classification of the ring of witt vectors and the necklace ring associated with the formal group law x y qxy abstract in this paper we classify the ring of witt vectors and the necklace ring associated with the formal group law x y q x y up to strict isomorphism as q varies over the set of integers we also introduce a q deformation of the grothendieck ring of formal power series with constant term 1 over rings in which q 1 is a unit various bijections related to witt vectors necklaces and formal power series are given as an application we provide a q analogue of the celebrated cyclotomic identity and its bijective proof

on equality of hausdorff and affinity dimensions via self affine measures on positive subsystems under mild conditions we show that the affinity dimension of a planar self affine set is equal to the supremum of the lyapunov dimensions of self affine measures supported on self affine proper subsets of the original set these self affine subsets may be chosen so as to have stronger separation properties and in such a way that the linear parts of their affinities are positive matrices combining this result with some recent breakthroughs in the study of self affine measures and their associated furstenberg measures we obtain new criteria under which the hausdorff dimension of a self affine set equals its affinity dimension for example applying recent results of barany hochman solomyak and rapaport we provide many new explicit examples of self affine sets whose hausdorff dimension equals its affinity dimension and for which the linear parts do not satisfy any positivity or domination assumptions

computing covariant lyapunov vectors oseledets vectors and dichotomy projectors a comparative numerical study abstract covariant lyapunov vectors or oseledets vectors are increasingly being used for a variety of model analyses in areas such as partial differential equations nonautonomous differentiable dynamical systems and random dynamical systems these vectors identify spatially varying directions of specific asymptotic growth rates and obey equivariance principles in recent years new computational methods for approximating oseledets vectors have been developed motivated by increasing model complexity and greater demands for accuracy in this numerical study we introduce two new approaches based on singular value decomposition and exponential dichotomies and comparatively review and improve two recent popular approaches of ginelli et al 2007 36 and wolfe and samelson 2007 37 we compare the performance of the four approaches via three case studies with very different dynamics in terms of symmetry spectral separation and dimension we also investigate which methods perform well with limited data

perturbation theory for the approximation of stability spectra by qr methods for sequences of linear operators on a hilbert space abstract in this paper we develop a perturbation analysis for stability spectra lyapunov exponents and sacker sell spectrum for products of operators on a hilbert space both real and complex based upon the discrete qr technique error bounds are obtained in both the integrally separated and non integrally separated cases that correspond to distinct and multiple eigenvalues respectively for a single linear operator we illustrate our results using a linear parabolic partial differential equation in which the strength of the integral separation the time varying analogue of gaps between eigenvalues determines the sensitivity of the stability spectra to perturbation

witt vectors and the algebra of necklaces where u is the classical mobius function m a n is called a necklace polynomial we shall be concerned with some remarkable identities satisfied by the polynomials m a n which apparently have not been previously noticed specifically we prove that m a n can be expressed as a quadratic polynomial in m a i and mj that m a n can be expressed linearly and integrally in terms of m a i the proofs given below of these identities are entirely combinatorial theorems l 6

exponential separation exponential dichotomy and almost periodicity of linear difference equations abstract in this paper we study the relation between the concept of exponential separation and those of reducibility exponential dichotomy and almost periodicity of linear difference equations

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

what are lyapunov exponents and why are they interesting taking as inspiration the fields medal work of artur avila i d like to introduce you to lyapunov exponents my plan is to show how lyapunov exponents play a key role in three areas in which avila s results lie smooth ergodic theory billiards and translation surfaces and the spectral theory of 1 dimensional schrodinger operators but first what are lyapunov exponents let s begin by viewing them in one of their natural habitats the barycentric subdivision of a triangle is a collection of 6 smaller triangles obtained by joining the midpoints of the sides to opposite vertices here s what happens when you start with an equilateral triangle and repeatedly barycentrically subdivide

properties of sufficiency and statistical tests 1 in a previous paper dealing with the importance of properties of sufficiency in the statistical theory of small samples attention was mainly confined to the theory of estimation in the present paper the structure of small sample tests whether these are related to problems of estimation and fiducial distributions or are of the nature of tests of goodness of fit is considered further

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

characterization of dominated splittings for operator cocycles acting on banach spaces abstract versions of the oseledets multiplicative ergodic theorem for cocycles acting on infinite dimensional banach spaces have been investigated since the pioneering work of ruelle in 1982 and are a topic of continuing research interest for a cocycle to induce a continuous splitting in which the growth in one subbundle exponentially dominates the growth in a complementary subbundle requires additional assumptions a necessary and sufficient condition for the existence of such a dominated splitting also known as exponential separation was recently given by j bochi and n gourmelon for invertible finite dimensional cocycles in discrete time we extend this result to cocycles of injective bounded and possibly noncompact linear maps acting on banach spaces in both discrete and continuous time using an essentially geometric approach based on a notion of approximate singular value decomposition in banach spaces our method is constructive and in the finite dimensional case yields explicit growth estimates on the dominated splitting which may be of independent interest

ledrappier young formula and exact dimensionality of self affine measures in this paper we solve the long standing open problem on exact dimensionality of self affine measures on the plane we show that every self affine measure on the plane is exact dimensional regardless of the choice of the defining iterated function system in higher dimensions under certain assumptions we prove that self affine and quasi self affine measures are exact dimensional in both cases the measures satisfy the ledrappier young formula

a generalized cyclotomic identity abstract in this paper we prove a generalization of the cyclotomic identity for each group g satisfying suitable finiteness conditions we prove this generalized cyclotomic identity by counting coloured g set structures the cyclotomic identity corresponds to taking g the free abelian group of rank one for the free abelian group of rank two for example we get an analogous identity involving the generating function of the partition function interalia we show that studying g set structures gives a unified and conceptually simple approach to other classical combinatorial identities not to the author s knowledge previously linked in the literature

r analogue of the burnside ring of profinite groups and free lie algebras abstract let r be a finite dimensional torsion free special ring in this paper we generalize the results in dress and siebeneicher adv in math 70 1988 89 78 1989 1 by constructing r analogue r g of the burnside ring of profinite groups g in particular we remark that the grothendieck lie module denominator identity of free lie algebras in oh necklace rings and logarithmic functions preprint kias 2003 is closely related to the canonical isomorphism between r g and grothendieck s ring of formal power series with coefficients in r and constant term 1

q analog of the mobius function and the cyclotomic identity associated to a profinite group abstract let g be a profinite group and q an indeterminate in this paper we introduce and study a q analog of the mobius function and the cyclotomic identity arising from the lattice of open subgroups of g when q is any integer we show that they have close connections with the functors w g q nr g q and nr g q introduced in y t oh q deformation of witt burnside rings math z 257 2007 151 191 in particular we interpret the multiplicative property of the inverse of the table of marks and the mobius function of g as a composition property of certain functors classification of w g q nr g q and nr g q up to strict natural isomorphism as q varies over the set of integers and its application will be dealt with too

q deformation of witt burnside rings in this paper we construct a q deformation of the witt burnside ring of a profinite group over a commutative ring where q ranges over the set of integers when q 1 it coincides with the witt burnside ring introduced by dress and siebeneicher adv math 70 87 132 1988 to achieve our goal we first show that there exists a q deformation of the necklace ring of a profinite group over a commutative ring as in the classical case i e the case q 1 q deformed witt burnside rings and necklace rings always come equipped with inductions and restrictions we also study their properties as a byproduct we prove a conjecture due to lenart j algebra 199 703 732 1998 finally we classify mathbb w g q up to strict natural isomorphism in case where g is an abelian profinite group

hilbert space lyapunov exponent stability we study cocycles of compact operators acting on a separable hilbert space and investigate the stability of the lyapunov exponents and oseledets spaces when the operators are subjected to additive gaussian noise we show that as the noise is shrunk to 0 the lyapunov exponents of the perturbed cocycle converge to those of the unperturbed cocycle and the oseledets spaces converge in probability to those of the unperturbed cocycle this is to our knowledge the first result of this type with cocycles taking values in operators on infinite dimensional spaces the infinite dimensionality gives rise to a number of substantial difficulties that are not present in the finite dimensional case

nested witt vectors and their q deformation abstract let n n be a truncation set we study the ring of n nested witt vectors and its q deformation given two arbitrary integers q and r we provide a necessary and sufficient condition of a so that w n q a and w n r a should be strictly isomorphic to each other also an isomorphism of functors w n q w m q w m n q will be established for coprime truncation sets m and n as a byproduct we deal with interesting connections between nested witt vectors and other areas such as generalized mobius functions and numerical polynomials i e polynomials which take integral values at integer arguments

a volume preserving diffeomorphism with essential coexistence of zero and nonzero lyapunov exponents we show that there exists a c volume preserving topologically transitive diffeomorphism of a compact smooth riemannian manifold which is ergodic indeed is bernoulli on an open and dense subset mathcal g of not full volume and has zero lyapunov exponent on the complement of mathcal g

temporal intermittency in a shell model turbulence in terms of covariant lyapunov vectors we study temporal intermittency in the goy shell model of turbulence by using the covariant lyapunov vectors clvs we find that physical and isolated modes in lyapunov modes proposed by yang et al 2009 phys rev lett 102 074102 exist in the shell model turbulence furthermore we analyze differences between burst and laminar periods in the turbulent attractor in terms of the clvs in particular we study the burst and the laminar periods by using two kinds of dynamical quantities defined by the clvs one is a set of finite time lyapunov exponents which correspond respectively to the clvs we find that the order of magnitudes of the finite time lyapunov exponents in the burst period changes more frequently with time than in the laminar period moreover we find that an effective dimension defined by the number of lyapunov modes of which the order of magnitude changes in the burst period is higher than in the laminar period the other quantity is an angle between the stable and the unstable manifolds along the turbulence attractor we obtain the result that an orbit in the burst period passes near nonhyperbolic points more frequently than in the laminar period from this result we conjecture that the tangency structure of the stable and the unstable manifolds is related to the intermittency this article is part of a special issue of journal of physics a mathematical and theoretical devoted to lyapunov analysis from dynamical systems theory to applications

goodness of fit tests for large sparse multinomial distributions abstract a goodness of fit statistic d 2 is introduced for use in multinomial distributions pearson s x2 and d 2 are both approximately normally distributed when the sample size n is not large relative to the number of multinomial categories k under sequences of local alternative hypotheses the test based on d 2 exhibits moderate power when the x2 test is biased application is made to the analysis of large sparse contingency tables a theorem is presented that describes the likelihood ratio for testing a simple multinomial distribution against a mixture of multinomial distributions a wide variety of mixing distributions is considered and the d 2 statistic is a special case of log when testing for dirichlet mixtures of multinomial distributions in the case where n is large relative to k x2 and d 2 k behave approximately as chi squared random variables and differ by a very small amount under the null hypothesis in this situation x2 and d 2 k will generally yield the same inference with a l

dichotomy spectra and morse decompositions of linear nonautonomous differential equations abstract recently the existence of morse decompositions for nonautonomous dynamical systems was shown for three different time domains the past the future and in the linear case the entire time in this article notions of exponential dichotomy are discussed with respect to the three time domains it is shown that an exponential dichotomy gives rise to an attractor repeller pair in the projective space which is a building block of a morse decomposition moreover based on the notions of exponential dichotomy dichotomy spectra are introduced and it is proved that the corresponding spectral manifolds lead to morse decompositions in the projective space

generic dynamics of 4 dimensional c2 hamiltonian systems we study the dynamical behaviour of hamiltonian flows defined on 4 dimensional compact symplectic manifolds we find the existence of a c2 residual set of hamiltonians for which there is an open mod 0 dense set of regular energy surfaces each either being anosov or having zero lyapunov exponents almost everywhere this is in the spirit of the bochi mane dichotomy for area preserving diffeomorphisms on compact surfaces 2 and its continuous time version for 3 dimensional volume preserving flows 1

decomposition of the witt burnside ring and burnside ring of an abelian profinite group abstract let g h be abelian profinite groups whose orders are coprime and assume that q ranges over the set of integers the aim of this paper is to establish an isomorphism of functors w g q w h q w g h q where w g q denotes the q deformed witt burnside ring functor of g introduced in y t oh q deformation of witt burnside rings math z 207 1 2007 151 191 to do this we first establish an isomorphism of functors b g q b h q b g h q where b g q denotes the q deformed burnside ring functor of g which was also introduced in y t oh q deformation of witt burnside rings math z 207 1 2007 151 191 as an application we derive a pseudo multiplicative property of the q mobius function associated to the lattice of open subgroups of the direct sum of g and h

continuity properties of the lower spectral radius 2014 london mathematical society the lower spectral radius or joint spectral subradius of a set of real d times d matrices is defined to be the smallest possible exponential growth rate of long products of matrices drawn from that set the lower spectral radius arises naturally in connection with a number of topics including combinatorics on words the stability of linear inclusions in control theory and the study of random cantor sets in this article we apply some ideas originating in the study of dominated splittings of linear cocycles over a dynamical system to characterize the points of continuity of the lower spectral radius on the set of all compact sets of invertible d times d matrices as an application we exhibit open sets of pairs of 2 times 2 matrices within which the analogue of the lagarias wang finiteness property for the lower spectral radius fails on a residual set and discuss some implications of this result for the computation of the lower spectral radius

the morse spectrum of linear flows on vector bundles for a linear flow on a vector bundle e s a spectrum can be defined in the following way for a chain recurrent component m on the projective bundle pe consider the exponential growth rates associated with finite time e t chains in m and define the morse spectrum mo m over m as the limits of these growth rates as e 0 and t the morse spectrum mo of is then the union over all componentsm pe this spectrum is a synthesis of the topological approach of selgrade and salamon zehnder with the spectral concepts based on exponential growth rates such as the oseledec spectrum or the dichotomy spectrum of sacker sell it turns out that mo contains all lyapunov exponents of for arbitrary initial values and the mo m are closed intervals whose boundary points are actually lyapunov exponents using the fact that is cohomologous to a subflow of a smooth linear flow on a trivial bundle one can prove integral representations of all morse and all lyapunov exponents via smooth ergodic theory a comparison with other spectral concepts shows that in general the morse spectrum is contained in the topological spectrum and the dichotomy spectrum but the spectral sets agree if the induced flow on the base space is chain recurrent however even in this case the associated subbundle decompositions of e may be finer for the morse spectrum than for the dynamical spectrum if one can show that the closure of the floquet spectrum i e the lyapunov spectrum based on periodic trajectories in pe agrees with the morse spectrum then one obtains equality for the floquet the entire oseledec the lyapunov and the morse spectrum we present an example flows induced by c vector fields with hyperbolic chain recurrent components on the projective bundle where this fact can be shown using a version of bowen s shadowing lemma

exponential dichotomy integral separation and diagonalizability of linear systems of ordinary differential equations abstract this paper is concerned with linear systems of ordinary differential equations a criterion for integral separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for integral separation and the new result that an upper triangular system on a half line is integrally separated if and only if the system corresponding to its diagonal is we then show that a diagonal system on a half line is integrally separated if and only if a certain perturbed system is diagonalizable using this result we are able to deduce that the interior of the set of diagonalizable systems on a half line is the set of systems with integral separation and that the interior for the whole line is the set of systems which are integrally separated on both half lines

nonuniform hyperbolicity global dominated splittings and generic properties of volume preserving diffeomorphisms we study generic volume preserving diffeomorphisms on compact manifolds we show that the following property holds generically in the c1 topology either there is at least one zero lyapunov exponent at almost every point or the set of points with only nonzero exponents forms an ergodic component moreover if this nonuniformly hyperbolic component has positive measure then it is essentially dense in the manifold that is it has a positive measure intersection with any nonempty open set and there is a global dominated splitting for the proof we establish some new properties of independent interest that hold cr generically for any r 1 namely the continuity of the ergodic decomposition the persistence of invariant sets and the l1 continuity of lyapunov exponents

on the error in qr integration an important change of variables for a linear time varying system dot x a t x t ge 0 is that induced by the qr factorization of the underlying fundamental matrix solution x qr with q orthogonal and r upper triangular with positive diagonal to find this change of variable one needs to solve a nonlinear matrix differential equation for q practically this means finding a numerical approximation to q by using some appropriate discretization scheme whereby one attempts to control the local error during the integration our contribution in this work is to obtain global error bounds for the numerically computed q these bounds depend on the local error tolerance used to integrate for q and on structural properties of the problem itself but not on the length of the interval over which we integrate this is particularly important since in principle q may need to be found on the half line t ge 0

a characterization of exponential separation in terms of reducibility for linear difference equations abstract the main aim of this paper is to show that a linear difference equation is exponentially separated if and only if all neighboring equations are reducible

cumulant methods for analysing independence of r way contingency tables and goodness of fit frequency data cumulant methods for analysisng the independence of r way contingency tables and goodness of fit frequency data are presented these methods are based on the exact first three cumulants of both the classical pearson chi squared statistic and a recent advantageous modification of this statistic these approximate methods provide major distributional improvements over existing asymptotic methods

exact sample size needed to detect dependence in 2 2 2 tables summary in the georgia centenarian study poon et al exceptional longevity 2006 centenarian cases and young controls are classified according to three categories age ethnic origin and single nucleotide polymorphisms snps of candidate longevity genes where each factor has two possible levels here we provide methodologies to determine the minimum sample size needed to detect dependence in 2 2 2 tables based on fisher s exact test evaluated exactly or by markov chain monte carlo mcmc assuming only the case total l and the control total n are known while our mcmc method uses serial computing parallel computing techniques are employed to solve the exact sample size problem these tools will allow researchers to design efficient sampling strategies and to select informative snps we apply our tools to 2 2 2 tables obtained from a pilot study of the georgia centenarians study and the sample size results provided important information for the subsequent major study a comparison between the results of an exact method and those of a mcmc method showed that the mcmc method studied needed much less computation time on average 10 16 times faster on average for situations examined with s e 2 60 but its sample size results were only valid as a rule for larger sample sizes in the hundreds

necklace rings and logarithmic functions in this paper we develop the theory of the necklace ring and the logarithmic function regarding the necklace ring we introduce the necklace ring functor nr from the category of special rings into the category of special rings and then study the associated adams operators as far as the logarithmic function is concerned we generalize the results in bryant s paper free lie algebras and formal power series j algebra 253 1 2002 167 188 to the case of graded lie super algebras with a group action by applying the euler poincare principle

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

exponential separation exponential dichotomy and spectral theory for linear systems of ordinary differential equations abstract this paper is concerned with linear nonautonomous systems of ordinary differential equations a criterion for exponential separation in terms of exponential dichotomy is given as corollaries we obtain the roughness theorem for exponential separation and the new result that an upper triangular system on a half line is exponentially separated if and only if the system corresponding to its diagonal is a minimal decomposition into exponentially separated subspaces is defined it turns out that it is in general finer than the sacker sell spectral decomposition but that the two decompositions coincide for almost periodic systems

conformality for a robust class of non conformal attractors in this paper we investigate the hausdorff dimension of limit sets of anosov representations in this context we revisit and extend the framework of hyperconvex representations and establish a convergence property for them analogue to a differentiability property as an application of this convergence we prove that the hausdorff dimension of the limit set of a hyperconvex representation is equal to a suitably chosen critical exponent in the appendix in collaboration with m bridgeman we extend a classical result on the hessian of the hausdorff dimension on purely imaginary directions

non uniformly hyperbolic diffeomorphisms derived from the standard map we prove that the system resulting from coupling the standard map with a fast hyperbolic system is robustly non uniformly hyperbolic

aperiodic rings necklace rings and witt vectors in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 metropolis and rota show that the cyclotomic identity provides a linkage between the necklace ring and the ring of witt vectors in this paper we study this linkage let a be the class of rings a with the property that the additive group of a is torsion free we will prove that the ring of witt vectors over a for any a a is isomorphic to a suitably defined subring of the necklace ring of the rationalization a q of a alternatively to a subring of the aperiodic ring of a a major part of this paper is concerned with a development of the properties and relationships involving the aperiodic ring which parallels the treatment of the necklace ring given in n metropolis and gian carlo rota witt vectors and the algebra of necklaces adv in math 50 1983 95 125 in addition we translate the combinatorial proof of the cyclotomic identity given by metropolis and rota n metropolis and gian carlo rota the cyclotomic identity contemp math 34 1984 19 27 into the language of species in the sense of joyal thereby making clear the naturality of the identity

exponential dichotomy on the real line svd and qr methods abstract in this work we show when and how techniques based on the singular value decomposition svd and the qr decomposition of a fundamental matrix solution can be used to infer if a system enjoys or not exponential dichotomy on the whole real line

a functorial property of nested witt vectors for coprime truncation sets m n subset of n we establish an isomorphism of functors w n circle w m similar or equal to w mn where w n a denotes the ring of n witt vectors over a ring a further we note that this isomorphism can under certain restrictions on a be expressed in terms of artin hasse exponentials c 2002 elsevier science usa all rights reserved

almost all cocycles over any hyperbolic system have nonvanishing lyapunov exponents we prove that for any s 0 the majority of c s linear cocycles over any hyperbolic uniformly or not ergodic transformation exhibit some nonzero lyapunov exponent this is true for an open dense subset of cocycles and actually vanishing lyapunov exponents correspond to codimension 1 this open dense subset is described in terms of a geometric condition involving the behavior of the cocycle over certain heteroclinic orbits of the transformation

continuity of the lyapunov exponent for quasiperiodic operators with analytic potential we study regularity properties of the lyapunov exponent l of one frequency quasiperiodic operators with analytic potential under no assumptions on the diophantine class of the frequency we prove joint continuity of l in frequency and energy at every irrational frequency

conditions for kinematical similarity of a linear difference equation to a block upper triangular equation abstract in this paper we consider a linear difference equation in which the coefficient matrix is variable and depends on a positive parameter under some conditions we prove that the above equation is kinematically similar to a block upper triangular equation

the burnside ring of profinite groups and the witt vector construction abstract the classical construction of witt vectors can be understood as a special case of a construction which can be defined relative to any given profinite group g in terms of its completed burnside ring lde g it specializes to the classical situation in case g is infinite cyclic

the burnside ring of the infinite cyclic group and its relations to the necklace algebra λ rings and the universal ring of witt vectors abstract it is shown that well known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the burnside ring of the infinite cyclic group on the one hand and grothendieck s ring of formal power series with constant term 1 as well as the universal ring of witt vectors on the other hand

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