering are here shown to be related to diffusion in the chain

anderson localization for random schrodinger operators with long range interactions we prove pure point spectrum with exponentially decaying eigenfunctions at all band edges for schrodinger operators with a periodic potential plus a random potential of the form v w x q i w f x i where f decays at infinity like x m for m 4d resp m 3d depending on the regularity of f the random variables q i are supposed to be independent and identically distributed we assume that their distribution has a bounded density of compact support

large deviations for non uniformly expanding maps we obtain large deviation bounds for non uniformly expanding maps with non flat singularities or criticalities and for partially hyperbolic non uniformly expanding attracting sets that is given a continuous function we consider its space average with respect to a physical measure and compare this with the time averages along orbits of the map showing that the lebesgue measure of the set of points whose time averages stay away from the space average tends to zero exponentially fast with the number of iterates involved as easy by products we deduce escape rates from subsets of the basins of physical measures for these types of maps the rates of decay are naturally related to the metric entropy and pressure function of the system with respect to a family of equilibrium states

from ruelle s transfer operator to the schro dinger operator abstract we show that ruelle s transfer operator defined for an appropriate class of strongly chaotic mappings reduces to the schrodinger operator with imaginary time in an appropriate scaling limit we also derive the equations for the next order corrections to the schrodinger behavior if the scaling limit is not completely performed in the simplest case the first order corrections turns out to obey a schrodinger equation with a source term this means a possible chaotic dynamics underlying the feynman kac formula on a very small time scale e g the planc scale leads generalized quantum mechanics with an additional source term

crossed products by finite index endomorphisms and kms states abstract given a unital c algebra a an injective endomorphism a a preserving the unit and a conditional expectation e from a to the range of we consider the crossed product of a by relative to the transfer operator l 1e when e is of index finite type we show that there exists a conditional expectation g from the crossed product to a which is unique under certain hypothesis we define a gauge action on the crossed product algebra in terms of a central positive element h and study its kms states the main result is if h 1 and e ab e ba for all a b a e g when a is commutative then the kms states are precisely those of the form g where is a trace on a satisfying the identity a l h ind e a where ind e is the jones kosaki watatani index of e

correlation spectrum of quenched and annealed equilibrium states for random expanding maps we show that the integrated transfer operators for positively weighted independent identically distributed smooth expanding systems give rise to annealed equilibrium states for a new variational principle the unique annealed equilibrium state coincides with the unique annealed gibbs state using work of ruelle 1990 and fried 1995 on generalised fredholm determinants for transfer operators we prove that the discrete spectrum of the transfer operators coincides with the correlation spectrum of these invariant measures yielding exponential decay of correlations and with the poles of an annealed zeta function defined also for complex weights a modified integrated transfer operator is introduced which describes the relativised quenched states studied e g by kifer 1992 and conditions including srb ensuring coincidence of quenched and annealed states are given for small random perturbations we obtain stability results on the quenched and annealed measures and spectra by applying perturbative results of young and the author 1993

entropy volume growth and srb measures for banach space mappings we consider c 2 frechet differentiable mappings of banach spaces leaving invariant compactly supported borel probability measures and study the relation between entropy and volume growth for a natural notion of volume defined on finite dimensional subspaces srb measures are characterized as exactly those measures for which entropy is equal to volume growth on unstable manifolds equivalently the sum of positive lyapunov exponents of the map in addition to numerous difficulties incurred by our infinite dimensional setting a crucial aspect to the proof is the technical point that the volume elements induced on unstable manifolds are regular enough to permit distortion control of iterated determinant functions the results here generalize previously known results for diffeomorphisms of finite dimensional riemannian manifolds and are applicable to dynamical systems defined by large classes of dissipative parabolic pdes

one dimensional gibbs states and axiom a diffeomorphisms on etudie la mecanique statistique en equilibre de systemes reticules classiques a une dimension a interactions a decroissance exponentielle pour de tels systemes les transformees de fourier des fonctions de correlation de paires sont meromorphes dans une bande et les residus des poles peuvent s exprimer en termes de distributions de gibbs on applique ces resultats aux diffeomorphismes de l axiome a de smale

the thermodynamic formalism for expanding maps letf x x be an expanding map of a compact space small distances are increased by a factor 1 a generating function z is defined which countsf periodic points with a weight one can express in terms of nonstandard fredholm determinants of certain transfer operators which can be studied by methods borrowed from statistical mechanics in this paper we review the spectral properties of the transfer operators and the corresponding analytic properties of z gibbs distributions and applications to julia sets are also discussed some new results are proved and some natural conjectures are proposed

correlation analysis of dynamical chaos abstract we study correlation and spectral properties of chaotic self sustained oscillations of different types it is shown that some classical models of stochastic processes can be used to describe behavior of autocorrelation functions of chaos the influence of noise on chaotic systems is also considered

r torsion and zeta functions for analytic anosov flows on 3 manifolds we improve previous results relating r torsion for an acyclic representation of the fundamental group with a special value of the torsion zeta function of an analytic anosov flow on a 3 manifold by using the new techniques of rugh and fried we get rid of the unpleasent assumptions about the regularity of the invariant foliations

physical measures for multivalued inverse iterates near hyperbolic repellors we study new invariant probability measures describing the distribution of mul tivalued inverse iterates i e of different local inverse iterates for a non invertible smooth function f which is hyperbolic but not necessarily expanding on a repellor the methods for the higher dimensional non expanding and non invertible case are different than the ones for diffeomorphisms due to the lack of a nice unstable foliation local unstable manifolds depend on prehistories and may intersect each other both inand outside and the fact that markov partitions may not exist on we obtain that for lebesgue almost all points z in a neighbourhood v of the normalized averages of dirac measures on the consecu tive preimage sets of z converge weakly to an equilibrium measure on this implies that is a physical measure for the local inverse iterates of f it turns out that is an inverse srb measure in the sense that it is the only invariant measure satisfying a pesin type formula for the negative lyapunov exponents also we show that has absolutely continuous conditional measures on local stable manifolds by using the above convergence of measures we prove then that f b b cannot be one sided bernoulli although it is an exact endomorphism of lebesgue spaces several classes of ex amples of hyperbolic non invertible and non expanding repellors with their inverse srb measures are given in the end

sharp estimates in ruelle theorems for matrix transfer operators a matrix coefficient transfer operator on the space of sections of an m dimensional vector bundle over n dimensional compact manifold is considered the spectral radius of is estimated by para and the essential spectral radius by here is the set of ergodic f invariant measures and for is the measure theoretic entropy of f is the largest lyapunov exponent of the cocycle over f generated by and is the smallest lyapunov exponent of the differential of f

the integrated density of states for some random operators with nonsign definite potentials we study the integrated density of states of random anderson type additive and multiplicative perturbations of deterministic background operators for which the single site potential does not have a fixed sign our main result states that under a suitable assumption on the regularity of the random variables the integrated density of states of such random operators is locally holder continuous at energies below the bottom of the essential spectrum of the background operator for any nonzero disorder and at energies in the unperturbed spectral gaps provided the randomness is sufficiently small the result is based on a proof of a wegner estimate with the correct volume dependence the proof relies upon the lp theory of the spectral shift function for p 1 comm math phys 218 2001 113 130 and the vector field methods of klopp comm math phys 167 1995 553 569 we discuss the application of this result to schrodinger operators with random magnetic fields and to band edge localization

action principle in nonequilibrium statistical dynamics we introduce a variational method for approximating distribution functions of dynamics with a liouville operator l in terms of a nonequilibriumaction functional for two independent left and right trial states the method is valid for deterministic or stochastic markov dynamics and for stationary or time dependent distributions a practical rayleigh ritz procedure is advanced whose inputs are a finitely parametrized ansatz for the trial states leading to a parametric action for their evolution the euler lagrange equations of the action principle are hamiltonian in form generally noncanonical this permits a simple identification of fixed points as critical points of the parametric hamiltonian we also establish a variational principle for low order statistics such as mean values and correlation functions by means of the least effective action the latter is a functional of the given variable which is positive and convex as a consequence of h older realizability inequalities its value measures the cost for a fluctuation from the average to occur and in a weak noise limit it reduces to the onsager machlup action in general the effective action is shown to arise from the nonequilibrium action functional by a constrained variation this result provides a rayleigh ritz scheme for calculating just the desired low order statistics with internal consistency checks less demanding than for the full distribution textcopyright 1996 the american physical society

spectrum of the ruelle operator and exponential decay of correlations for open billiard flows the paper deals with the billiard flow in the exterior of several strictly convex disjoint domains in the plane with smooth boundaries satisfying an additional visibility condition using a modification of the technique of dolgopyat we get spectral estimates for the ruelle operator related to a markov family for the nonwandering trapping set of the flow similar to those of dolgopyat in the case of transitive anosov flows on compact manifolds with smooth jointly non integrable horocycle foliations as a consequence we get exponential decay of correlation for holder continuous potentials on the nonwandering set combining the spectral estimate for the ruelle operator with an argument of pollicott and sharp we also derive the existence of a meromorphic continuation of the dynamical zeta function of the billiard flow to a half plane re s h t e where h t is the topological entropy of the billiard flow and an asymptotic formula with an error term for the number of closed orbits of least period 0

a deformed quantum su 2 superalgebra a quantum q deformation of the su 2 superalgebra is obtained which contains as an even part the quantum suq 2 algebra the finite dimensional irreducible representations of the deformed quantum superalgebra are found

a two parametric quantization of sl 2 a two parametric solution of the constant yang baxter equation is used for the construction of structures corresponding to quantized groups the corresponding hopf algebra is generated by five generators

a proof of thurston s topological characterization of rational functions the criterion proved in this paper was stated by thurston in november 1982 thurston lectured on its proof on several occasions notably at the nsf summer conference in duluth 1983 where one of the authors jhh was present using the notes of various attendants at these lectures we have reconstructed a proof that we have made as precise as we could since this required a certain amount of work on our part we thought it might be of some use to present this proof to the reader we thank dennis sullivan for useful conversations and fritz yon haesseler and especially ben wittner for help with the writing and valuable suggestions after the first version was written clifford earle pointed out that better estimates than what we had were to be found in b

meromorphic extensions of generalised zeta functions in this paper we give a full description of the spectrum of the ruelle perron frobenius operator acting on the banach space of holder continuous functions on a subshift of finite type theorem 1 these results are then used to extend the meromorphic domain of generalised zeta functions theorem 2 the most important application of these results is to the domain of the smale zeta function for axiom a flows theorem 3 in the course of this paper we settle questions raised by ruelle and sunada

lifshitz tail for schrodinger operator with random magnetic field we study the behavior of the density states at the lower edge of the spectrum for schrodinger operators with random magnetic fields we use a new estimate on magnetic schrodinger operators which is similar to the avron herbst simon estimate but the bound is always nonnegative

two parametric solution to graded yang baxter equation and two parametric uuvgl 1 mod 1 algebra as a hopf algebra the authors discuss a two parametric solution of the graded yang baxter equation ybe and perform the yang baxterization to obtain the solution to the quantum ybe in the formalism developed previously they present the two parametric quantized superalgebra uuvgl 1 mod 1 and prove that this algebra is a hopf algebra with the hopf operations explicitly provided