on the observability of continuous time switched linear systems under partially unknown inputs this technical note addresses the problem of observability of continuous time switched linear systems sls subject to unknown disturbances unknown switching signals and unconstrained nonzero dwell time for this class of hybrid systems a geometric characterization of the observability properties is presented in particular attention is focused on the estimation of the sls state from the continuous measurements using the well known concept of a b invariant subspaces three main problems are addressed and solved namely the observability for every nonzero state trajectory the observability for almost every control input and the observability for almost every state trajectory it is also shown that the approach herein presented subsumes some of the classical results on observability of sls previously reported in the literature and allows expressing them in a common framework

partial regularity of a generalized solution to the navier stokes equations in exterior domain in this note we prove two regularity theorems for solutions to the navier stokes equations of an i b v p in exterior domains namely we prove that the sets of the singular points of a solution if not empty has at most 1 hausdorff measureh1 s 0 moreover the sets is enclosed in a sphere of rayr for anyt 0 these results are obtained as corollaries to the partial regularity results furnished in 2

generic results in classes of ultradifferentiable functions abstract let e be a denjoy carleman class of ultradifferentiable functions of beurling type on the real line that strictly contains another class f of roumieu type we show that the set s of functions in e that are nowhere in the class f is large in the topological sense it is residual in the measure theoretic sense it is prevalent and that s 0 contains an infinite dimensional linear subspace it is lineable consequences for the gevrey classes are given similar results are also obtained for classes of ultradifferentiable functions defined imposing conditions on the fourier laplace transform of the function

state space reconstruction and spatio temporal prediction of lattice dynamical systems this paper addresses the problems of state space reconstruction and spatio temporal prediction for lattice dynamical systems it is shown that the state space of any finite lattice dynamical system can be embedded into a reconstruction space for almost every in the sense of prevalence smooth measurement mapping as long as the dimension of the reconstruction space is larger than twice the size of the lattice based on this result an input output spatio temporal dynamical relation for each site within the lattice is derived and used for spatio temporal prediction of the system in the case of infinite lattice dynamical systems an approach based on constructing local lattice dynamical systems is proposed it is shown that the finite dimensional results can be directly applied to the local modelling and spatio temporal prediction for infinite lattice dynamical systems two numerical examples are provided to demonstrate the proposed theory and approach

complexities of differentiable dynamical systems we define the notion of localizable property for a dynamical system then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems these notions are the fast growth of the number of periodic points the positive entropy and the high emergence we finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence

prevalence an addendum to prevalence a translation invariant almost every on infinite dimensional spaces bull amer math soc n s 27 1992 no 2 217 238 mr1161274 93k 28018

regularity criteria for almost every function in sobolev spaces in this paper we determine the multifractal nature of almost every function in the prevalence setting in a given sobolev or besov space according to different regularity exponents these regularity criteria are based on local l p regularity or on wavelet coefficients and give a precise information on pointwise behavior

how far is the borel map from being surjective in quasianalytic ultradifferentiable classes abstract the borel map j takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0 in the literature it is well known that the restriction of j to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space in this paper we are interested in studying how large the image of j is and we investigate the size and the structure of this image by using different approaches baire residuality prevalence and lineability we give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices which contains as particular cases the classes defined by a single weight sequence or by a weight function

null sets and essentially smooth lipschitz functions in this paper we extend the notion of a lebesgue null set to a notion which is valid in any completely metrizable abelian topological group we then use this definition to introduce and study the class of essentially smooth functions these are roughly speaking those lipschitz functions which are smooth in each direction almost everywhere

monotonic decreasing and asymptotic behavior of the kinetic energy for weak solutions of the navier stokes equations in exterior domains existence of weak solutions of the three dimensional navier stokes problem was first proved by j leray in the case of the cauchy problem cf leray 1934 as is well known these solutions are important in that they are the only solutions which so far are known to exist for all times without restriction on the data unfortunately however the question of whether they are classical in an ordinary sense is still open even though partial conclusions regarding regularity are avilable leray 1934 scheffer 1976 1980 caffarelli kohn nirenberg 1982 subsequently e hopf 1951 using a different technique constructed weak solutions for a general initial boundary value problem however hopf s solution even for the cauchy problem has weaker properties than leray s solution among others we refer to the energy inequality for the velocity field u x t i e ntimits mega u 2 x t dx ntimits mega u 2 x s sx eqq 2ntimits s t ntimits mega abla u abla udxdau i for s 0 for almost all s 0 and for all t s

pattern formation in a nonlinear model for animal coats abstract several models have been proposed for describing the formation of animal coat patterns we consider reaction diffusion models due to murray which rely on a turing instability for the pattern selection in this paper we describe the early stages of the pattern formation process for large domain sizes this includes the selection mechanism and the geometry of the patterns generated by the nonlinear system on one two and three dimensional base domains these results are obtained by an adaptation of results explaining the occurrence of spinodal decomposition in materials science as modeled by the cahn hilliard equation we use techniques of dynamical systems viewing solutions of the reaction diffusion model in terms of nonlinear semiflows our results are applicable to any parabolic system exhibiting a turing instability

on the testability of identification in some nonparametric models with endogeneity this paper examines three distinct hypothesis testing problems that arise in the context of identification of some nonparametric models with endogeneity the first hypothesis testing problem we study concerns testing necessary conditions for identification in some nonparametric models with endogeneity involving mean independence restrictions these conditions are typically referred to as completeness conditions the second and third hypothesis testing problems we examine concern testing for identification directly in some nonparametric models with endogeneity involving quantile independence restrictions for each of these hypothesis testing problems we provide conditions under which any test will have power no greater than size against any alternative in this sense we conclude that no nontrivial tests for these hypothesis testing problems exist publication abstract

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

hausdorff and packing dimension of fibers and graphs of prevalent continuous maps abstract the notions of shyness and prevalence generalize the property of being zero and full haar measure to arbitrary not necessarily locally compact polish groups the main goal of the paper is to answer the following question what can we say about the hausdorff and packing dimension of the fibers of prevalent continuous maps let k be an uncountable compact metric space we prove that a prevalent f c k r d has many fibers with almost maximal hausdorff dimension this generalizes a theorem of dougherty and yields that a prevalent f c k r d has graph of maximal hausdorff dimension generalizing a result of bayart and heurteaux we obtain similar results for the packing dimension we show that for a prevalent f c 0 1 m r d the set of y f 0 1 m for which dim h f 1 y m contains a dense open set having full measure with respect to the occupation measure m f 1 where dim h and m denote the hausdorff dimension and the m dimensional lebesgue measure respectively we also prove an analogous result when 0 1 m is replaced by any self similar set satisfying the open set condition we cannot replace the occupation measure with lebesgue measure in the above statement we show that the functions f c 0 1 for which positively many level sets are singletons form a non shy set in c 0 1 in order to do so we generalize a theorem of antunovic burdzy peres and ruscher as a complementary result we prove that the functions f c 0 1 for which dim h f 1 y 1 for all y min f max f form a non shy set in c 0 1 we also prove sharper results in which large hausdorff dimension is replaced by positive measure with respect to generalized hausdorff measures which answers a problem of fraser and hyde

parametrising the attractor of the two dimensional navier stokes equations with a finite number of nodal values abstract we consider the solutions lying on the global attractor of the two dimensional navier stokes equations with periodic boundary conditions and analytic forcing we show that in this case the value of a solution at a finite number of nodes determines elements of the attractor uniquely proving a conjecture due to foias and temam our results also hold for the complex ginzburg landau equation the kuramoto sivashinsky equation and reaction diffusion equations with analytic nonlinearities

unexpectedly linear behavior for the cahn hilliard equation this paper gives theoretical results on spinodal decomposition for the cahn hillard equation we prove a mechanism which explains why most solutions for the cahn hilliard equation starting near a homogeneous equilibrium within the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r namely most solutions are driven into a region of phase space in which linear behavior dominates for much longer than expected the cahn hilliard equation depends on a small parameter epsilon modeling the atomic scale interaction length we quantify the behavior of solutions as epsilon rightarrow 0 specifically we show that most solutions starting close to the homogeneous equilibrium remain close to the corresponding solution of the linearized equation with relative distance o epsilon 2 n 2 up to a ball of radius r in the h 2 omega norm where r is proportional to epsilon 1 rho n 4 as epsilon to 0 here n in 1 2 3 denotes the

local behavior of traces of besov functions prevalent results let 1 d d and p q s satisfying 0 p 0 q 0 s d p in this article we study the global and local regularity properties of traces on affine subsets of rd of functions belonging to the besov space bp qs rd given a d dimensional subspace h rd for almost all functions in bp qs rd in the sense of prevalence we are able to compute the singularity spectrum of the traces fa of f on affine subspaces of the form a h for lebesgue almost every a rd d in particular we prove that for lebesgue almost every a rd d these traces fa are more regular than what could be expected from standard trace theorems and that fa enjoys a multifractal behavior

observing lyapunov exponents of infinite dimensional dynamical systems can lyapunov exponents of infinite dimensional dynamical systems be observed by projecting the dynamics into n using a typical nonlinear projection map we answer this question affirmatively by developing embedding theorems for compact invariant sets associated with c1 maps on hilbert spaces examples of such discrete time dynamical systems include time t maps and poincare return maps generated by the solution semigroups of evolution partial differential equations we make every effort to place hypotheses on the projected dynamics rather than on the underlying infinite dimensional dynamical system in so doing we adopt an empirical approach and formulate checkable conditions under which a lyapunov exponent computed from experimental data will be a lyapunov exponent of the infinite dimensional dynamical system under study provided the nonlinear projection map producing the data is typical in the sense of prevalence

amenability free subgroups and haar null sets in non locally compact groups the paper has two objectives on the one hand we study left haar null sets a measure theoretic notion of smallness on polish not necessarily locally compact groups on the other hand we introduce and investigate two classes of polish groups which are closely related to this notion and to amenability we show that left haar null sets form a sigma ideal and have the steinhaus property on polish groups which are amenable at the identity and that they lose these two properties in the presence of appropriately embedded free subgroups as an application we prove an automatic continuity result for universally measurable homomorphisms from inverse limits of sequences of amenable locally compact second countable groups to second countable groups

generalizations of srb measures to nonautonomous random and infinite dimensional systems we review some developments that are direct outgrowths of or closely related to the idea of srb measures as introduced by sinai ruelle and bowen in the 1970s these new directions of research include the emergence of strange attractors in periodically forced dynamical systems random attractors in systems defined by stochastic differential equations srb measures for infinite dimensional systems including those defined by large classes of dissipative pdes quasi static distributions for slowly varying time dependent systems and surviving distributions in leaky dynamical systems

detection of symmetry of attractors from observations i theory abstract barany et al 1993 proposed a method for determining the symmetries of attractors of equivariant systems by averaging certain classes of equivariant maps we use an idea in barany et al to re cast definitions of symmetry detectives assuming that we only have access to equivariant observations from the system detecting from observations allows one to perform averaging in spaces that may have much lower dimension than the phase space this paper generalises and develops their suggestion among the generalisations we consider is the use of nonpolynomial detectives and we show using the notion of prevalence of hunt et al 1992 that our detectives and the detectives of barany et al 1993 dellnitz et al 1994 and golubitsky and nicol 1995 give the correct symmetry of attractors almost certainly in a measure theoretic sense we show that detectives can persistently give incorrect symmetries at isolated points in parametrised systems and discuss how to overcome this we show how one can find the symmetry of an attractor from examination of a poincare section in part ii of this article ashwin and tomes apply these results to find symmetries of attractors in a physical system of four coupled electronic oscillators with s4 symmetry

how often do simple dynamical processes have infinitely many coexisting sinks this work concerns the nature of chaotic dynamical processes sheldon newhouse wrote on dynamical processes depending on a parameter xx 1 t xn wherex is in the plane such as might arise when studying poincare return maps for autonomous differential equations in ir3 he proved that if the system is chaotic there will very often be existing parameter values for which there are infinitely many periodic attractors coexisting in a bounded region of the plane and that such parameter values would be dense in some interval the fact that infinitely many coexisting sinks can occur brings into question the very nature of the foundations of chaotic dynamical processes we prove for an apparently typical situation that newhouse s construction yields only a set of parameter values of measure zero

prevalence a translation invariant almost every on infinite dimensional spaces we present a measure theoretic condition for a property to hold almost everywhere on an infinite dimensional vector space with particularemphasis on function spaces such as c k and l p like the concept of lebesgue almost every on an infinite dimensional spaces our notion of prevalence is translation invariant instead of using a specific measure on the entire space we define prevalence in terms of the class of all probability measures with compact support prevalence is a more appropriate condition than the topological concepts of open and dense or generic when one desires a probabilistic result on the likelihood of a given property on a function space

a high regularity result of solutions to modified p stokes equations abstract this paper is concerned with a special quasilinear elliptic system which can be seen as a perturbed p laplacean p 1 2 in the whole space r n for its shape it is close to the p stokes system however our quasilinear second order differential operator is given by means of u and not by its symmetric part so that our system cannot be considered as a p stokes system hence it is called modified p stokes system we look for the high regularity of the solutions u in the sense of d 2 u l q r n q n in particular we get u c 0 r n as far as we know such a result of high regularity is the first concerning the coupling of unknowns u however our result also holds for the p laplacean and it is the first high regularity result in an unbounded domain

basic control for the viscous moore greitzer partial differential equation the notions of basic controllability and basic control are defined a quadratic optimal control of the linearized viscous moore greitzer equation is presented and it is confirmed that stall is uncontrollable in this model a basic control is constructed for the nonlinear viscous moore greitzer equation which can control both surge and stall some extensions of this construction are discussed numerical simulations of the basic control are presented and its performance is compared to the performance of a backstepping control constructed by banaszuk hauksson and mezic siam j control optim 37 1999 pp 1503 1537 it is shown that the viscous moore greizer equation with throttle control is not basically controllable but under certain conditions adding air injection control will make the equation basically controllable

closed form solution of visual inertial structure from motion this paper investigates the visual inertial structure from motion problem a simple closed form solution to this problem is introduced special attention is devoted to identify the conditions under which the problem has a finite number of solutions specifically it is shown that the problem can have a unique solution two distinct solutions and infinite solutions depending on the trajectory on the number of point features and on their layout and on the number of camera images the investigation is also performed in the case when the inertial data are biased showing that in this latter case more images and more restrictive conditions on the trajectory are required for the problem resolvability

on weighted norm estimates for nonstationary incompressible navier stokes flows in a 3d exterior domain abstract we study the time decay of weighted norms of weak and strong solutions to the navier stokes equations in a 3d exterior domain moment estimates for weak solutions and weighted l q estimates for strong solutions are deduced both of which seem to be optimal the relation is discussed between the space time decay and the vanishing of the total net force exerted by the fluid to the body a class of initial data is given so that the total net force associated to the corresponding fluid flows does not vanish

genericity and robustness of full surplus extraction we study whether priors that admit full surplus extraction fse are generic an issue that becomes a gauge to evaluate the validity of the current mechanism design paradigm we consider the space of priors on the universal type space and thereby relax the assumption of a fixed finite number of types made by cremer and mclean 1988 we show that fse priors are topologically generic contrary to the result of heifetz and neeman 2006 that fse is generically impossible both geometrically and measure theoretically instead of using the bdp approach or convex combinations of priors adopted in heifetz and neeman 2006 we prove our genericity results by showing a robustness property of cremer mclean mechanisms

monte carlo simulations for spinodal decomposition this paper addresses the phenomenon of spinodal decomposition for the cahn hilliard equation namely we are interested in why most solutions to the cahn hilliard equation which start near a homogeneous equilibrium u 0 in the spinodal interval exhibit phase separation with a characteristic wavelength when exiting a ball of radius r in a hilbert space centered at u 0 there are two mathematical explanations for spinodal decomposition due to grant and to maier paape and wanner in this paper we numerically compare these two mathematical approaches in fact we are able to synthesize the understanding we gain from our numerics with the approach of maier paape and wanner leading to a better understanding of the underlying mechanism for this behavior with this new approach we can explain spinodal decomposition for a longer time and larger radius than either of the previous two approaches a rigorous mathematical explanation is contained in a separate paper our approach is to use monte carlo simulations to examine the dependence of r the radius to which spinodal decomposition occurs as a function of the parameter e of the governing equation we give a description of the dominating regions on the surface of the ball by estimating certain densities of the distributions of the exit points we observe and can show rigorously that the behavior of most solutions originating near the equilibrium is determined completely by the linearization for an unexpectedly long time we explain the mechanism for this unexpectedly linear behavior and show that for some exceptional solutions this cannot be observed we also describe the dynamics of these exceptional solutions

absolute continuity of stable foliations for mappings of banach spaces we prove the absolute continuity of stable foliations for mappings of banach spaces satisfying conditions consistent with time t maps of certain classes of dissipative pdes this property is crucial for passing information from submanifolds transversal to the stable foliation to the rest of the phase space it is also used in proofs of ergodicity absolute continuity of stable foliations is well known in finite dimensional hyperbolic theory on banach spaces the absence of nice geometric properties poses some additional difficulties

on the asymptotic behaviour of the l 2 norm of suitable weak solutions to the navier stokes equations in three dimensional exterior domains we provel2 decay rates of suitable weak solutions to the navier stokes equations in exterior domains the results for the order of decay are the same as for the solutions to the cauchy problem of the navier stokes equations finally in the case of r3 the decay rate order is sharp in the class of solutions considered by us

some analogies between haar meager sets and haar null sets in abelian polish groups abstract in the paper we would like to pay attention to some analogies between haar meager sets and haar null sets among others we show that 0 int a a for each borel non haar meager set a in an abelian polish group moreover we define d measurability as a topological analog of christensen measurability and prove that each d measurable homomorphism is continuous finally we show easy constructions of a haar meager non haar null set and conversely a meager haar null set which is not haar meager in spaces of sequences our results refer to the papers 1 2 4

avoiding σ porous sets in hilbert spaces abstract we give a constructive proof that any porous subset of a hilbert space has lebesgue measure zero on typical c 1 curves further we discover that this result does not extend to all forms of porosity we find that even power p porous sets may meet many c 1 curves in positive measure

probability of local bifurcation type from a fixed point a random matrix perspective results regarding probable bifurcations from fixed points are presented in the context of general dynamical systems real random matrices time delay dynamical systems companion matrices and a set of mappings known for their properties as universal approximators neural networks the eigenvalue spectrum is considered both numerically and analytically using previous work of edelman et al based upon the numerical evidence various conjectures are presented the conclusion is that in many circumstances most bifurcations from fixed points of large dynamical systems will be due to complex eigenvalues nevertheless surprising situations are presented for which the aforementioned conclusion does not hold e g real random matrices with gaussian elements with a large positive mean and finite variance

stable coexistence and bi stability for competitive systems on ordered banach spaces abstract autonomous and asymptotically autonomous semiflows modeling two species competition are studied which are strongly order preserving on a convex subset of a banach space with competitive order conditions are derived for stable coexistence bi stability and competitive exclusion a complete classification of all possible outcomes is obtained in case that there is at most one positive steady state representing coexistence of both species

on the generic im possibility of full surplus extraction in mechanism design a number of studies most notably cremer and mclean 1985 1988 have shown that in harsanyi type spaces of a fixed finite size it is generically possible to design mechanisms that extract all the surplus from players and as a consequence implement any outcome as if the players private information were commonly known in contrast we show that within the set of common priors on the universal type space the subset of priors that permit the extraction of the players full surplus is shy shyness is an otion of smallness for convex subsets of infinite dimensional topological vector spaces in our case the set of common priors which generalizes the usual notion of zero lebesgue measure in finite dimensional spaces

convergence and divergence of wavelet series multifractal aspects we study the convergence and divergence of the wavelet expansion of a function in a sobolev or a besov space from a multifractal point of view in particular we give an upper bound for the hausdorff and for the packing dimension of the set of points where the expansion converges or diverges at a given speed and we show that generically these bounds are optimal

advanced topology on the multiscale sequence spaces s nu abstract we pursue the study of the multiscale spaces s introduced by jaffard in the context of multifractal analysis we give the necessary and sufficient condition for s to be locally p convex and exhibit a sequence of p norms that defines its natural topology the strong topological dual of s is identified to another sequence space depending on endowed with an inductive limit topology as a particular case we describe the dual of a countable intersection of besov spaces

how often surface diffeomorphisms have infinitely many sinks and hyperbolicity of periodic points near a homoclinic tangency abstract here we study an amazing phenomenon discovered by newhouse s newhouse non density of axiom a a on s 2 in proc sympos pure math vol 14 amer math soc 1970 pp 191 202 s newhouse diffeomorphisms with infinitely many sinks topology 13 1974 9 18 s newhouse the abundance of wild hyperbolic sets and nonsmooth stable sets of diffeomorphisms publ math inst hautes etudes sci 50 1979 101 151 it turns out that in the space of c r smooth diffeomorphisms diff r m of a compact surface m there is an open set u such that a baire generic diffeomorphism f u has infinitely many coexisting sinks in this paper we make a step towards understanding how often does a surface diffeomorphism have infinitely many sinks our main result roughly says that with probability one for any positive d a surface diffeomorphism has only finitely many localized sinks either of cyclicity bounded by d or those whose period is relatively large compared to its cyclicity it verifies a particular case of palis conjecture saying that even though diffeomorphisms with infinitely many coexisting sinks are baire generic they have probability zero one of the key points of the proof is an application of newton interpolation polynomials to study the dynamics initiated in v kaloshin b hunt a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms i ann of math in press 92 pp v kaloshin a stretched exponential bound on the rate of growth of the number of periodic points for prevalent diffeomorphisms ii preprint 85 pp

kupka smale theorem for polynomial automorphisms of ℂ2 and persistence of heteroclinic intersections a map is kupka smale if all periodic points are hyperbolic and the stable and unstable manifolds of any two saddle points are transverse here we prove that kupka smale maps form a residual set of full lebesgue measure in the space mathcal p d of polynomial automorphisms of 2 of fixed dynamical degree d 2 we also prove that a heteroclinic point of two saddle periodic orbits may be continued over almost the entire parameter space for this set of maps this is one of the first persistence theorems proved in holomorphic dynamics in several variables

topological properties of the sequence spaces s nu abstract we define sequence spaces based on the distributions of the wavelet coefficients in the spirit of s jaffard beyond besov spaces part i distributions of wavelet coefficients j fourier anal appl 10 2004 221 246 we study their topology and especially show that they can be endowed with a unique complete metric for which compact sets can be explicitly described and we study properties of this metric we also give relationships with besov spaces

global theory of one frequency schrodinger operators we study schrodinger operators with a one frequency analytic potential focusing on the transition between the two distinct local regimes characteristic respectively of large and small potentials from the dynamical point of view the transition signals the emergence of non uniform hyperbolicity so the dependence of the lyapunov exponent with respect to parameters plays a central role in the analysis though often ill behaved by conventional measures we show that the lyapunov exponent is in fact remarkably regular in a stratified sense which we define the irregularity comes from the matching of nice analytic or smooth functions along sets with complicated geometry this result allows us to establish that the critical set for the transition lies within countably many codimension one subvarieties of the infinite dimensional parameter space a more refined renormalization based analysis shows that the critical set is rather thin within those subvarieties and allows us to conclude that a typical potential has no critical energies such acritical potentials also form an open set and have several interesting properties only finitely many phase transitions may happen but never at any specific point in the spectrum and the lyapunov exponent is minorated in the region of the spectrum where it is positive on the other hand we do show that the number of phase transitions can be arbitrarily large

abundance of fast growth of the number of periodic points in 2 dimensional area preserving dynamics we prove that there exists an open subset of the set of real analytic hamiltonian diffeomorphisms of the two dimensional torus in which diffeomorphisms exhibiting fast growth of the number of periodic points are dense we also prove that there exists an open subset of the set of smooth area preserving diffeomorphisms of a closed surface in which typical diffeomorphisms exhibit fast growth of the number of periodic points

orthogonal sequences and regularity of embeddings into finite dimensional spaces abstract this paper focuses on the regularity of linear embeddings of finite dimensional subsets of hilbert and banach spaces into euclidean spaces we study orthogonal sequences in a hilbert space h whose elements tend to zero and similar sequences in the space c 0 of null sequences the examples studied show that the results due to hunt and kaloshin regularity of embeddings of infinite dimensional fractal sets into finite dimensional spaces nonlinearity 12 1999 1263 1275 and robinson linear embeddings of finite dimensional subsets of banach spaces into euclidean spaces nonlinearity 22 2009 711 728 for subsets of hilbert and banach spaces with finite box counting dimension are asymptotically sharp an analogous argument allows us to obtain a lower bound for the power of the logarithmic correction term in an embedding theorem proved by olson and robinson almost bi lipschitz embeddings and almost homogeneous sets trans amer math soc 362 1 2010 145 168 for subsets x of hilbert spaces when x x has finite assouad dimension

wild hyperbolic sets yet no chance for the coexistence of infinitely many klus simple newhouse attracting sets the phenomenon of the coexistence of infinitely many sinks for two dimensional dissipative diffeomorphisms is a result due to newhouse ne1 ne2 in fact for each parameter value at which a homoclinic tangency is formed nondegenerately there exist intervals in the parameter space containing dense sets of parameter values for which there are infinitely many coexisting sinks robinson r the structure of the sinks constructed by newhouse is limited simple newhouse parameter values are values at which there are infinitely many sinks having some special well defined property concerning the structure a result due to tedeschini lalli and yorke ty says that the lebesgue measure of the set of simple newhouse parameter values is zero when the tangencies are due to the standard affine horseshoe map it is argued in ty and pr that a more general derivation of this measure zero result would be desirable the main result of this paper is that the lebesgue measure of the set of klus simple parameter values including the simple newhouse parameter values is zero for saddle hyperbolic basic sets forming tangencies

smoothness of projections bernoulli convolutions and the dimension of exceptions erd s 1939 1940 studied the distribution of the random series p 0 n and showed that is singular for infinitely many 1 2 1 and absolutely continuous for a e in a small interval 1 1 solomyak 1995 proved a conjecture made by garsia 1962 that is absolutely continuous for a e 1 2 1 in order to sharpen this result we have developed a general method that can be used to estimate the hausdorff dimension of exceptional parameters in several contexts in particular we prove for any 0 1 2 the set of 0 1 such that is singular has hausdorff dimension strictly less than 1 for any borel set a rd with hausdorff dimension dim a d 1 2 there are points x a such that the pinned distance set x y y a has positive lebesgue measure moreover the set of x where this fails has hausdorff dimension at most d 1 dim a let k denote the middle cantor set for 1 2 and let k r be any compact set peres and solomyak 1998 showed that for a e 0 1 2 such that dim k dimk 1 the sum k k has positive length we show that the set of exceptional in this statement has hausdorff dimension at most 2 dim k dimk 0 for any borel set e rd with dim e 2 almost all orthogonal projections of e onto lines through the origin have nonempty interior and the exceptional set of lines where this fails has dimension at most d 1 dim e if is a borel probability measure on rd with correlation dimension greater than m 2 then for a prevalent set of c1 maps f rd rm in the sense described by hunt sauer and yorke 1992 the image of under f has a density with at least fractional derivatives in l2 rm

the limits of ex post implementation the sensitivity of bayesian implementation to agents beliefs about others suggests the use of more robust notions of implementation such as ex post implementation which requires that each agent s strategy be optimal for every possible realization of the types of other agents we show that the only deterministic social choice functions that are ex post implementable in generic mechanism design frameworks with multidimensional signals interdependent valuations and transferable utilities are constant functions in other words deterministic ex post implementation requires that the same alternative must be chosen irrespective of agents signals the proof shows that ex post implementability of a nontrivial deterministic social choice function implies that certain rates of information substitution coincide for all agents this condition amounts to a system of differential equations that are not satisfied by generic valuation functions

almost bi lipschitz embeddings and almost homogeneous sets this paper is concerned with embeddings of homogeneous spaces into euclidean spaces we show that any homogeneous metric space can be em bedded into a hilbert space using an almost bi lipschitz mapping bi lipschitz to within logarithmic corrections the image of this set is no longer homo geneous but almost homogeneous we therefore study the problem of em bedding an almost homogeneous subset x of a hilbert space h into a finite dimensional euclidean space in fact we show that if x is a compact subset of a banach space and x x is almost homogeneous then for n sufficiently large a prevalent set of linear maps from x into r n are almost bi lipschitz between x and its image we are then able to use the kuratowski embedding of x d into l x to prove a similar result for compact metric spaces

on the number of mather measures of lagrangian systems in 1996 ricardo ricardo mane discovered that mather measures are in fact the minimizers of a universal infinite dimensional linear programming problem this fundamental result has many applications of which one of the most important is to the estimates of the generic number of mather measures mane obtained the first estimation of that sort by using finite dimensional approximations recently we were able with gonzalo contreras to use this method of finite dimensional approximation in order to solve a conjecture of john mather concerning the generic number of mather measures for families of lagrangian systems in the present paper we obtain finer results in that direction by applying directly some classical tools of convex analysis to the infinite dimensional problem we use a notion of countably rectifiable sets of finite codimension in banach and frechet spaces which may deserve independent interest

analog computation with dynamical systems a b s t r a c t physical systems exhibit various levels of complexity their long term dynamics may converge to fixed points or exhibit complex chaotic behavior this paper presents a theory that enables to interpret natural processes as special purpose analog computers since physical systems are naturally described in continuous time a definition of computational complexity for continuous time systems is required in analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems discrete or continuous in time on the basis of an intrinsic time scale of the system dissipative dynamical systems are classified into the computational complexity classes pd co rpd npd and exp t corresponding to their standard counterparts according to the complexity of their long term behavior the complexity of chaotic attractors relative to regular ones leads to the conjecture pa fi npj continuous time flows have been proven useful in solving various practical problems our theory provides the tools for an algorithmic analysis of such flows as an example we analyze the continuous hopfield network 1998 elsevier science b v all rights reserved

complex analytic dynamics on the riemann sphere dichotomie dynamique de fatou et julia points periodiques consequences du theoreme de montel l ensemble de julia est la fermeture de l ensemble des points periodiques repulseurs resultats classiques sur l ensemble de fatou classification de sullivan de l ensemble de fatou une condition pour le developpement sur l ensemble de julia la dynamique des polynomes l ensemble de mandelbrot et le travail de douady et hubbard le theoreme d application de riemann mesurable et la dynamique analytique

asymptotically mean stationary channels a necessary and sufficient condition for a source to satisfy the ergodic theorem and the shannon mcmillan theorem the two basic mathematical tools of the shannon theory is that it be asymptotically mean stationary ams a channel is defined here to be ams if whenever an ams input source is connected to the channel the resulting input output process is ams we develop several characterizations and properties of ams channels that resemble those of ams sources as an application we show that these ideas are useful in characterizing composite sources and in particular that there exist sources that exhibit distinct short term and long term stationarity properties thus locally stationary or quasi stationary processes such as those used to model speech waveforms may also be stationary in addition some preliminary results on coding for ams channels are presented

on the conicity of eigenvalues intersections for parameter dependent self adjoint operators motivated by recent controllability results for the bilinear schrodinger equation based on the existence of conical intersections in this paper we identify two physically interesting families of parameter dependent hamiltonians that admit residual and prevalent subfamilies for which all double eigenvalues are conical in order to obtain such a result we exploit a characterization of conical intersections in terms of a transversality condition which allows to apply a suitable transversality theorem

bifurcation to infinitely many sinks this paper considers one parameter families of diffeomorphisms ft in two dimensions which have a curve of dissipative saddle periodic pointspt i e ftn pt pt and detdftn pt 1 the family is also assumed to create new homoclinic intersections of the stable and unstable manifolds ofpt as the parameter varies throught0 gavirlov and silnikov proved that if the new homoclinic intersections are created nondegenerately att0 then there is an infinite cascade of periodic sinks i e there are parameter valuestn accumulating att0 for which there is a sink of periodn gs2 sect 4 we show that this result is true for real analytic diffeomorphisms even if the homoclinic intersection is created degenerately we give computer evidence to show that this latter result is probably applicable to the henon map fora near 1 392 andb equal 0 3