topological stability for conservative systems we prove that the c1 interior of the set of all topologically stable c1 incompressible flows is contained in the set of anosov incompressible flows moreover we obtain an analogous result for the discrete time case

quantum algebras as quantizations of dual poisson lie groups a systematic computational approach for the explicit construction of any quantum hopf algebra uz g z starting from the lie bialgebra g that gives the first order deformation of the coproduct map z is presented the procedure is based on the well known quantum duality principle namely the fact that any quantum algebra can be viewed as the quantization of the unique poisson lie structure g g on the dual group g which is obtained by exponentiating the lie algebra g defined by the dual map from this perspective the coproduct for uz g is just the pull back of the group law for g and the poisson analogues of the quantum commutation rules for uz g are given by the unique poisson lie structure g on g whose linearization is the poisson analogue of the initial lie algebra g this approach is shown to be a very useful technical tool in order to solve the lie bialgebra quantization problem explicitly since once a lie bialgebra g is given the full dual poisson lie group g can be obtained either by applying standard poisson lie group techniques or by implementing the algorithm presented here with the aid of symbolic manipulation programs as a consequence the quantization of g will give rise to the full uz g quantum algebra provided that ordering problems are appropriately fixed through the choice of certain local coordinates on g whose coproduct fulfils a precise quantum symmetry property the applicability of this approach is explicitly demonstrated by reviewing the construction of several instances of quantum deformations of physically relevant lie algebras such as the 2 1 anti de sitter algebra so 2 2 and the poincare algebra in 3 1 dimensions

new criteria for ergodicity and nonuniform hyperbolicity in this work we obtain a new criterion to establish ergodicity and nonuniform hyperbolicity of smooth measures of diffeomorphisms of closed connected riemannian manifolds this method allows us to give a more accurate description of certain ergodic components the use of this criterion in combination with topological devices such as blenders lets us obtain global ergodicity and abundance of nonzero lyapunov exponents in some contexts in the partial hyperbolicity context we obtain that stably ergodic diffeomorphisms are c1 dense among volume preserving partially hyperbolic diffeomorphisms with 2 dimensional center bundle this is motivated by a well known conjecture of pugh and shub

equivariant quantizations of symmetric algebras abstract let g be a lie bialgebra and let v be a finite dimensional g module we study deformations of the symmetric algebra of v which are equivariant with respect to an action of the quantized enveloping algebra u h g resp u q g we investigate in particular such quantizations obtained from the quantization of certain lie bialgebra structures on the semidirect product of g and v we classify these structure in the important special case when g is complex simple with quasitriangular lie bialgebra structure and v is a simple g module we then introduce a more general notion co poisson module algebras and their quantizations to further address the problem and show that many known examples of quantized symmetric algebras can be described in this language

homoclinic tangencies and hyperbolicity for surface diffeomorphisms we prove here that in the complement of the closure of the hyperbolic surface diffeomorphisms the ones exhibiting a homoclinic tangency are c 1 dense this represents a step towards the global understanding of dynamics of surface diffeomorphisms

singular robustly chain transitive sets are singular volume partial hyperbolic for diffeomorphisms or for non singular flows there are many results relating properties persistent under mathcal c 1 perturbations and a global structures for the dynamics such as hyperbolicity partial hyperbolicity dominated splitting however a difficulty appears when a robust property of a flow holds on a set containing recurrent orbits accumulating a singular point in bonatti star flows and multisingular hyperbolicity arxiv 1705 05799 2017 with christan bonatti we propose a a general procedure for adapting the usual hyperbolic structures to the singularities using this tool we recover the results in bonatti et al ann math 158 2 355 418 2003 for flows showing that robustly chain transitive sets have a weak form of hyperbolicity allowing us to conclude as well the kind of hyperbolicity carried by the examples in bonatti et al j inst math jussieu 12 3 449 501 2013 a robust chain transitive singular attractor with periodic orbits of different indexes along with the results in 8 this shows that the way we propose to interpret the effect of singularities has the potential to adapt to other settings in which there is coexistence of singularities and regular orbits with the goal of re obtaining the results that we already know for diffeomorphisms

maps close to identity and universal maps in the newhouse domain given an n dimensional c r diffeomorphism g its renormalized iteration is an iteration of g restricted to a certain n dimensional ball and taken in some c r coordinates in which the ball acquires radius 1 we show that for any r 1 the renormalized iterations of c r close to identity maps of an n dimensional unit ball b n n 2 form a residual set among all orientation preserving c r diffeomorphisms b n r n in other words any generic n dimensional dynamical phenomenon can be obtained by iterations of c r close to identity maps with the same dimension of the phase space as an application we show that any c r generic two dimensional map that belongs to the newhouse domain i e it has a so called wild hyperbolic set so it is not uniformly hyperbolic nor uniformly partially hyperbolic and that neither contracts nor expands areas is c r universal in the sense that its iterations after an appropriate coordinate transformation c r approximate every orientation preserving two dimensional diffeomorphism arbitrarily well in particular every such universal map has an infinite set of coexisting hyperbolic attractors and repellers

formality theorem for lie bialgebras and quantization of twists and coboundary r matrices abstract let g be a lie bialgebra let u g a quantization of g through etingof kazhdan functor we prove the existence of a l morphism between the lie algebra c g g and the tensor algebra without unit t u t u g 1 with lie algebra structure given by the gerstenhaber bracket when s is a twist for g we deduce from the formality morphism the existence of a quantum twist f when g r is a coboundary lie bialgebra we get the existence of a quantization r of r

semi continuity of the entropy of sinai probability measures for partially hyperbolic diffeomorphisms abstract we establish sufficient conditions for the upper semicontinuity and the continuity of the entropy of sinai probability measures invariant by partially hyperbolic diffeomorphisms and discuss their application in several examples

abundance of strange attractors in 1976 he h non performed a numerical study of the family of diffeomorphisms of the plane ha b x y 1 ax2 y bx and detected for parameter values a l 4 b 0 3 what seemed to be a non trivial attractor with a highly intricate geometric structure this family has since then been the subject of intense research both numerical and theoretical but its dynamics is still far from being completely understood in particular one could not exclude the possibility that the attractor observed by h6non were just a periodic orbit with a very high period recently in a remarkable paper bc2 benedicks and carleson were able to show that this is not the case at least for a positive lebesgue measure set of parameter values near a 2 b o more precisely they showed that if b 0 is small enough then for a positive measure set of a values near a 2 the corresponding diffeomorphism ha b exhibits a strange attractor their argument is a very creative extension of the techniques they had previously developed in bull for the study of the quadratic family on the real line and no doubt it will be important for the understanding of several other situations of complicated nonhyperbolic dynamics when acquainted in 1985 with the work by benedicks and carleson then in progress palls suggested that one should in this context think of the h6non family as a particular although important model for the creation of a horseshoe and that the emphasis should be put on the occurrence of unfoldings of homoclinic tangencies he proposed that the correct setting for benedicks carleson s results is within this more general framework of homoclinic bifurcations and stated the following

a characterization of singular hyperbolicity via the linear poincare flow abstract singular hyperbolicity is a fundamental notion in dynamical systems that describes in particular lorenz like dynamics it is defined using the tangent flow and concerns both the regular and singular parts of the dynamics in this paper we give a characterization of singular hyperbolicity using the linear poincare flow which by definition is defined on the regular part usually non compact of the dynamics only

stable ergodicity of certain linear automorphisms of the torus we find a class of ergodic linear automorphisms of tn that are stably ergodic this class includes all non anosov ergodic automorphisms when n 4 as a corollary we obtain the fact that all ergodic linear automorphism of tn are stably ergodic when n iu 5

symbolic extensions and smooth dynamical systems let f x x be a homeomorphism of the compact metric space x a symbolic extension of f x is a subshift on a finite alphabet g y which has f as a topological factor we show that a generic c1 non hyperbolic i e non anosov area preserving diffeomorphism of a compact surface has no symbolic extensions for r 1 we exhibit a residual subset mathcal r of an open set mathcal u of cr diffeomorphisms of a compact surface such that if f in mathcal r then any possible symbolic extension has topological entropy strictly larger than that of f these results complement the known fact that any c diffeomorphism has symbolic extensions with the same entropy we also show that cr generically on surfaces homoclinic closures which contain tangencies of stable and unstable manifolds have hausdorff dimension two

generic diffeomorphisms with superexponential growth of number of periodic orbits let m be a smooth compact manifold of dimension at least 2 and diff r m be the space of c r smooth diffeomorphisms of m associate to each diffeomorphism f 2 diff r m the sequence pn f of the number of isolated periodic points for f of period n in this paper we exhibit an open set n in the space of diffeomorphisms diff r m such for a baire generic diffeomorphism f 2 n the number of periodic points pnf grows with a period n faster than any following sequence of numbers fangn2zc along a subsequence i e pn f a ni for some ni 1 with i 1 in the cases of surface diffeomorphisms i e dim m d 2 an open set n with a supergrowth of the number of periodic points is a newhouse domain a proof of the man result is based on the gontchenko shilnikov turaev theorem gst a complete proof of that theorem is also presented

the bernoulli property for weakly hyperbolic systems a dynamical system is called partially hyperbolic if it exhibits three invariant directions one unstable expanding one stable contracting and one central direction somewhere in between the other two we prove that topologically mixing partially hyperbolic diffeomorphisms whose central direction is non uniformly contracting negative lyapunov exponents almost everywhere have the bernoulli property the system is equivalent to an i i d independently identically distributed random process in particular these systems are mixing correlations of integrable functions go to zero as time goes to infinity we also extend this result in two different ways firstly for 3 dimensional diffeomorphisms if one requires only non zero instead of negative lyapunov exponents then one still gets a quasi bernoulli property secondly if one assumes accessibility any two points are joined by some path whose legs are stable segments and unstable segments then it suffices to requires the mostly contracting property on a positive measure subset to obtain the same conclusions

on the finiteness of uniform sinks abstract we study the finiteness of uniform sinks for flows precisely we prove that for 0 and t 0 if a vector field x has only hyperbolic singularities or sectionally dissipative singularities then x can have only finitely many t uniform sinks this is a generalized version of a theorem of liao 4

infinitesimal lyapunov functions for singular flows we present an extension of the notion of infinitesimal lyapunov function to singular flows and from this technique we deduce a characterization of partial sectional hyperbolic sets in absence of singularities we can also characterize uniform hyperbolicity these conditions can be expressed using the space derivative dx of the vector field x together with a field of infinitesimal lyapunov functions only and are reduced to checking that a certain symmetric operator is positive definite at the tangent space of every point of the trapping region

on the existence of attractors on every compact 3 manifold we build a non empty open set u of diff m such that for every r 1 every cr generic diffeomorphism f u diffr m has no topological attractors on higher dimensional manifolds one may require that f has neither topological attractors nor topological repellers our examples have finitely many quasi attractors for flows we may require that these quasi attractors contain singular points finally we discuss alternative definitions of attractors which may be better adapted to generic dynamics

homoclinic tangencies versus uniform hyperbolicity for conservative 3 flows abstract we prove that a volume preserving three dimensional flow can be c 1 approximated by a volume preserving anosov flow or else by another volume preserving flow exhibiting a homoclinic tangency this proves the conjecture of palis for conservative 3 flows and with respect to the c 1 topology

pathological foliations and removable zero exponents the ergodic theory of uniformly hyperbolic or axiom a diffeomorphisms has been studied extensively beginning with the pioneering work of anosov sinai ruelle and bowen an si ru bo while uniformly hyperbolic systems enjoy strong mixing properties they are not dense among c diffeomorphisms sm as using the concept of lyapunov exponents pesin introduced a weaker form of hyperbolicity which he termed nonuniform hyperbolicity nonuniformly hyperbolic diffeomorphisms share several mixing properties with uniformly hyperbolic ones our construction of the diffeomorphisms in this paper was motivated by the question of whether nonuniform hyperbolicity is dense among a large class of diffeomorphisms as a curious by product of our construction we prove that a pathological feature of central foliations the complete failure of absolute continuity can exist in a c open set of volume preserving diffeomorphisms we recall the definition of a nonuniformly hyperbolic diffeomorphism a real number is a lyapunov exponent of the diffeomorphism g m m if there exists a nonzero vector v tm such that

the lyapunov exponents of generic volume preserving and symplectic maps we show that the integrated lyapunov exponents of c1 volume preserving diffeomorphisms are simultaneously continuous at a given diffeomorphism only if the corresponding oseledets splitting is trivial all lyapunov exponents are equal to zero or else dominated uniform hyperbolicity in the projective bundle almost everywhere we deduce a sharp dichotomy for generic volume preserving diffeomorphisms on any compact manifold almost every orbit either is projectively hyperbolic or has all lyapunov exponents equal to zero similarly for a residual subset of all c1 symplectic diffeomorphisms on any compact manifold either the diffeomorphism is anosov or almost every point has zero as a lyapunov exponent with multiplicity at least 2 finally given any set s gl d satisfying an accessibility condition for a residual subset of all continuous s valued cocycles over any measure preserving homeomorphism of a compact space the oseledets splitting is either dominated or trivial the condition on s is satisfied for most common matrix groups and also for matrices that arise from discrete schr nodinger operators

robust criterion for the existence of nonhyperbolic ergodic measures we give explicit c 1 open conditions that ensure that a diffeomorphism possesses a nonhyperbolic ergodic measure with positive entropy actually our criterion provides the existence of a partially hyperbolic compact set with one dimensional center and positive topological entropy on which the center lyapunov exponent vanishes uniformly

on some universal algebras associated to the category of lie bialgebras in our previous work math 0008128 we studied the set quant k of all universal quantization functors of lie bialgebras over a field k of characteristic zero compatible with the operations of taking duals and doubles we showed that quant k is canonically isomorphic to a product g0 k k where g0 k is a universal group and k is a quotient set of a set b k of families of lie polynomials by the action of a group g k we prove here that g0 k is equal to the multiplicative group 1 k so quant k is as close as it can be to k we also prove that the only universal derivations of lie bialgebras are multiples of the composition of the bracket with the cobracket finally we prove that the stabilizer of any element of b k is reduced to the 1 parameter subgroup generated by the corresponding square of the antipode

some characterizations of domination we show that a cocycle has a dominated splitting if and only if there is a uniform exponential gap between singular values of its iterates then we consider sets in rm gl d mathbb r with the property that any cocycle with values in has a dominated splitting we characterize these sets in terms of existence of invariant multicones thus extending a two dimensional result by avila bochi and yoccoz we give an example showing how these multicones can fail to have convexity properties

the dynamics of the henon map the purpose of this paper is to develop machinery which is suitable for a study of the long time behavior of systems such as the henon map with expansion combined with strong contraction our study is modeled on the treatment of the one dimensional system x 1 ax 2 and we study the perturbation of a from the value a 2 and b small a surprising number of complications arise when b 0 the computer simulations give some indication of the complicated structure of the attractor but it can safely be stated that the real difficulties do not show up the central aim of the paper is to introduce what we consider the relevant concepts and techniques needed to study mappings such as the henon one

topological pressure via saddle points let a be a compact locally maximal invariant set of a c 2 diffeomorphism f m m on a smooth riemannian manifold m in this paper we study the topological pressure p top with respect to the dynamical system f for a wide class of holder continuous potentials and analyze its relation to dynamical as well as geometrical properties of the system we show that under a mild non uniform hyperbolicity assumption the topological pressure of is entirely determined by the values of on the saddle points of f in a moreover it is enough to consider saddle points with large lyapunov exponents we also introduce a version of the pressure for certain non continuous potentials and establish several variational inequalities for it finally we deduce relations between expansion and escape rates and the dimension of a our results generalize several well known results to certain non uniformly hyperbolic systems

stability and bifurcations for dissipative polynomial automorphisms of mathbb c 2 we study stability and bifurcations in holomorphic families of polynomial automorphisms of mathbb c 2 we say that such a family is weakly stable over some parameter domain if periodic orbits do not bifurcate there we first show that this defines a meaningful notion of stability which parallels in many ways the classical notion of j stability in one dimensional dynamics define the bifurcation locus to be the complement of the weak stability locus in the second part of the paper we prove that under an assumption of moderate dissipativity the parameters displaying homoclinic tangencies are dense in the bifurcation locus this confirms one of palis conjectures in the complex setting the proof relies on the formalism of semi parabolic bifurcation and the construction of critical points in semi parabolic basins which makes use of the classical denjoy carleman ahlfors and wiman theorems

modular quantizations of lie algebras of cartan type k via drinfeld twists of jordanian type we construct explicit drinfel d twists of jordanian type for the generalized cartan type k lie algebras in characteristic 0 and obtain the corresponding quantizations especially their integral forms by making modular reductions including modulo p and modulo p restrictedness reduction and base changes we derive certain modular quantizations of the restricted universal enveloping algebra u k 2n 1 1 for the restricted simple lie algebra of cartan type k in characteristic p they are new families of noncommutative and noncocommutative hopf algebras of dimension pp2n 1 1 if 2n 4 0 modp or pp2n 1 if 2n 4 0 modp over a truncated p polynomials ring which also contain the well known radford algebras see 20 as hopf subalgebras some open questions are proposed

srb measures for partially hyperbolic systems whose central direction is mostly expanding we construct sinai ruelle bowen srb measures supported on partially hyperbolic sets of diffeomorphisms the tangent bundle splits into two invariant subbundles one of which is uniformly contracting under the assumption that the complementary subbundle is non uniformly expanding if the rate of expansion lyapunov exponents is bounded away from zero then there are only finitely many srb measures our techniques extend to other situations including certain maps with singularities or critical points as well as diffeomorphisms having only a dominated splitting and no uniformly hyperbolic subbundle

sinai bowen ruelle measures for certain henon maps we study maps t a b 2 2 defined by t a b left x y right left 1 a x 2 y bx right 0 0

absolutely continuous invariant measures for one parameter families of one dimensional maps given a one parameter familyf x of maps of the interval 0 1 we consider the set of parameter values for whichf has an invariant measure absolutely continuous with respect to lebesgue measure we show that this set has positive measure for two classes of maps i f x f x where 0 4 andf x is a functionc3 near the quadratic mapx 1 x and ii f x f x mod 1 wheref isc3 f 0 f 1 0 andf has a unique nondegenerate critical point in 0 1

symbolic extensions and dominated splittings for generic c 1 diffeomorphisms let mathrm diff 1 m be the set of all c 1 diffeomorphisms f m rightarrow m where m is a compact boundaryless d dimensional manifold d ge 2 we prove that there is a residual subset mathfrak r of mathrm diff 1 m such that if f in mathfrak r and if h p is the homoclinic class associated with a hyperbolic periodic point p then either h p admits a dominated splitting of the form e oplus f 1 oplus dots oplus f k oplus g where f i is not hyperbolic and one dimensional or f h p has no symbolic extensions

equivariant quantization of poisson homogeneous spaces and kostant s problem we find a partial solution to the longstanding problem of kostant concerning description of the so called locally finite endomorphisms of highest weight irreducible modules the solution is obtained by means of its reduction to a far reaching extension of the quantization problem while the classical quantization problem consists in finding product deformations of the commutative algebras of functions we consider the case when the initial object is already a noncommutative algebra the algebra of functions within q calculus

robustly topological mixing of kan s map abstract in 1994 i kan constructed a smooth map on the annulus admitting two physical measures whose basins are intermingled in this paper we prove that kan s map is c 2 robustly topologically mixing

particle filtering with invertible particle flow a key challenge when designing particle filters in high dimensional state spaces is the construction of a proposal distribution that is close to the posterior distribution recent advances in particle flow filters provide a promising avenue to avoid weight degeneracy particles drawn from the prior distribution are migrated in the state space to the posterior distribution by solving partial differential equations numerous particle flow filters have been proposed based on different assumptions concerning the flow dynamics approximations are needed in the implementation of all of these filters as a result the particles do not exactly match a sample drawn from the desired posterior distribution past efforts to correct the discrepancies involve expensive calculations of importance weights in this paper we present new filters which incorporate deterministic particle flows into an encompassing particle filter framework the valuable theoretical guarantees concerning particle filter performance still apply but we can exploit the attractive performance of the particle flow methods the filters we describe involve a computationally efficient weight update step arising because the embedded particle flows we design possess an invertible mapping property we evaluate the proposed particle flow particle filters performance through numerical simulations of a challenging multitarget multisensor tracking scenario and complex high dimensional filtering examples

on the dynamics of dominated splitting let f m m be a c 2 dieomorphism of a compact surface we give a complete description of the dynamics of any compact invariant set having dominated splitting in particular we prove a spectral decomposition theorem for the limit set l f under the assumption of dominated splitting moreover we describe all the bifurcations that these systems can exhibit and the dierent types of dynamics that could follow for small c r perturbations

beyond topological hyperbolicity the l shadowing property abstract in this paper we further explore the l shadowing property defined in 20 for dynamical systems on compact spaces we prove that structurally stable diffeomorphisms and some pseudo anosov diffeomorphisms of the two dimensional sphere satisfy this property homeomorphisms satisfying the l shadowing property have a spectral decomposition where the basic sets are either expansive or contain arbitrarily small topological semi horseshoes periodic sets where the restriction is semiconjugate to a shift to this end we characterize the l shadowing property using local stable and unstable sets and the classical shadowing property we exhibit homeomorphisms with the l shadowing property and arbitrarily small topological semi horseshoes without periodic points at the end we show that positive finite expansivity jointly with the shadowing property imply that the space is finite

the specification property for flows from the robust and generic viewpoint abstract we prove that if x has the weak specification property robustly where is an isolated set then is a hyperbolic topologically mixing set and as a consequence if x is a vector field that has the weak specification property robustly on a closed manifold m then the flow x t is a topologically mixing anosov flow also we prove that there exists a residual subset r x 1 m so that if x r and x has the weak specification property then x t is an anosov flow

lyapunov exponents and rigidity of anosov automorphisms and skew products abstract in this paper we obtain local rigidity results for linear anosov diffeomorphisms in terms of lyapunov exponents more specifically we show that given an irreducible linear hyperbolic automorphism l with simple real eigenvalues with distinct absolute values any small perturbation preserving the volume and with the same lyapunov exponents is smoothly conjugate to l we also obtain rigidity results for skew products over anosov diffeomorphisms given a volume preserving partially hyperbolic skew product diffeomorphism f 0 over an anosov automorphism of the 2 torus we show that for any volume preserving perturbation f of f 0 with the same average stable and unstable lyapunov exponents the center foliation is smooth

on some lie bialgebra structures on polynomial algebras and their quantization we study classical twists of lie bialgebra structures on the polynomial current algebra mathfrak g u where mathfrak g is a simple complex finite dimensional lie algebra we focus on the structures induced by the so called quasi trigonometric solutions of the classical yang baxter equation it turns out that quasi trigonometric r matrices fall into classes labelled by the vertices of the extended dynkin diagram of mathfrak g we give the complete classification of quasi trigonometric r matrices belonging to multiplicity free simple roots which have coefficient 1 in the decomposition of the maximal root we quantize solutions corresponding to the first root of mathfrak sl n

essential hyperbolicity and homoclinic bifurcations a dichotomy phenomenon mechanism for diffeomorphisms we prove that any diffeomorphism of a compact manifold can be approximated in the c 1 topology by another diffeomorphism exhibiting a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by one which is essentially hyperbolic has a finite number of transitive hyperbolic attractors with open and dense basin of attraction

quantization of γ lie bialgebras abstract we introduce the notion of lie bialgebras where is a group these objects give rise to cocommutative co poisson bialgebras for which we construct quantization functors this enlarges the class of co poisson algebras for which a quantization is known our result relies on our earlier work where we showed that twists of lie bialgebras can be quantized we complement this work by studying the behavior of this quantization under compositions of twists

a cohomological construction of quantization functors of lie bialgebras abstract we propose a variant to the etingof kazhdan construction of quantization functors we construct the twistor j associated to an associator using cohomological techniques we then introduce a criterion ensuring that the left hopf algebra of a quasitriangular que algebra is flat we prove that this criterion is satisfied at the universal level this gives a construction of quantization functors equivalent to the etingof kazhdan construction

partial hyperbolicity or dense elliptic periodic points for c 1 generic symplectic diffeomorphisms we prove that if a symplectic diffeomorphism is not partially hyperbolic then with an arbitrarily smal c 1 perturbation we can create a totally elliptic periodic point inside any given open set as a consequence a c 1 generic symplectic diffeomorphism is either partially hyperbolic or it has dense elliptic periodic points this extends the similar results of s new house in dimension 2 and m c arnaud in dimension 4 another interesting consequence is that stably ergodic symplectic diffeomorphisms must be partially hyperbolic a conserve to shub pugh s stable ergodicity conjecture for the symplectic case

some large deviation results for dynamical systems we prove some large deviation estimates for continuous maps of compact metric spaces and apply them to attractors in differentiable dynamics rate of escape problems and to shift spaces

hyperbolicity sinks and measure in one dimensional dynamics letf be ac2 map of the circle or the interval and let f denote the complement of the basins of attraction of the attracting periodic orbits we prove that f is a hyperbolic expanding set if and obviously only if every periodic point is hyperbolic and f doesn t contain the critical point this is the real one dimensional version of fatou s hyperbolicity criteria for holomorphic endomorphisms of the riemann sphere we also explore other applications of the techniques used for the result above proving for instance that for everyc2 immersionf of the circle i e a map of the circle onto itself without critical points either its julia set has measure zero or it is the whole circle and thenf is ergodic i e positively invariant borel sets have zero or full measure

partial hyperbolicity far from homoclinic bifurcations abstract we prove that any diffeomorphism of a compact manifold can be c 1 approximated by a diffeomorphism which exhibits a homoclinic bifurcation a homoclinic tangency or a heterodimensional cycle or by a diffeomorphism which is partially hyperbolic its chain recurrent set splits into partially hyperbolic pieces whose centre bundles have dimensions less or equal to two we also study in a more systematic way the central models introduced in crovisier in press 10

implicit sampling for particle filters we present a particle based nonlinear filtering scheme related to recent work on chainless monte carlo designed to focus particle paths sharply so that fewer particles are required the main features of the scheme are a representation of each new probability density function by means of a set of functions of gaussian variables a distinct function for each particle and step and a resampling based on normalization factors and jacobians the construction is demonstrated on a standard ill conditioned test problem

particle filtering with path sampling and an application to a bimodal ocean current model this paper introduces a recursive particle filtering algorithm designed to filter high dimensional systems with complicated non linear and non gaussian effects the method incorporates a parallel marginalization pmmc step in conjunction with the hybrid monte carlo hmc scheme to improve samples generated by standard particle filters parallel marginalization is an efficient markov chain monte carlo mcmc strategy that uses lower dimensional approximate marginal distributions of the target distribution to accelerate equilibration as a validation the algorithm is tested on a 2516 dimensional bimodal stochastic model motivated by the kuroshio current that runs along the japanese coast the results of this test indicate that the method is an attractive alternative for problems that require the generality of a particle filter but have been inaccessible due to the limitations of standard particle filtering strategies

density of hyperbolicity for classes of real transcendental entire functions and circle maps we prove density of hyperbolicity in spaces of i real transcendental entire functions bounded on the real line whose singular set is finite and real and ii transcendental self maps of the punctured plane which preserve the circle and whose singular set apart from zero and infinity is contained in the circle in particular we prove density of hyperbolicity in the famous arnol d family of circle maps and its generalizations and solve a number of other open problems for these functions including three conjectures by de melo salom ao and vargas we also prove density of real hyperbolicity for certain families as in i but without the boundedness condition our results apply in particular when the functions in question have only finitely many critical points and asymptotic singularities or when there are no asymptotic values and the degree of critical points is uniformly bounded

following a moving target monte carlo inference for dynamic bayesian models markov chain monte carlo mcmc sampling is a numerically intensive simulation technique which has greatly improved the practicality of bayesian inference and prediction however mcmc sampling is too slow to be of practical use in problems involving a large number of posterior target distributions as in dynamic modelling and predictive model selection alternative simulation techniques for tracking moving target distributions known as particle filters which combine importance sampling importance resampling and mcmc sampling tend to suffer from a progressive degeneration as the target sequence evolves we propose a new technique based on these same simulation methodologies which does not suffer from this progressive degeneration

improved particle filters for multi target tracking we present a novel approach for improving particle filters for multi target tracking the suggested approach is based on drift homotopy for stochastic differential equations drift homotopy is used to design a markov chain monte carlo step which is appended to the particle filter and aims to bring the particle filter samples closer to the observations while at the same time respecting the target dynamics we have used the proposed approach on the problem of multi target tracking with a nonlinear observation model the numerical results show that the suggested approach can improve significantly the performance of a particle filter

on c1 robust transitivity of volume preserving flows we prove that a divergence free and c1 robustly transitive vector field has no singularities moreover if the vector field is smooth enough then the linear poincare flow associated to it admits a dominated splitting over m

degenerate behavior in non hyperbolic semigroup actions on the interval fast growth of periodic points and universal dynamics we consider semigroup actions on the unit interval generated by strictly increasing c r maps we assume that one of the generators has a pair of fixed points one attracting and one repelling and a heteroclinic orbit that connects the repeller and attractor we also assume that the other generators form a robust blender which can bring the points from a small neighborhood of the attractor to an arbitrarily small neighborhood of the repeller this is a model setting for partially hyperbolic systems with one central direction we show that under additional conditions on frac f f and the schwarzian derivative the above semigroups exhibit c r generically for any r ge 3 arbitrarily fast growth of the number of periodic points as a function of the period we also show that a c r generic semigroup from the class under consideration supports an ultimately complicated behavior called universal dynamics

perturbation of the lyapunov spectra of periodic orbits we describe all lyapunov spectra that can be obtained by perturbing the derivatives along periodic orbits of a diffeomorphism the description is expressed in terms of the finest dominated splitting and lyapunov exponents that appear in the limit of a sequence of periodic orbits and involves the majorization partial order among the applications we give a simple criterion for the occurrence of universal dynamics

quantization of coboundary lie bialgebras we show that any coboundary lie bialgebra can be quantized for this we prove that a etingof kazhdan quantization functors are compatible with lie bialgebra twists and b if such a quantization functor corresponds to an even associator then it is also compatible with the operation of taking coopposites we also use the relation between the etingof kazhdan construction of quantization functors and the alternative approach to this problem which was established in a previous work

partial hyperbolicity and robust transitivity throughout this paper m denotes a three dimensional boundaryless compact manifold and diff m the space of gl diffeomorphisms defined on m endowed with the usual cl topology a invariant set a is transitive if a w x for some xea here w x is the forward limit set of x the accumulation points of the positive orbit of x the maximal invariant set of in an open set u denoted by a u is the set of points whose whole orbit is contained in u i e a u i e z i u the set a u is robustly transitive if ar is transitive for every diffeomorphism r clclose to a diffeomorphism ediff m is transitive if m w x for some xem i e if a m m is transitive analogously is robustly transitive if every r glclose to also is transitive i e if a m m is robustly transitive in this paper we focus our attention on forms of hyperbolicity uniform partial and strong partial of a maximal invariant set a u derived from its robust transitivity observe that u can be equal to m and then is robustly transitive on one hand in dimension one there do not exist robustly transitive diffeomorphisms the diffeomorphisms with finitely many hyperbolic periodic points morse smale are open and dense in diff s1 on the other hand for two dimensional diffeomorphisms every robustly transitive set a u is a basic set i e a u is hyperbolic transitive and the periodic points of are dense in a u in particular every robustly transitive surface diffeomorphism is anosov and the unique surface which supports such diffeomorphisms is the torus t 2 these assertions follow from m3 and m4 in dimension bigger than or equal to three besides anosov hyperbolic diffeomorphisms there are robustly transitive diffeomorphisms of nonhyperbolic type as far as we know three types of such diffeomorphisms have been constructed skew products

homoclinic points in symplectic and volume preserving diffeomorphisms letmn be a compactn dimensional manifold and be a symplectic or volume form onmn let be ac1 diffeomorphism onmn that preserves and letp be a hyperbolic periodic point of we show that genericallyp has a homoclinic point and moreover the homoclinic points ofp is dense on both stable manifold and unstable manifold ofp takens 11 obtained the same result forn 2

existence and non existence of length averages for foliations since the pioneering work of ghys et al it has been known that several methods of dynamical systems theory can be adopted to study of foliations our aim in this paper is to investigate complexity of foliations by generalising existence problem of time averages in dynamical systems theory to foliations it has recently been realised that a positive lebesgue measure set of points without time averages only appears for complicated dynamical systems such as dynamical systems with heteroclinic connections or homoclinic tangencies in this paper we introduce the concept of length averages to singular foliations and attempt to collect interesting examples with without length averages in particular we demonstrate that length averages exist everywhere for any codimension one mathscr c 1 orientable singular foliation without degenerate singularities on a compact surface under a mild condition on quasi minimal sets of the foliation which is in strong contrast to time averages of surface flows

solution of the basin problem for henon like attractors for a large class of non uniformly hyperbolic attractors of dissipative diffeomorphisms we prove that there are no holes in the basin of attraction stable manifolds of points in the attractor fill in a full lebesgue measure subset then lebesgue almost every point in the basin is generic for the srb sinai ruelle bowen measure of the attractor this solves a problem posed by sinai and by ruelle for this class of systems

renormalization in the henon family ii the heteroclinic web we study highly dissipative henon maps f c b x y mapsto c x 2 by x with zero entropy they form a region in the parameter plane bounded on the left by the curve w of infinitely renormalizable maps we prove that morse smale maps are dense in but there exist infinitely many different topological types of such maps even away from w we also prove that in the infinitely renormalizable case the average jacobian bf on the attracting cantor set mathcal o f is a topological invariant these results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory along these lines we show that the unstable manifolds of the periodic points form a lamination outside mathcal o f if and only if there are no heteroclinic tangencies

abundance of c 1 robust homoclinic tangencies a diffeomorphism f has a c 1 robust homoclinic tangency if there is a c 1 neighbourhood u of f such that every diffeomorphism in g u has a hyperbolic setg depending contin uously on g such that the stable and unstable manifolds ofg have some non transverse intersection for every manifold of dimension greater than or equal to three we exhibit a lo cal mechanism blender horseshoes generating diffeomorphisms with c 1 robust homoclinic tangencies using blender horseshoes we prove that homoclinic classes of c 1 generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings of appropriate dimensions display c 1 robust homoclinic tangencies

birth of homoclinic intersections a model for the central dynamics of partially hyperbolic systems creation d intersection homoclines un modele pour la dynamique centrale des systemes partiellement hyperboliques nous montrons une conjecture de j palis tout diffeomorphisme d une variete compacte peut etre approche en topologie c 1 par un diffeomorphisme morse smale ou par un diffeomorphisme ayant une intersection homocline transverse

a c 1 generic dichotomy for diffeomorphisms weak forms of hyperbolicity or infinitely many sinks or sources we show that for every compact n dimensional manifold n 1 there is a residual subset of diff m of diffeomorphisms for which the homoclinic class of any periodic saddle of f verifies one of the following two possibilities either it is contained in the closure of an infinite set of sinks or sources newhouse phenomenon or it presents some weak form of hyperbolicity called dominated splitting this is a generalization of a bidimensional result of mafine ma3 in particular we show that any cl robustly transitive diffeomorphism admits a dominated splitting

lower central series of free algebras in symmetric tensor categories abstract we continue the study of the lower central series of a free associative algebra initiated by feigin and shoikhet 2007 fs we generalize via schur functors the constructions of the lower central series to any symmetric tensor category specifically we compute the modified first quotient b 1 and second and third quotients b 2 and b 3 of the series for a free algebra t v in any symmetric tensor category generalizing the main results of feigin and shoikhet 2007 fs and arbesfeld and jordan 2010 aj in the case a m n t c m n we use these results to compute the explicit hilbert series finally we prove a result relating the lower central series to the corresponding filtration by two sided associative ideals confirming a conjecture from etingof et al 2009 ekm and another one from arbesfeld and jordan 2010 aj as corollaries