stability of statistical properties in two dimensional piecewise hyperbolic maps we investigate the statistical properties of a piecewise smooth dynamical system by studying directly the action of the transfer operator on appropriate spaces of distributions we accomplish such a program in the case of two dimensional maps with uniformly bounded second derivative but we are confident that the present approach can be successful in much greater generality we hope including higher dimensional billiards for the class of systems at hand we obtain a complete description of the srb measures their statistical properties and their stability with respect to many types of perturbations including deterministic and random perturbations and holes

on a jumping problem for quasilinear elliptic equations let us denote by k the eigenvalues of the operator with homogeneous dirichlet condition first studied in 1 when 1 2 this problem has been widely investigated in the case that some eigenvalue k belongs to the interval see e g 14 13 10 and references therein the methods used are often a combination of topological and variational techniques in this paper we will be concerned with a jumping problem associated to a quasilinear elliptic equation of the form n

multiple solutions for p laplacian type problems with asymptotically p linear terms via a cohomological index theory abstract the aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem div a x u u p 2 u 1 p a t x u u p f x u in u 0 on where r n is a bounded domain n 2 p 1 a is a given function which admits partial derivative a t x t a t x t and f is asymptotically p linear at infinity under suitable hypotheses both at the origin and at infinity and if a x is even while f x is odd by using variational tools a cohomological index theory and a related pseudo index argument we prove a multiplicity result if p n in the non resonant case

quasi fuchsian seifert surfaces let k subset s 3 be a non fibered knot with hyperbolic complement given a seifert surface of minimal genus for k we prove that it corresponds to a quasi fuchsian group by showing it has no accidental parabolics in particular this shows that any lift of the seifert surface to the universal cover is a quasi disk and its limit set is a quasi circle in the sphere at infinity

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

on a singular elliptic system with quadratic growth in the gradient abstract in this paper we deal with the following singular elliptic system u u 2 u p p q a x v q u p 2 u f x r n v v 2 v q p q a x u p v q 2 v g x r n where n 3 n 2 4 p q 1 and p q n 2 n 2 we show through the sub and supersolutions method the existence of a nonnegative solution for an approximated system the limit of the approximated solution is a positive solution in the case 0 p q and f g we prove the uniqueness of a solution among others we prove some existence and uniqueness results for some auxiliary problems by using the comparison principle a minimization method and with the help of nehari manifold the proofs rely on the concentration compactness principle

on a pde involving the ap laplace operator abstract this paper establishes the existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present this operator represents a generalization of the p laplace operator i e p u div u p 2 u where p is a continuous function the proof of the main result is based on schauder s fixed point theorem combined with adequate variational arguments the function space setting used here makes appeal to the variable exponent lebesgue and sobolev spaces

a nondifferentiable extension of a theorem of pucci and serrin and applications abstract we study the multiplicity of critical points for functionals which are only differentiable along some directions we extend to this class of functionals the three critical point theorem of pucci and serrin and we apply it to a one parameter family of functionals j i r under suitable assumptions we locate an open subinterval of values in i for which j possesses at least three critical points applications to quasilinear boundary value problems are also given

limit theorems for fast slow partially hyperbolic systems we prove several limit theorems for a simple class of partially hyperbolic fast slow systems we start with some well know results on averaging then we give a substantial refinement of known large and moderate deviation results and conclude with a completely new result a local limit theorem on the distribution of the process determined by the fluctuations around the average the method of proof is based on a mixture of standard pairs and transfer operators that we expect to be applicable in a much wider generality

statistical properties of dynamical systems with some hyperbolicity this paper is about the ergodic theory of attractors and conservative dynamical systems with hyperbolic properties on large parts though not necessarily all of their phase spaces the main results are for discrete time systems to put this work into context recall that for axiom a attractors the picture has been fairly complete since the 1970 s see s1 b r2 since then much progress has been made on two fronts there is a general nonuniform theory that deals with properties common to all diffeomorphisms with nonzero lyapunov exponents o p1 ka ly and there are detailed analyses of specific kinds of dynamical systems including for example billiards 1 dimensional and henon type maps s2 bsc hk j bc2 by1 statistical properties such as exponential decay of correlations are not enjoyed by all diffeomorphisms with nonzero lyapunov exponents the goal of this paper is a systematic understanding of these and other properties for a class of dynamical systems larger than axiom a this class will not be defined explicitly but it includes some of the much studied examples by looking at regular returns to sets with good hyperbolic properties one could give systems in this class a simple dynamical representation conditions for the existence of natural invariant measures exponential mixing and central limit theorems are given in terms of the return times these conditions can be checked in concrete situations giving a unified way of proving a number of results some new and some old among the new results are the exponential decay of correlations for a class of scattering billiards and for a positive measure set of henon type maps

random products of standard maps we develop a general geometric method to establish the existence of positive lyapunov exponents for a class of skew products the technique is applied to show non uniform hyperbolicity of some conservative partially hyperbolic diffeomorphisms having as center dynamics coupled products of standard maps notably for skew products whose fiber dynamics is given by a continuum of parameters in the froeschle family these types of coupled systems appear as some induced maps in models for the study of arnold diffusion consequently we are able to present new examples of partially hyperbolic diffeomorphisms having rich high dimensional center dynamics the methods are also suitable for studying cocycles over shift spaces and do not demand any low dimensionality condition on the fiber

the volume of hyperbolic alternating link complements if a hyperbolic link has a prime alternating diagram d then we show that the link complement s volume can be estimated directly from d we define a very elementary invariant of the diagram d its twist number t d and show that the volume lies between v 3 t d 2 2 and v 3 10t d 10 where v 3 is the volume of a regular hyperbolic ideal 3 simplex as a consequence the set of all hyperbolic alternating and augmented alternating link complements is a closed subset of the space of all complete finite volume hyperbolic 3 manifolds in the geometric topology

compact locally maximal hyperbolic sets for smooth maps fine statistical properties compact locally maximal hyperbolic sets are studied via geometrically defined functional spaces that take advantage of the smoothness of the map in a neighborhood of the hyperbolic set this provides a self contained theory that not only reproduces all the known classical results but also gives new insights on the statistical properties of these systems

existence of radial bounded solutions for some quasilinear elliptic equations in rn abstract we study the quasilinear equation p div a x u u p 2 u 1 p a t x u u p u p 2 u g x u i n r n with n 3 p 1 where a x t a t x t a t x t and g x t are caratheodory functions on r n r suitable assumptions on a x t and g x t set off the variational structure of p and its related functional j is c 1 on the banach space x w 1 p r n l r n in order to overcome the lack of compactness we assume that the problem has radial symmetry then we look for critical points of j restricted to x r subspace of the radial functions in x following an approach which exploits the interaction between x and the norm on w 1 p r n we prove the existence of at least one weak bounded radial solution of p by applying a generalized version of the ambrosetti rabinowitz mountain pass theorem

variational methods for non differentiable functionals and their applications to partial differential equations the interest in extending the theory of nonlinear partial differential equations pde to pde with discontinuous nonlinearities dnde has been increasing daily the reasons are twofold 1 many free boundary problems may be reduced to boundary value problems of dnde cf chang 16 71 chang and jiang 9 and fleishman and mahar 121 2 sometimes in dealing with a pde problem it is beneficial to put the original pde into a large category for instance dnde cf chang 7 simply speaking a dnde is as follows

critical points for multiple integrals of the calculus of variations in this paper we deal with the existence of critical points of functional defined on the sobolev space w01 p p 1 by j u int limits omega vartheta x u du dx text where is a bounded open subset of n even for very simple examples in n the differentiability of j u can fail to overcome this difficulty we prove a suitable version of the ambrosetti rabinowitz mountain pass theorem applicable to functionals which are not differentiable in all directions existence and multiplicity of nonnegative critical points are also studied through the use of this theorem

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

unbounded critical points for a class of lower semicontinuous functionals for a general class of lower semicontinuous functionals we prove existence and multiplicity of critical points which turn out to be unbounded solutions to the associated euler equation we apply a nonsmooth critical point theory developed in 10 12 13 and applied in 8 9 20 to treat the case of continuous functionals

mountain pass solutions for quasi linear equations via a monotonicity trick abstract we obtain the existence of symmetric mountain pass solutions for quasi linear equations without the typical assumptions which guarantee the boundedness of an arbitrary palais smale sequence this is done through a recent version of the monotonicity trick proved in squassina in press 22 the main results are new also for the p laplacian operator

hyperbolic knots spanning accidental seifert surfaces of arbitrarily high genus a method for constructing hyperbolic knots each of which bounds accidental incompressible seifert surfaces of arbitrarily high genus is given

decay of correlations for piecewise expanding maps this paper investigates the decay of correlations in a large class of non markov one dimensional expanding maps the method employed is a special version of a general approach recently proposed by the author explicit bounds on the rate of decay of correlations are obtained

local limit theorem for nonuniformly partially hyperbolic skew products and farey sequences we study skew products of the form x omega mapsto tx omega phi x where t is a nonuniformly expanding map on a space x preserving a possibly singular probability measure tilde mu and phi x to s 1 is a c 1 function under mild assumptions on tilde mu and phi we prove that such a map is exponentially mixing and satisfies the central and local limit theorems these results apply to a random walk related to the farey sequence thereby answering a question of guivarc h and raugi

statistical properties of mostly contracting fast slow partially hyperbolic systems we consider a class of mathcal c 4 partially hyperbolic systems on mathbb t 2 described by maps f varepsilon x theta f x theta theta varepsilon omega x theta where f cdot theta are expanding maps of the circle for sufficiently small varepsilon and omega generic in an open set we precisely classify the srb measures for f varepsilon and their statistical properties including exponential decay of correlation for holder observables with explicit and nearly optimal bounds on the decay rate

soliton solutions for quasilinear schrodinger equations ii abstract for a class of quasilinear schrodinger equations we establish the existence of ground states of soliton type solutions by a variational method

crosscap numbers and the jones polynomial abstract we give sharp two sided linear bounds of the crosscap number non orientable genus of alternating links in terms of their jones polynomial our estimates are often exact and we use them to calculate the crosscap numbers for several infinite families of alternating links and for several alternating knots with up to twelve crossings we also discuss generalizations of our results for classes of non alternating links

subexponential decay of correlations we describe a method for proving subexponential lower bounds for correlations functions and apply it to study decay of correlations for maps with countable markov partitions one result is that ls young s upper estimates y2 are optimal in many situations our method is based on a general result concerning the asymptotics of renewal sequences of bounded operators acting on banach spaces which we apply to the iterates of the transfer operator

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

fast slow partially hyperbolic systems versus freidlin wentzell random systems we consider a simple class of fast slow partially hyperbolic dynamical systems and show that the properly rescaled behaviour of the slow variable is very close to a freidlin wentzell type random system for times that are rather long but much shorter than the metastability scale also we show the possibility of a sink with all the lyapunov exponents positive a phenomenon that turns out to be related to the lack of absolutely continuity of the central foliation

homogenization for deterministic maps and multiplicative noise a recent paper of melbourne stuart a note on diffusion limits of chaotic skew product flows nonlinearity 24 2011 1361 1367 gives a rigorous proof of convergence of a fast slow deterministic system to a stochastic differential equation with additive noise in contrast to other approaches the assumptions on the fast flow are very mild in this paper we extend this result from continuous time to discrete time moreover we show how to deal with one dimensional multiplicative noise this raises the issue of how to interpret certain stochastic integrals it is proved that the integrals are of stratonovich type for continuous time and neither

spike solutions for a class of singularly perturbed quasilinear elliptic equations abstract by means of a penalization scheme due to del pino and felmer we prove the existence of single peaked solutions for a class of singularly perturbed quasilinear elliptic equations associated with functionals which lack of smoothness we do not require neither uniqueness assumptions on the limiting autonomous equation nor monotonicity conditions on the nonlinearity compared with the semilinear case some difficulties arise and the study of concentration of the solutions needs a somewhat involved analysis in which the pucci serrin variational identity plays an important role

limit theorems for partially hyperbolic systems we consider a large class of partially hyperbolic systems containing among others affine maps frame flows on negatively curved manifolds and mostly contracting diffeomorphisms if the rate of mixing is sufficiently high the system satisfies many classical limit theorems of probability theory

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

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

multibump solutions for quasilinear elliptic equations abstract the current paper is concerned with constructing multibump type solutions for a class of quasilinear schrodinger type equations including the modified nonlinear schrodinger equations our results extend the existence results on multibump type solutions in coti zelati and rabinowitz 1992 17 to the quasilinear case our work provides a theoretic framework for dealing with quasilinear problems which lack both smoothness and compactness by using more refined variational techniques such as gluing techniques morse theory lyapunov schmidt reduction etc

multibump solutions for quasilinear elliptic equations with critical growth the current paper is concerned with constructing multibump solutions for a class of quasilinear schrodinger equations with critical growth this extends the classical results of coti zelati and rabinowitz commun pure appl math 45 1217 1269 1992 for semilinear equations as well as recent work of liu wang and guo j funct anal 262 4040 4102 2012 for quasilinear problems with subcritical growth the periodicity of the potentials is used to glue ground state solutions to construct multibump bound state solutions

existence of nontrivial solutions for quasilinear elliptic equations at critical growth abstract in this paper we study the existence of nontrivial weak solutions for the following quasilinear elliptic equation with natural growth d j a ij x u d i u 1 2 s a ij x u d i ud j u f x u in a bounded smooth domain r n where aij x u are functions possibly unbounded with respect to u

nontrivial solutions of some quasilinear problems via a cohomological local splitting abstract this paper deals with a generalization of the p laplacian type boundary value problem div p a x u u p 2 u a t x u u p g x u in u 0 on being a bounded domain in r n under suitable assumptions and if p n the existence of a nontrivial solution can be proved by means of variational tools and a cohomological local splitting

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

decay of correlations dedicated to micheline ishay i would like to thank p boyland l chierchia v donnay g de martino c gole j l lebowitz m lyubich m rychlik i g schwarz s vaienti and especially g gallavotti for helpful and enlightening discussions particularly warm thanks go to n chernov for carefully reading and finding a mistake in an early version the present paper benefits from several improvements due to his sharp criticism in addition i am indebted to p collet and most of all m wojtkowski for introducing me to the magical world of cones finally i thank j milnor director of the institute for mathematical sciences at stony brook university where i was visiting during part of this work and the italian c n r gnfm for providing travel funds

multiple critical points for nondifferentiable functionals involving hardy potentials abstract in this paper we study general functionals of the calculus of variations with the presence of a hardy potential we will improve several results obtained in the semilinear framework we will first prove a general weak lower semicontinuity result which will imply the existence of a minimum point whenever the functional is coercive then we will demonstrate existence and multiplicity results of critical points even if our functional is not differentiable we will apply a nonsmooth critical point theory developed in corvellec et al nonlinear anal 1 1993 151 and degiovanni and marzocchi ann mat pura appl 167 1994 73