gibb s functionals on subshifts we consider functionals on one dimensional subshifts which have prescribed randon nikodym derivative under transportation by conjugating homeomorphisms and investigate their relation to ruelle s transfer operator in particular we show that two sided functionals essentially are products of a functional which are supported on stable and unstable leaves we also prove the meromorphicity of the fourier transform of correlation functions for axioma follows in a more general setting

area expanding fancyscript c 1 alpha suspension semiflows we study a large class of suspension semiflows which contains the lorenz semiflow this is a class with low regularity the return map is piecewise fancyscript c 1 alpha and the return time is piecewise fancyscript c alpha and where the return time is unbounded we establish the functional analytic framework which is typically employed to study rates of mixing the laplace transform of the correlation function is shown to admit a meromorphic extension to a strip about the imaginary axis as part of this argument we give a new result of independent interest concerning the quasi compactness of weighted transfer operators for piecewise fancyscript c 1 alpha expanding interval maps

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

succinct representation of regular languages by boolean automata ii abstract boolean automata are a generalization of finite automata in the sense that the next state the result of the transition function given a state and a letter is not just a single state deterministic automata or a set of states nondeterministic automata but a boolean function of the states boolean automata accept precisely the regular languages also they correspond in a natural way to certain language equation involving complementation as well as to sequential networks in a previous note we showed that for every n 1 there exists a boolean automaton b n with n states such that the smallest deterministic automaton for the same language has 2 2 n states in the present note we will show a precisely attainable lower bound on the succinctness of representing regular languages by boolean automata namely we will show that for every n 1 there exists a reduced automaton d n with n states such that the smallest boolean automaton accepting the same language has also n states

fick s law and fractality of nonequilibrium stationary states in a reversible multibaker map nonequilibrium stationary states are studied for a multibaker map a simple reversible chaotic dynamical system the probabilistic description is extended by representing a dynamical state in terms of a measure instead of a density function the equation of motion for the cumulative function of this measure is derived and stationary solutions are constructed with the aid of derham type functional equations to select the physical states the time evolution of the distribution under a fixed boundary condition is investigated for an open multibaker chain of scattering type this system corresponds to a diffusive flow experiment through a slab of material for long times any initial distribution approaches the stationary one obeying fick s law at stationarity the intracell distribution is singular in the stable direction and expressed by the takagi function which is continuous but has no finite derivatives the result suggests that singular measures play an important role in the dynamical description of non equilibrium states

spectral decomposition of chebyshev maps abstract we construct a spectral decomposition of the frobenius perron operator for the chebyshev polynomials of the first kind we defined a suitable dual pair or rigged hilbert space which provides mathematical meaning to the spectral decomposition the spectra of the even chebyshev maps do not contain the odd powers of 1 m and the odd eigenfunctions are in the null space of the frobenius perron operator moreover the odd chebyshev maps have degenerate spectra without jordan blocks the eigenvalues in the decompositions are the resonances of power spectrum and have magnitudes less than one as in the case of the family of tent maps

on contact anosov flows exponential decay of correlations for co 4 contact anosov flows is established this implies in particular exponential decay of correlations for all smooth geodesic flows in strictly negative curvature

dynamical l functions and homology of closed orbits soit t un flot d anosov transitif lisse sur une variete fermee x on suppose que t a la propriete de melange faible on classe les orbites fermees du flot a l aide des classes d homologie et on les denombre

hypermeander of spirals local bifurcations and statistical properties in both experimental studies and numerical simulations of waves in excitable media rigidly rotating spiral waves are observed to undergo transitions to complicated spatial dynamics with long term brownian like motion of the spiral tip this phenomenon is known as hypermeander in this paper we review a number of recent results on dynamics with noncompact group symmetries and make the case that hypermeander may occur at a codimension two bifurcation from a rigidly rotating spiral wave our predictions are based on center bundle reduction sandstede scheel and wulff and on central limit theorems and invariance principles for group extensions of hyperbolic dynamical systems these predictions are confirmed by numerical simulations of the center bundle equations

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

on classes of tractable unrestricted regular expressions abstract it is known that the equivalence problem for the class ure of unrestricted regular expressions is not elementary recursive this implies that constructing the corresponding reduced automaton is not tractable for some expressions in ure we say these expressions are of intractable complexity however the closure of boolean automata under efficient complementation suggests that there are large nontrivial sub classes of ure with arbitrary level of complementation and tractable complexity in this paper we introduce the class c l corresponding in a natural way to boolean automata we also consider other related subclasses of ure all these classes have tractable complexity these results suggest a revised explanation why certain unrestricted regular expressions are so hard

algebraic decay in hierarchical graphs we study the algebraic decay of the survival probability in open hierarchical graphs we present a model of a persistent random walk on a hierarchical graph and study the spectral properties of the frobenius perron operator using a perturbative scheme we derive the exponent of the classical algebraic decay in terms of two parameters of the model one parameter defines the geometrical relation between the length scales on the graph and the other relates to the probabilities for the random walker to go from one level of the hierarchy to another the scattering resonances of the corresponding hierarchical quantum graphs are also studied the width distribution shows the scaling behavior p 1

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

central limit theorems and invariance principles for time one maps of hyperbolic flows we give a general method for deducing statistical limit laws in situations where rapid decay of correlations has been established as an application of this method we obtain new results for time one maps of hyperbolic flows

rates of mixing for non markov infinite measure semiflows we develop an abstract framework for obtaining optimal rates of mixing and higher order asymptotics for infinite measure semiflows previously such results were restricted to the situation where there is a first return poincare map that is uniformly expanding and markov as illustrations of the method we consider semiflows over non markov pomeau manneville intermittent maps with infinite measure and we also obtain mixing rates for semiflows over collet eckmann maps with nonintegrable roof function

stability of mixing and rapid mixing for hyperbolic flows we obtain general results on the stability of mixing and rapid mixing superpolynomial decay of correlations for hyperbolic flows amongst c r axiom a flows r 2 we show that there is a c 2 open c r dense set of flows for which each nontrivial hyperbolic basic set is rapid mixing this is the first general result on the stability of rapid mixing or even mixing for axiom a

classical nonlinear response of a chaotic system i collective resonances we develop a general semiquantitative picture of nonlinear classical response in strongly chaotic systems in contrast to behavior in integrable or almost integrable systems the nonlinear classical response in chaotic systems vanishes at long times the exponential decay of the response functions in the case of strong chaos is attributed to both exponentially decaying and growing elements in the stability matrices we calculate the linear and second order response in one of the simplest chaotic systems free classical motion on a compact surface of constant negative curvature the response reveals certain features of collective resonances which do not correspond to any periodic classical trajectories we demonstrate the relevance of the model for the interpretation of spectroscopic experiments

existence of a nontrivial solution for the p q laplacian in rn without the ambrosetti rabinowitz condition abstract in this paper we prove the existence of at least one nonnegative nontrivial weak solution in d 1 p r n d 1 q r n for the equation p u q u a x u p 2 u b x u q 2 u f x u x r n where 1 q p q n q n q p n m u div u m 2 u is the m laplacian operator the coefficients a and b are continuous coercive and positive functions and the nonlinearity f is a caratheodory function satisfying some hypotheses which do not include the ambrosetti rabinowitz condition

lyapunov exponents and anomalous diffusion of a lorentz gas with infinite horizon using approximate zeta functions we compute the lyapunov exponent the generalized lyapunov exponents and the diffusion constant for a lorentz gas on a square lattice thus having infinite horizon approximate zeta functions written in terms of probabilities rather than periodic orbits are used in order to avoid the convergence problems of cycle expansions the emphasis is on the relation between the analytic structure of the zeta function where a branch cut plays an important role and the asymptotic dynamics of the system the lyapunov exponent for the corresponding map agrees with the conjectured limit map 2 log r c o r and we derive an approximate value for the constantc in good agreement with numerical simulations we also find a diverging diffusion constantd t logt and a phase transition for the generalized lyapunov exponents

robust exponential decay of correlations for singular flows we construct open sets of ck k 2 vector fields with singularities that have robust exponential decay of correlations with respect to the unique physical measure in particular we prove that the geometric lorenz attractor has exponential decay of correlations with respect to the unique physical measure

decay of correlations for slowly mixing flows we show that polynomial decay of correlations is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of a class of nonuniformly hyperbolic dieomorphisms for which young proved polynomial decay of correlations roughly speaking in situations where the decay rate o 1 n has previously been proved for dieomorphisms we establish the decay rate o 1 t for flows applications include certain classes of semidispersing billiards as well as dispersing billiards with vanishing curvature in addition we obtain results for suspension flows with unbounded roof functions in particular the classical planar lorentz flow with a doubly periodic array of circular scatterers has decay rate 1 t as anticipated by physicists

on generalized language equations a system of generalized language equations over an alphabet a is a set of n equations in n variables xi gi x1 xn i 1 n where the gi are functions from p a n into p a i 1 n p a denoting the set of all languages over a furthermore the gi are expressible in terms of set operations concatenations and stars which involve the variable xi as well as certain mixed languages in this note we investigate existence and uniqueness of solutions of a certain subclass of generalized language equations furthermore we show that a solution is regular if all fixed languages are regular

locating resonances for axiom a dynamical systems for a class of differentiable dynamical systems called axiom a systems it has been shown by pollicott and the author that correlation functions have fourier transforms which are meromorphic in a strip the poles or resonances are however not easy to locate this note reviews the results which are known and discusses a simple model where the position of resonances can be estimated the effect of noise is also discussed

deterministic homogenization for fast slow systems with chaotic noise consider a fast slow system of ordinary differential equations of the form x a x y e 1b x y y e 2g y where it is assumed that b averages to zero under the fast flow generated by g we give conditions under which solutions x to the slow equations converge weakly to an it o diffusion x as e 0 the drift and diffusion coefficients of the limiting stochastic differential equation satisfied by x are given explicitly our theory applies when the fast flow is anosov or axiom a as well as to a large class of nonuniformly hyperbolic fast flows including the one defined by the well known lorenz equations and our main results do not require any mixing assumptions on the fast flow

stable resonances and signal propagation in a chaotic network of coupled units we apply the linear response theory developed by ruelle j stat phys 95 393 1999 to analyze how a periodic signal of weak amplitude superimposed upon a chaotic background is transmitted in a network of nonlinearly interacting units we numerically compute the complex susceptibility and show the existence of specific poles stable resonances corresponding to the response to perturbations transverse to the attractor contrary to the poles of correlation functions they depend on the pair emitting receiving units this dynamic differentiation induced by nonlinearities exhibits the different ability that units have to transmit a signal in this network

locating pollicott ruelle resonances in chaotic dynamical systems a class of numerical schemes a class of numerical methods to determine pollicott ruelle resonances in chaotic dynamical systems is proposed this is achieved by relating some existing procedures that make use of the pade approximants and interpolating exponentials to both the memory function techniques used in the theory of relaxation and the filter diagonalization method used in the harmonic inversion of time correlation functions this relationship leads to a theoretical framework in which all these methods become equivalent and which allows for new and improved numerical schemes

anosov flows and dynamical zeta functions we study the ruelle and selberg zeta functions for c r anosov ows r 2 on a compact smooth manifold we prove several results the most remarkable being a for c 1 ows the zeta function is meromorphic on the entire complex plane b for contact ows satisfying a bunching condition e g geodesic ows on manifolds of negative curvature better than 1 9 pinched the zeta function has a pole at the topological entropy and is analytic in a strip to its left c under the same hypotheses as in b we obtain sharp results on the number of periodic orbits our arguments are based on the study of the spectral properties of a transfer operator acting on suitable banach spaces of anisotropic currents

on solving star equations abstract a star equation in x is a language equation x f x where f is a function mapping languages to languages which can be expressed using the operations union concatenation from the left by a constant language and unrestricted star we show existence of solutions as well as uniqueness under certain additional conditions also the equation is studied whether a star equation has a regular solution if all its constant languages are regular

transport in polygonal billiards we present in this work a numerical study of the dynamics of ensembles of point particles within a polygonal billiard chain this billiard is a system with no exponential instability our numerical results suggest that some members of the family exhibit normal diffusive behavior while others present anomalous diffusion our conclusions are drawn from the numerical evaluation of the mean square displacement the velocity autocorrelation function and its spectral analysis furthermore we analyze the properties of the incoherent scattering function the super burnett coefficient seems to be ill defined in all systems the multifractal analysis of the spectrum of the velocity autocorrelation functions is also reported finally we study the heat conduction in our polygonal chain

symbolic dynamics for geodesic floes the subject of symbolic dynamics is of central importance in the modern theory of dynamical systems the use of symbolic sequences to study dynamical properties of geodesies originates in the work of koebe 21 22 and morse 24 25 and is already foreshadowed in hadamard 15 and jordan 19 the method of koebe and of morse is to code a geodesic on a surface m of negative curvature by recording the order in which it traverses a given set of labelled curves on m the treatment of morse allows variable curvature but assumes at least two boundary components whereas koebe assumes constant curvature but allows infinite connectivity and nonorientability and treats the more difficult case of a closed surface this last case is handled by recording crossings of a fixed pants decomposition anticipating the thurston parameterization of simple curves as described in 11 both koebe and morse used their codings to demonstrate the existence of countably many closed geodesies and of everywhere dense transitive geodesies morse further constructed the first nonsynthetic example of a recurrent nonperiodic discontinuous motion in modern terminology a minimal nowhere dense set later 26 morse treated the special closed surfaces of genus g associated to tesselations of the disc d by regular 4 gons each edge of a region in such a tesselation may be labelled by the isometry which glues it to another side of the same region in forming the quotient sin face m the set of isometries appearing as labels generate tti m thus any geodesic is coded as a doubly infinite sequence of generators of iri m the same method applies quite generally to tesselations associated to any fuchsian group we call the sequences thus obtained cutting sequences and refer to generating sets of this kind as geometric the difficulty of course is to determine precisely the class of sequences which occur for a surface with boundary one obtains exactly reduced sequences in the generators in generalthe problem4s gompligated1 henge the difficulties encountered in 26 see also theorem 3 below there is another method of coding geodesies using certain boundary expansions for points at infinity in the universal cover of m for the modular surface h sl 2 2 the appropriate expansions are continued fractions and for the symmetrical genus g surfaces above they are the nielsen boundary expansions of 27

resonances in chaotic dynamics we present a discussion and some numerical results on the actual possibility of making accessible by numerical techniques the complex singularities of the power spectrum resonances for a chaotic signal henon s transformation is investigated in detail showing that the position of the leading resonance in the complex frequency plane determines the kind of mixing rate in the time evolution

decay of correlations for maps with uniformly contracting fibers and logarithm law for singular hyperbolic attractors we consider maps preserving a foliation which is uniformly contracting and a one dimensional associated quotient map having exponential convergence to equilibrium iterates of lebesgue measure converge exponentially fast to physical measure we prove that these maps have exponential decay of correlations over a large class of observables we use this result to deduce exponential decay of correlations for suitable poincare maps of a large class of singular hyperbolic flows from this we deduce a logarithm law for these flows

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

perturbations of geodesic flows on surface of constant negative curvature and their mixing properties we consider one parameter analytic hamiltonian perturbations of the geodesic flows on surfaces of constant negative curvature we find two different necessary and sufficient conditions for the canonical equivalence of the perturbed flows and the non perturbed ones one condition says that the hamilton jacobi equation introduced in this work for the conjugation problem should admit a solution as a formal power series not necessarily convergent in the perturbation parameter the alternative condition is based on the identification of a complete set of invariants for the canonical conjugation problem the relation with the similar problems arising in the kam theory of the perturbations of quasi periodic hamiltonian motions is briefly discussed as a byproduct of our analysis we obtain some results on the livscic guillemin kazhdan equation and on the fourier series for the sl 2 group we also prove that the analytic functions on the phase space for the geodesic flow of unit speed have a mixing property with respect to the geodesic flow and to the invariant volume measure which is exponential with a universal exponent independent on the particular function equal to the curvature of the surface divided by 2 this result is contrasted with the slow mixing rates that the same functions show under the horocyclic flow in this case we find that the decay rate is the inverse of the time up to logarithms

a huygens principle for diffusion and anomalous diffusion in spatially extended systems we present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media for anisotropic systems strong chaos leads to diffusive behavior brownian motion with drift and weak chaos leads to superdiffusive behavior levy processes with drift for isotropic systems the drift term vanishes and strong chaos again leads to brownian motion we establish the existence of a nonlinear huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions

discordance between quantum and classical correlation moments for chaotic systems for systems whose classical orbits are chaotic a set of quantum expectation values qr is constructed which vanish for all unlike their classical counterparts cr which are finite this behaviour is not paradoxical because qr and cr are moments of time correlation functions which are dominated by the long time limit where the quantum and classical evolution disagree

leading pollicott ruelle resonances for chaotic area preserving maps recent investigations in nonlinear sciences show that not only hyperbolic but also mixed dynamical systems may exhibit exponential relaxation in the chaotic regime the relaxation rates which lead the decay of probability distributions and correlation functions are related to the classical evolution resolvent perron frobenius operator pole logarithm the so called pollicott ruelle resonances in this brief report the leading pollicott ruelle resonances are calculated analytically for a general class of area preserving maps besides the leading resonances related to the diffusive modes of momentum dynamics slow rate we also calculate the leading faster rate related to the angular correlations the analytical results are compared to the existing results in the literature

linear response susceptibility and resonances in chaotic toy models abstract we consider simple examples illustrating some new features of the linear response theory developed by ruelle for dissipative and chaotic systems d ruelle smooth dynamics and new theoretical ideas in nonequilibrium statistical mechanics j stat phys 95 1999 393 468 in this theory the concepts of linear response susceptibility and resonance which are familiar to physicists have been revisited due to the dynamical contraction of the whole phase space onto attractors in particular the standard framework of the fluctuation dissipation theorem breaks down and new resonances can show up outside the power spectrum in previous papers we proposed and used new numerical methods to demonstrate the presence of the new resonances predicted by ruelle in a model of chaotic neural networks in this article we deal with simpler models which can be worked out analytically in order to gain more insights into the genesis of the stable resonances and their consequences on the linear response of the system we consider a class of two dimensional time discrete maps describing simple rotator models with a contracting radial dynamics onto the unit circle and a chaotic angular dynamics t 1 2 t mod 2 a generalisation of this system to a network of interconnected rotators is also analysed and related with our previous studies b cessac j a sepulchre stable resonances and signal propagation in a chaotic network of coupled units phys rev e 70 2004 056111 b cessac j a sepulchre transmitting a signal by amplitude modulation in a chaotic network chaos 16 2006 013104 1 13104 12 these models permit us to classify the different types of resonances in the susceptibility and to discuss in particular the relation between the relaxation time of the system to equilibrium with the mixing time given by the decay of the correlation functions also it enables one to propose some general mechanisms responsible for the creation of stable resonances with arbitrary frequencies widths and dependence on the pair of perturbed observed variables

resonances in a chaotic attractor crisis of the lorenz flow local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions lyapunov exponents and their associated covariant lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems however it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle this behavior is investigated here for a boundary crisis in the lorenz flow for which neither the lyapunov exponents nor the covariant lyapunov vectors provide a criterion for the crisis instead the convergence of the time evolution of probability densities to the invariant measure governed by the semigroup of transfer operators is expected to slow down at the approach of the crisis such convergence is described by the eigenvalues of the generator of this semigroup which can be divided into two families referred to as the stable and unstable ruelle pollicott resonances respectively the former describes the convergence of densities to the attractor or escape from a repeller and is estimated from many short time series sampling the state space the latter is responsible for the decay of correlations or mixing and can be estimated from a long times series invoking ergodicity it is found numerically for the lorenz flow that the stable resonances do approach the imaginary axis during the crisis as is indicative of the loss of global stability of the attractor on the other hand the unstable resonances and a fortiori the decay of correlations do not flag the proximity of the crisis thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises

markov approximations and decay of correlations for anosov flows we develop markov approximations for very general suspension flows based on this we obtain a stretched exponential bound on time correlation functions for 3 d anosov flows that verify uniform nonintegrability of foliations these include contact anosov flows and geodesic flows on compact surfaces of variable negative curvature our bound on correlations is stable under small smooth perturbations

spectral properties of a piecewise linear intermittent map for a piecewise linear intermittent map 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 operator p are explicitly derived the evolution of the averages are shown to be a superposition of the contributions from two simple eigenvalues 1 and d 1 0 and a continuous spectrum on the unit interval 0 1 of p power law decay of correlations are controlled by the continuous spectrum also the non normalizable invariant measure in the non stationary regime is shown to determine the strength of the power law decay

classical nonlinear response of a chaotic system ii langevin dynamics and spectral decomposition the spectrum of a strongly chaotic system consists of discrete complex ruelle pollicott rp resonances we interpret the rp resonances as eigenstates and eigenvalues of the fokker planck operator obtained by adding an infinitesimal diffusion term to the first order liouville operator we demonstrate how the deterministic expression for the linear response is reproduced in the limit of vanishing noise for the second order response function we establish an equivalence of the spectral decomposition in the limit of vanishing noise and the long time asymptotic expansion in the deterministic case

entropy production in thermostats ii we show that an arbitrary anosov gaussian thermostat close to equilibrium has positive entropy poduction unless the external field e has a global potential the configuration space is allowed to have any dimension and magnetic forces are also allowed we also show the following non perturbative result suppose a gaussian thermostat satisfies k w sigma frac14 e sigma 2 0 for every 2 plane where kw is the sectional curvature of the associated weyl connection and e sigma is the orthogonal projection of e onto then the entropy production of any srb measure is positive unless e has a global potential a related non perturbative result is also obtained for certain generalized thermostats on surfaces

classical and quantum resonances for hyperbolic surfaces for compact and for convex co compact oriented hyperbolic surfaces we prove an explicit correspondence between classical ruelle resonant states and quantum resonant states except at negative integers where the correspondence involves holomorphic sections of line bundles

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

on decay of correlations in anosov flows there is some disagreement about the meaning of the phrase chaotic flow however there is no doubt that mixing anosov flows provides an example of such systems anosov systems were introduced and extensively studied in his classical memoir a among other things he proved the following fact known now as anosov alternative for flows either every strong stable and strong unstable manifold is everywhere dense or the flow gt is a suspension over an anosov diffeomorphism by a constant roof function if the first alternative holds gt is mixing with respect to every gibbs measure see pp2 now the natural question is what is the estimated rate of mixing this is certainly one of the simplest questions concerning correlation decay in continuous time systems nevertheless the only results obtained until recently dealt with the case when the system discussed had an additional algebraic structure the easier case of anosov diffeomorphisms can be treated by the methods of thermodynamic formalism of sinai ruelle and bowen b2 namely one uses markov partitions to construct an isomorphism between the diffeomorphism and a subshift of finite type and then proves that all such subshifts are exponentially mixing this method would succeed also for flows if any suspension over a subshift of finite type had exponentially decaying correlations however the simplest example suspensions with locally constant roof functions never have such a property ri one can use the above observation to produce examples of axiom a flows with arbitrary slow correlation decay it became clear therefore that some additional geometric properties should be taken into account in recent work chernov chl ch2 has employed a uniform nonintegrability condition to get a subexponential estimate for correlation functions for geodesic flows on surfaces of variable negative curvature his method relies on the technique of markov approximation developed in chl the aim of this paper is to combine geometric considerations of chernov with the thermodynamic formalism approach the later method seems to be more appropriate than markov approximations since it gives simpler

pollicott ruelle resonant states and betti numbers given a closed orientable hyperbolic manifold of dimension ne 3 we prove that the multiplicity of the pollicott ruelle resonance of the geodesic flow on perpendicular one forms at zero agrees with the first betti number of the manifold additionally we prove that this equality is stable under small perturbations of the riemannian metric and simultaneous small perturbations of the geodesic vector field within the class of contact vector fields for more general perturbations we get bounds on the multiplicity of the resonance zero on all one forms in terms of the first and zeroth betti numbers furthermore we identify for hyperbolic manifolds further resonance spaces whose multiplicities are given by higher betti numbers

rapid decay of correlations for nonuniformly hyperbolic flows we show that superpolynomial decay of correlations rapid mixing is prevalent for a class of nonuniformly hyperbolic flows these flows are the continuous time analogue of the class of nonuniformly hyperbolic maps for which young proved exponential decay of correlations the proof combines techniques of dolgopyat and operator renewal theory it follows from our results that planar periodic lorentz flows with finite horizons and flows near homoclinic tangencies are typically rapid mixing

microscopic chaos and reaction diffusion processes in the periodic lorentz gas we apply the hypothesis of microscopic chaos to diffusion controlled reaction which we study in a reactive periodic lorentz gas the relaxation rate of the reactive eigenmodes is obtained as eigenvalue of the frobenius perron operator which determines the reaction rate the cumulative functions of the eigenstates of the frobenius perron operator are shown to be generalizations of lebesgue s singular continuous functions for small enough densities of catalysts the reaction is controlled by the diffusion a random walk model of this diffusion controlled reaction process is presented which is used to study the dependence of the reaction rate on the density of catalysts

closed geodesics in homology classes on surfaces of variable negative curvature soit m une variete de riemann compacte de courbure negative variable on etudie le nombre de geodesiques fermees de longueurs t pour une classe d homologie m n t m on decrit le comportement asymptotique

generalized spectral decomposition of the β adic baker s transformation and intrinsic irreversibility abstract we construct a generalized spectral decomposition of the frobenius perron operator of the adic baker s transformation using a general iterative operator method applicable in principle for any mixing dynamical system the eigenvalues in the decomposition are related to the decay rates of the autocorrelation functions and have magnitudes less than one we explicitly define appropriate generalized function spaces which provide mathematical meaning to the formally obtained spectral decomposition the unitary frobenius perron evolution of densities when extended to the generalized function spaces splits into two semigroups one decaying in the future and the other in the past this split which reflects the asymptotic evolution of the forward and backward k partitions shows the instrinsic irreversibility of the baker s transformation

decay of correlations and invariance principles for dispersing billiards with cusps and related planar billiard flows following recent work of chernov markarian and zhang it is known that the billiard map for dispersing billiards with zero angle cusps has slow decay of correlations with rate 1 n since the collisions inside a cusp occur in quick succession it is reasonable to expect a much faster decay rate in continuous time in this paper we prove that the flow is rapid mixing correlations decay faster than any polynomial rate a consequence is that the flow admits strong statistical properties such as the almost sure invariance principle even though the billiard map does not the techniques in this paper yield new results for other standard examples in planar billiards including bunimovich flowers and stadia

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

margulis distributions for anosov flows for the strong unstable foliation or horocycle foliation of an anosov flow there exists a unique transverse measure called the margulis measure in this paper we extend margulis results to more general transverse distributions for the foliation as an application we derive our main result the non zero analytic extension to a strip of the selberg zeta function for compact surfaces of constant negative curvature persists under small perturbations in the metric there is an equivalent formulation in terms of the fourier transform of the correlation function

generalized language equations with multiple solutions abstract a generalized language equation x f x is an equation in a variable x over an alphabet a where the language function f is expressible in terms of arbitrary set operations including complementation and concatenation from the left by a constant i e a language in previous work these equations were studied provided an easily verifiable condition was met in this case solutions always exist and are always unique in the present paper the condition is dropped as a consequence our equations may have zero one or more up to uncountably infinitely many solutions we give a representation theorem for these equations and use it to classify them for equations in one variable a complete classification is given including parametric representations of all solutions for systems of m equations in m variables the notion of 1 reducible substitution graph is introduced and employed to reduce the problem of existence of solutions of the entire system to the same problem for a single equation the paper concludes with a discussion of the implications if one assumes that all constants are regular languages and poses some open problems

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

trace formula for noisy flows a trace formula is derived for the fokker planck equation associated with ito stochastic differential equations describing noisy time continuous nonlinear dynamical systems in the weak noise limit the trace formula provides estimations of the eigenvalues of the fokker planck operator on the basis of the pollicott ruelle resonances of the noiseless deterministic system which is assumed to be non bifurcating at first order in the noise amplitude the effect of noise on a periodic orbit is given in terms of the period and the derivative of the period with respect to the pseudo energy of the onsager machlup freidlin wentzell scheme