equivariant deformations of the affine multicone over a flag variety we prove that the invariant hilbert scheme parameterising the equivariant deformations of the affine multicone over a flag variety is under certain hypotheses an affine space more specifically we obtain that the isomorphism classes of equivariant deformations of such a multicone are in correspondence with the orbits of a well determined wonderful variety

the total coordinate ring of a wonderful variety abstract we study the cone of effective divisors and the total coordinate ring of wonderful varieties with applications to their automorphism group we show that the total coordinate ring of any spherical variety is obtained from that of the associated wonderful variety by a base change of invariant subrings

entropy formula for random transformations we exhibit random strange attractors with random sinai bowen ruelle measures for the composition of independent random diffeomorphisms

dimension formula for random transformations we consider compositions of random diffeomorphisms and show that the dimension of sample measures equals lyapunov dimension as conjectured in the nonrandom case by yorke et al

on the coordinate ring of spherical conjugacy classes let g be a simple algebraic group over an algebraically closed field k of characteristic zero and mathcal o be a spherical conjugacy class of g we determine the decomposition of the coordinate ring k mathcal o of mathcal o into simple g modules

pesin s entropy formula for systems between c 1 and c 1 alpha in this article we give a new observation of pesin s entropy formula motivated from mane s proof of ergod theory dyn sys 1 95 102 1981 let m be a compact riemann manifold and f m rightarrow m be a c 1 diffeomorphism on m if mu is an f invariant probability measure which is absolutely continuous relative to lebesgue measure and nonuniformly h ddot text o lder continuous see definition 1 1 then we have pesin s entropy formula i e the metric entropy h mu f satisfies begin aligned h mu f int sum lambda i x 0 lambda i x d mu end aligned where lambda 1 x ge lambda 2 x ge cdots ge lambda dim m x are the lyapunov exponents at x with respect to mu nonuniformly h ddot text o lder continuous is a new notion from probabilistic perspective weaker than c 1 alpha

simple immersions of wonderful varieties let g be a semisimple connected linear algebraic group over and x a wonderful g variety we study the possibility of realizing x as a closed subvariety of the projective space of a simple g module we describe the wonderful varieties having this property as well as the linear systems giving rise to such immersions we also prove that any ample line bundle on a wonderful variety is very ample

an introduction to wonderful varieties with many examples of type f4 we give an introduction to the theory of wonderful g varieties with many examples when g is simple of type f4 we present results and open problems about these varieties on their classification on their isotropy groups on morphisms between them and on their relations with the representation theory of g

some amazing properties of spherical nilpotent orbits let g be a simple algebraic group defined over an algebraically closed field k of characteristic zero write g for its lie algebra let x g be a nilpotent element and g x g the corresponding nilpotent orbit the maximal number m such that adx 6 0 is called the height of x or of g x denoted ht x recall that an irreducible g variety x is called g spherical if a borel subgroup of g has an open orbit in x it was shown in pa1 that g x is g spherical if and only if adx 0 this means that the spherical nilpotent orbits are precisely the orbits of height 2 and 3 unfortunately whereas my proof for the orbits of height 2 and height 4 was completely general the argument for the orbits of height 3 explicitly used their classification in this paper we give a proof of sphericity that does not rely on the classification of nilpotent orbits see theorem 3 3 we begin with some properties of invariants of symplectic representations for instance we prove that 1 if h sp v is an irreducible representation without nonconstant invariants then h sp v and 2 if h has non constant invariants then it has an invariant of degree 4 applying these results to nilpotent orbits we prove that the centraliser zg x has a rather specific structure whenever ht x is odd from this description we deduce a conceptual proof of sphericity in case ht x 3 as another application we compute the index of zg x it will be shown that ind zg x rk g if ht x 3 this confirms elashvili s conjecture for such x see pa5 sect 3 about this conjecture in section 4 we prove that if the highest root of g is fundamental then g always has a specific orbit of height 3 which is denoted by o this orbit satisfies several arithmetical relations namely if g 3 i 3 g i is the z grading associated with an sl2 triple containing e o then dim g 3 2 and dim g 1 2 dim g 2 furthermore the weighted dynkin diagram of o can explicitly be described let be the unique simple root that is not orthogonal to and let i be all simple roots adjacent to on the dynkin diagram of g then one has to put 1 at all i s and 0 at all other simple roots theorem 4 5 it is curious that these properties of o enables us to give an intrinsic construction of g2 grading in each simple g whose highest root is fundamental

piecewise linear maps liapunov exponents and entropy abstract let l a f a x x is a partition of 0 1 be a class of piecewise linear maps associated with a transition matrix a in this paper we prove that if f a x l a then the liapunov exponent x of f a x is equal to a measure theoretic entropy h m a x of f a x where m a x is a markov measure associated with a and x the liapunov exponent and the entropy are computable by solving an eigenvalue problem and can be explicitly calculated when the transition matrix a is symmetric moreover we also show that max x x max x h m a x log 1 where 1 is the maximal eigenvalue of a

the metric entropy of diffeomorphisms part i characterization of measures satisfying pesin s entropy formula soit m une variete de riemann compacte soit f m m un diffeomorphisme et soit m une mesure de probabilite de borel f invariante sur m on identifie les mesures pour lesquelles l inegalite de margulis et ruelle qui relie l entropie aux exposants de lyapunov atteint l egalite on demontre une formule valable pour toutes les mesures invariantes

heckman kostant and steinberg formulas for symplectic manifolds abstract let m be a compact symplectic manifold on which a compact connected lie group k acts in a hamiltonian fashion in part i we derive a formula for j m which generalizes the heckman formula for co adjoint orbits here j is the moment map associated with the action of k on m and m the liouville measure on m in part ii we derive a quantum analogue of this formula which extends to the symplectic setting classical multiplicity formulas of kostant and steinberg

characteristic exponents and invariant manifolds in hilbert space the multiplicative ergodic theorem and the construction almost everywhere of stable and unstable manifolds pesin theory are extended to differentiable dynamical systems on hilbert manifolds under some compactness assumptions the results apply to partial differential equations of evolution and also to non invertible maps of compact manifolds

on the generalized linear response functions abstract to discuss the response of a system to an electromagnetic field the minimal coupling hamiltonian is assumed the quadratic term in the potential is kept the kubo formalism for perturbation is applied and perturbations to all orders in the internal fields are considered however only the first order perturbation for the external probe is taken thus the linear response function is calculated the inclusion of the quadratic term contributes terms to this function due to these terms the dyson like propagator equation is corrected this equation is developed to obtain the different susceptibilities it is found that the contributions from the density current and current current propagators must be taken into account in addition to the density density propagator when the polarizability is considered to obtain the correct result the polarizability of the system is then calculated in two approximations when the quadrupole contribution is considered the number of propagators increases they are given through a coupled set of generalized dyson like equations the different fields are defined and discussed and the interplay between them is manifested the simplest coupled set is considered and developed to give two coupled integral equations in an extension to the classical integral equation for the electric polarization field they are solved for a finite crystal and the refractive index is calculated for the propagating waves when the probing field is light the phenomena of birefringence and optical rotation are discussed through simplified models the extinction theorem is found to hold in a generalized form hence the dependence on the geometry of the system is discussed the effects of the quadrupole terms on these phenomena are shown and discussed some other optical phenomena and higher multipole contributions are briefly discussed

a non linear response theory ii generalised dispersion relations abstract the non linear response function approach is used to discuss the relationship in the non linear regime between the different polarisation fields and the impinging field the dispersion relation is obtained in a generalised form the relationship between the impinging field and the transmitted field is derived these relations generalise earlier results obtained in the electric dipole approximation

automorphisms root systems and compactifications of homogeneous varieties let g be a complex semisimple group and let h c g be the group of fixed points of an involutive automorphism of g then x g h is called a symmetric variety in cp de concini and procesi have constructed an equivariant compactification x which has a number of remarkable properties some of them being i the boundary is the union of divisors d1 dr ii there are exactly 2r orbits their closures are the intersections di1 n ndi even schematically in particular there is only one closed orbit iii in case g is of adjoint type all orbit closures are smooth it is called the wonderful embedding of x or a complete symmetric variety and is the foundation for most deeper results about x independently luna and vust developed in lv a general theory of equivariant compactifications of homogeneous varieties under a connected reductive group g in particular they realized the reason which makes symmetric varieties behave so nicely a borel subgroup b has an open dense orbit in g h varieties with this property are called spherical luna and vust were able to describe all equivariant compactifications of them in terms of combinatorial data very similar to torus embeddings which are actually a special case they obtained in particular that every spherical embedding has only finitely many orbits nevertheless the reason for the existence of a compactification with properties i iii remained mysterious then brion and pauer established a relation with the automorphism group they proved in bp a spherical variety x g h possesses an equivariant compactification with exactly one closed orbit if and only if autg x ng h h is finite in this case there is a unique one which dominates all others the wonderful compactification x they also showed that the orbits of x correspond to the faces of a strictly convex polyhedral cone z then properties i and ii above are equivalent to z being simplicial this fact is much deeper and was proved by brion in brl in fact he showed much more let f be the set of characters of b which are the characters of a rational b eigenfunction on x this is a finitely generated free abelian group then the cone z is a subset of the real vector space hom f r brion showed that there is a finite reflection group wx acting on f such that z is one of its weyl chambers in case of a symmetric variety wx is its little weyl group

a non linear response theory i general formulation abstract we present a generalisation of response theory to the non linear regime of the interaction of an electromagnetic field with an extended system of two level atoms the different propagators together with the equations for the fields with their interdependence are presented the optical extinction theorem is further generalised the formulated theory is utilised later to derive the dispersion relations relevant to the non linear effect of optical multistability

entropy formula of pesin type for noninvertible random dynamical systems in this paper we consider random dynamical systems abbreviated henceforth as rds s generated by compositions of random endomorphisms maybe noninvertible and with singularities of class c 2 of a compact manifold entropy formula of pesin type is proved for such rds s under some absolute continuity conditions on the associated invariant measures

uniqueness property for spherical homogeneous spaces let g be a connected reductive group recall that a homogeneous g space x is called spherical if a borel subgroup b g has an open orbit on x to x one assigns certain combinatorial invariants the weight lattice the valuation cone and the set of b stable prime divisors we prove that two spherical homogeneous spaces with the same combinatorial invariants are equivariantly isomorphic further we recover the group of g equivariant automorphisms of x from these invariants

random perturbations of axiom a basic sets in this paper we study small random diffeomorphism type perturbations of an axiom a basic set by means of the structural stability of such a basic set with respect to time dependent perturbations and by means of the markov partition of the basic set we apply the thermodynamic formalism of random subshifts of finite type to this situation obtaining some ergodic theoretic results concerning equilibrium states

on the spectra of randomly perturbed expanding maps we consider small random perturbations of expanding and piecewise expanding maps and prove the robustness of their invariant densities and rates of mixing we do this by proving the robustness of the spectra of their perron frobenius operators

primitive wonderful varieties we complete the classification of wonderful varieties initiated by d luna we review the results that reduce the problem to the family of primitive varieties and report the references where some of them have already been studied finally we analyze the rest case by case

manin s conjecture for certain spherical threefolds abstract we prove manin s conjecture on the asymptotic behavior of the number of rational points of bounded anticanonical height for a spherical threefold with canonical singularities and two infinite families of spherical threefolds with log terminal singularities moreover we show that one of these families does not satisfy a conjecture of batyrev and tschinkel on the leading constant in the asymptotic formula our proofs are based on the universal torsor method using brion s description of cox rings of spherical varieties

dimension theory for invariant measures of endomorphisms we establish the exact dimensional property of an ergodic hyperbolic measure for a c 2 non invertible but non degenerate endomorphism on a compact riemannian manifold without boundary based on this we give a new formula of lyapunov dimension of ergodic measures and show it coincides with the dimension of hyperbolic ergodic measures in a setting of random endomorphisms our results extend several well known theorems of barreira et al ann math 149 755 783 1999 and ledrappier and young commun math phys 117 4 529 548 1988 for diffeomorphisms to the case of endomorphisms

wonderful subgroups of reductive groups and spherical systems abstract let g be a semisimple complex algebraic group and h g a wonderful subgroup we prove several results relating the subgroup h to the properties of a combinatorial invariant s of g h called its spherical system it is also possible to consider a spherical system s as a datum defined by purely combinatorial axioms and under certain circumstances our results prove the existence of a wonderful subgroup h associated with s as a byproduct we reduce for any group g the proof of the classification of wonderful g varieties known as the luna conjecture to its verification on a small family of cases called primitive