on the thermodynamic formalism for the gauss map we study the generalized transfer operator open image in new window of the gauss maptx 1 x mod 1 on the unit interval this operator which for 1 is the familiar perron frobenius operator oft can be defined for re 1 2 as a nuclear operator either on the banach spacea d of holomorphic functions over a certain discd or on the hilbert space open image in new window of functions belonging to some hardy class of functions over the half planeh 1 2 the spectra of open image in new window on the two spaces are identical on the space open image in new window is isomorphic to an integral operator open image in new window with kernel the bessel function mathfrak f 2 beta 1 2 sqrt st and hence to some generalized hankel transform this shows that open image in new window has real spectrum for real 1 2 on the spacea d the operator open image in new window can be analytically continued to the entire plane with simple poles at beta beta k 1 k 2 k 0 1 2 and residue the rank 1 operator mathcal n k f tfrac 1 2 1 k f k 0 from this similar analyticity properties for the fredholm determinant open image in new window of open image in new window and hence also for ruelle s zeta function follow another application is to the function zeta m beta sum limits n 1 infty n beta where n denotes the irrational n n n2 4 1 2 2 m extends to a meromorphic function in the plane with the only poles at 1 both with residue 1

the spectrum of the period doubling operator in terms of cycles the eigenvalue spectrum of the period doubling operator is evaluated both in terms of unstable periodic orbits and by a diagonalization at the operator cycles up to length ten yield the twelve leading eigenvalues direct diagonalization yields some 50 eigenvalues the feigenbaum 6 the hausdorff dimension and the escape rate for the period doubling repeller are evaluated to high accuracy in this letter we apply the technique for the extraction of correlation exponents of i to evaluation of the spectrum of the period doubling renormalization operator the basic idea of relating the feigenbaum constant 6 to the scaling or the presentation function is due to sullivan 2 and is implemented as an eigenvalue spectrum calcula tion in 3 51 we refer the reader to 3 41 for an introduction to the cycle expansions in general and their application to the evaluation of the stability the dimension and the escape rate of the period doubling repeller in particular the fredholm determinant evaluation method and the transfer operator diagonalization 11 used here are superior to the method of locating zeros of finite products of dynamical 6 functions 6 of 3 4 7 81 and in that sense the present letter supersedes the above references the repeller we study here is the non wandering set of the period doubling presenta tion function 9

indecomposable higher chow cycles on jacobians we construct some natural indecomposable elements of ch g j c 1 with trivial regulator and in particular prove that ch g ind j c 1 mb q is uncountable for c a generic curve or a generic hyperelliptic curve of genus g geq 3

expanding direction of the period doubling operator we prove that the period doubling operator has an expanding direction at the fixed point we use the induced operator a perron frobenius type operator to study the linearization of the period doubling operator at its fixed point we then use a sequence of linear operators with finite ranks to study this induced operator the proof is constructive one can calculate the expanding direction and the rate of expansion of the period doubling operator at the fixed point

on the chow group of zero cycles of a generalized kummer variety abstract for a generalized kummer variety x of dimension 2n we will construct for each 0 i n some co isotropic subvarieties in x foliated by i dimensional constant cycle subvarieties these subvarieties serve to prove that the rational orbit filtration introduced by voisin on the chow group of zero cycles of a generalized kummer variety coincides with the induced beauville decomposition from the chow ring of abelian varieties as a consequence the rational orbit filtration is opposite to the conjectural bloch beilinson filtration for generalized kummer varieties

motives and filtrations on chow groups in this paper sequential to sa we discuss a philosophy of bloch and beilinson concerning filtrations on chow groups of algebraic vareties which provides us with a good understanding on the structure of at least certain portions of chow groups let k be a fixed ground field and let c be the category of smooth projective schemes over k with a fixed suitable weil cohomology theory cf notations x in mathcal c rightsquigarrow h x mathop oplus limits i in z h i x

murre s conjectures and explicit chow kunneth projectors for varieties with a nef tangent bundle in this paper we investigate murre s conjectures on the structure of rational chow groups and exhibit explicit chow kiinneth projectors for some examples more precisely the examples we study are the varieties which have a nef tangent bundle for surfaces and threefolds which have a nef tangent bundle explicit chow kunneth projectors are obtained which satisfy murre s conjectures and the motivic hard lefschetz theorem is verified

a note on the artuso aurell cvitanovic approach to the feigenbaum tangent operator in this note we explain the rigorous mathematical arguments underlying some recent work of artuso aurell and cvitanovic on the feigenbaum tangent operator in particular we attempt to clarify the advantages of introducing zeta functions through the ideas of ruelle and grothendieck

contractions of filippov algebras we introduce in this paper the contractions gc of n lie or filippov algebras g and show that they have a semidirect structure as their n 2 lie algebra counterparts as an example we compute the nontrivial contractions of the simple an 1 filippov algebras by using the inonu wigner and the generalized weimar woods contractions of ordinary lie algebras we compare in the g an 1 simple case the lie algebras liegc the lie algebra of inner endomorphisms of gc with certain contractions lie g iw and lie g w w of the lie algebra lieg associated with g

new generalized poisson structures new generalized poisson structures are introduced by using suitable skew symmetric contravariant tensors of even order the corresponding jacobi identities are provided by conditions on these tensors which may be understood as cocycle conditions as an example we provide the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

the zero section of the universal semiabelian variety and the double ramification cycle we study the chow ring of the boundary of the partial compactification of the universal family of principally polarized abelian varieties ppav we describe the subring generated by divisor classes and we compute the class of the partial compactification of the universal zero section which turns out to lie in this subring our formula extends the results for the zero section of the universal uncompactified family the partial compactification of the universal family of ppav can be thought of as the first two boundary strata in any toroidal compactification of ag our formula provides a first step in a program to understand the chow groups of a g especially of the perfect cone compactification by induction on genus by restricting to the image of mg under the torelli map our results extend the results of hain on the double ramification cycle answering eliashberg s question

correspondences to abelian varieties i when s is an algebraic scheme and x s and y s proper schemes over s we define the notion of correspondences from x to y over s and when y s is a relative abelian scheme and x is a normal variety we give a charac terization for a correspondence from x to y over s to be a graph of some morphism x y over s which is a generalization of the result for classical correspondences in 4

on foundation of the generalized nambu mechanics we outline basic principles of a canonical formalism for the nambu mechanics a generalization of hamiltonian mechanics proposed by yoichiro nambu in 1973 it is based on the notion of a nambu bracket which generalizes the poisson bracket a binary operation on classical observables on the phase space to the multiple operation of higher ordern 3 nambu dynamics is described by the phase flow given by nambu hamilton equations of motion a system of ode s which involvesn 1 hamiltonians we introduce the fundamental identity for the nambu bracket a generalization of the jacobi identity as a consistency condition for the dynamics we show that nambu bracket structure defines a hierarchy of infinite families of subordinated structures of lower order including poisson bracket structure which satisfy certain matching conditions the notion of nambu bracket enables us to define nambu poisson manifolds phase spaces for the nambu mechanics which turn out to be more rigid than poisson manifolds phase spaces for the hamiltonian mechanics we introduce the analog of the action form and the action principle for the nambu mechanics in its formulation dynamics of loops n 2 dimensional chains for the generaln ary case naturally appears we discuss several approaches to the quantization of nambu mechanics based on the deformation theory path integral formulation and on nambu heisenberg commutation relations in the latter formalism we present an explicit representation of the nambu heisenberg relation in then 3 case we emphasize the role ternary and higher order algebraic operations and mathematical structures related to them play in passing from hamilton s to nambu s dynamical picture

on weight truncations in the motivic setting abstract using punctual gluing of t structures we construct an analogue of s morel s weight truncation functors for certain weight profiles in the setting of motivic sheaves as an application we construct a canonical motivic analogue of the intersection complex for an arbitrary threefold over any field k we are also able to recover certain invariants of singularities motivically

on the tautological ring of mg we prove that any product of tautological classes of g of degreed in the chow ring of g with rational coefficients vanishes ford g 2 and is proportional to the class of the hyperelliptic locus in degreeg 2

explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions abstract we consider transfer operators acting on spaces of holomorphic functions and provide explicit bounds for their eigenvalues more precisely if is any open set in c d and l is a suitable transfer operator acting on bergman space a 2 its eigenvalue sequence n l is bounded by n l a exp a n 1 d where a a 0 are explicitly given

action of correspondences on filtrations on cohomology and 0 cycles of abelian varieties we prove that given a symmetrically distinguished correspondence of a suitable complex abelian variety which includes any abelian variety of dimension at most 5 powers of complex elliptic curves etc that vanishes as a morphism on a certain quotient of its middle singular cohomology then it vanishes as a morphism on the deepest part of a particular filtration on the chow group of 0 cycles of the abelian variety as a consequence we prove that an automorphism of such an abelian variety that acts as the identity on a certain quotient of its middle singular cohomology acts as the identity on the deepest part of this filtration on the chow group of 0 cycles of the abelian variety as an application we prove that for the generalized kummer variety associated to a complex abelian surface and the automorphism induced from a symplectic automorphism of the complex abelian surface the automorphism of the generalized kummer variety acts as the identity on a certain subgroup of its chow group of 0 cycles

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

contractions of lie algebras generalized inonu wigner contractions versus graded contractions recently a new notion of contraction has been introduced the purely algebraically defined graded contractions that fall into two disjoint classes continuous and discrete our main result is that all continuous graded contractions can be realized by generalized inonu wigner contractions furthermore we give a complete characterization of the discrete graded contractions via certain higher order identities inherent to the grading group

intermediate extension of chow motives of abelian type abstract in this article we give an unconditional construction of a motivic analogue of the intermediate extension in the context of chow motives of abelian type our main application concerns intermediate extensions of chow motives associated to kuga families to the baily borel compactification of a shimura variety

correlations for pairs of closed geodesics in this article we consider natural counting problems for closed geodesics on negatively curved surfaces we present asymptotic estimates for pairs of closed geodesics the differences of whose lengths lie in a prescribed family of shrinking intervals related pair correlation problems have been studied in both quantum chaos and number theory

some new examples in the theory of quadratic forms we construct a 6 dimensional anisotropic quadratic form phi and a 4 dimensional quadratic form psi over some fieldf such that phi becomes isotropic over the function field f psi but every proper subform of phi is still anisotropic over f psi it is an example of non standard isotropy with respect to some standard conditions of isotropy for 6 dimensional forms over function fields of quadrics known previously besides of that we produce an 8 dimensional quadratic form phi with trivial determinant such that the index of the clifford invariant of phi is 4 but phi can not be represented as a sum of two 4 dimensional forms with trivial determinants using this we find a 14 dimensional quadratic form with trivial discriminant and clifford invariant which is not similar to a difference of two 3 fold pfister forms the proofs are based on computations of the topological filtration on the grothendieck group of certain projective homogeneous varieties to do these computations we develop several methods covering a wide class of varieties and being to our mind of independent interest

the selberg zeta function for convex co compact schottky groups we give a new upper bound on the selberg zeta function for a convex co compact schottky group acting on the hyperbolic space n 1 in strips parallel to the imaginary axis the zeta function is bounded by exp c s where is the dimension of the limit set of the group this bound is more precise than the optimal global bound exp c s n 1 and it gives new bounds on the number of resonances scattering poles of n 1 the proof of this result is based on the application of holomorphic l2 techniques to the study of the determinants of the ruelle transfer operators and on the quasi self similarity of limit sets we also study this problem numerically and provide evidence that the bound may be optimal our motivation comes from molecular dynamics and we consider n 1 as the simplest model of quantum chaotic scattering

chow groups are finite dimensional in some sense when s is a surface with pg s 0 mumford proved that its chow group a s is not finite dimensional in some sense in this paper we propose another definition of finite dimensionality for the chow groups using this new definition at least the chow group of some surface s with pg s 0 for example the product of two curves becomes finite dimensional the finite dimensionality of the chow groups follows from the finite dimensionality of the chow motives it turns out that the finite dimensionality of the chow motives is a very strong property for example we can prove bloch s conjecture representability of the chow groups of surfaces with pg s 0 under the assumption that the chow motive of s is finite dimensional

the schouten nijenhuis bracket cohomology and generalized poisson structures newly introduced generalized poisson structures based on suitable skew symmetric contravariant tensors of even order are discussed in terms of the schouten nijenhuis bracket the associated jacobi identities are expressed as conditions on these tensors the cohomological contents of which is given in particular we determine the linear generalized poisson structures which can be constructed on the dual spaces of simple lie algebras

dirac operators in tensor categories and the motive of quaternionic modular forms we define a motive whose realizations afford modular forms of arbitrary weight on an indefinite division quaternion algebra this generalizes work of iovita spiess to odd weights in the spirit of jordan livne it also generalizes a construction of scholl to indefinite division quaternion algebras and provides the first motivic construction of new subspaces of modular forms

on the chow motive of an abelian scheme with non trivial endomorphisms let x be an abelian scheme over a base variety s with endomorphism algebra d we prove that the relative chow motive r x s has a canonical decomposition as a direct sum of motives r whereruns over an explicitly determined finite set of irreducible representations of the group d op such that r seen as a functor from chow motives to d op representations is isotypic our decomposition refines the motivic decomposition of deninger and murre as well as beauville s decomposition of the chow group the second main result is that we construct a canonical generalized motivic lefschetz decomposition inspired by work of looijenga and lunts in 17 the role of sl2 in the classical theory is here replaced by a larger lie algebra generated by all lefschetz and lambda operators associated to non degenerate line bundles we construct an action of this larger lie algebra on r x s and deduce from this a generalized lefschetz decomposition we also give a precise structure result for the lefschetz components that arise as an application of our techniques we provide a positive answer to a question of claire voisin concerning the analogue for abelian varieties of a conjecture of beauville ams 2010 mathematics subject classification 14c15 14k05