Multi Ping-Pong and an Entropy Estimate in Groups

We show that each weakly mixing group action on a dendrite must have a ping-pong game, and has positive geometric entropy when the acting group is finitely generated. As a corollary, we prove that no nilpotent group action on a dendrite is weakly mixing. At last, we show that each dendrite admits no chaotic group actions.

Let Z1, Z2,... be a sequence of i.i.d. random transformations (possibly discontinuous) of a compact metric space M, and let E denote the space of normalized mass distributions on M. Given μ in E, let μ n denote the random measure μ°(Z n °...°Z1)−1 (when well-defined). We construct the transition probability P of the E-valued Markov chain (μ n ), and give a necessary and sufficient condition for P to have a unique invariant measure concentrated on the degenerate mass distributions. Convergence to ‘statistical equilibrium’ of the associated discrete-time stochastic flow is investigated.

We solve a longstanding problem of classification of compact homogeneous spaces with invariant symplectic structures. We also give a splitting conjecture on compact homogeneous spaces with symplectic structures (which are not necessarily invariant under the group action) that makes the classification of this kind of manifolds possible.

The main topic of discussion in this survey is the geometric and metric properties of geodesic and horocycle flows on surfaces of constant curvature and their multidimensional analogues, the so-called G-induced actions (that is, restrictions to a suitable subgroup F of the natural action of G on a homogeneous space of it). One usually takes as F one-parameter or cyclic subgroups of G and the horospherical subgroups corresponding to them (the details are given in Sect. 1). However, in the course of the exposition of our main topic we sometimes mention results relating to the case when F is an n-parameter subgroup of G or a lattice. This class of dynamical systems consists precisely of group actions admitting an extension to a transitive action of a Lie group. We shall not be considering in our survey (or, at any rate, not systematically) the following themes which are close to our main subject matter: Topological dynamics and metric properties of general group actions (in particular, the work of Zimmer on semisimple group actions (Zimmer 1984); Group automorphisms; Affine transformations of homogeneous spaces; G-induced flows on spaces with an infinite invariant measure. The papers (Hopf 1939), (Sullivan 1981) and the recent book (Nicholls 1989) are devoted to the connections between G-induced flows (and also geodesic flows) and the theory of discrete groups of motions of a hyperbolic space; The actions of subgroups on the double cosets related to geodesic (for n > 2) and horocycle (for n > 3) flows on an n-dimensional manifold of constant negative curvature (see (Gel’fand and Fomin 1952) and, in particular, (Mautner 1957)). Flows of this type in somewhat more general form differing from geodesic and horocycle flows have not been investigated.

We present a complete study of rigidity for actions of quantum permutation groups on compact spaces. In particular, we describe explicitly all actions of the quantum permutation groups on classical compact spaces, and show that the defining action is the only non-trivial ergodic one. We then extend these results to all easy quantum groups associated with non-crossing partitions.

For the analysis and description of textures the orientation distribution function f(g) is used. Orientations g as points in “orientation space” are usually characterized by the Euler angles φ1, Θ, φ2. The symmetries of the function f(g) result from the symmetries of the crystal and those of the specimen.The symmetries in orientation space induced by both the point group symmetry of the crystal and of the specimen are analysed. Any combination of crystal and specimen symmetries yield a group of linear transformations in orientation space; this group corresponds to one of the 230 space groups.If the point group of the crystal or the point group of the specimen belong to the cubic system the resulting group of linear transformations does not contain all induced symmetries.Tables with the induced space groups are given for all combinations of crystal and specimen symmetries.

We have shown in an earlier work that, assuming a particular class of equations of state, the Euler equations of one-dimensional gas flow are invariant under an SU(3) group of transformations, and in fact admit of a Lie group of symmetry of infinite order; they, therefore, possess an infinite number of conservation laws. We show in the present work that the SU(3) symmetrical formalism still brings about tremendous simplification and analytical order in the most general case where the equation of state is arbitrary. The six characteristic equations assume a vector form and relate two conjugate, three-dimensional vectors U and X. The SU(3) symmetry is only broken to a minor extent through the occurrence of a multiplicative factor Γ in the equations. The conservation laws take the form of the Cauchy integrability condition for the elements of a traceless second rank tensor εij and, taken all together, form an SU(3) octet; in the most general case, however, there exist four conservation laws only (five if the gas is monatomic) as a result of the breaking of symmetry. Application of these results to the theory of self-similar flow is also discussed. Finally, we show the invariance of the equations of monatomic gas flow under Lorentz transformations in a three-dimensional Minkowski space; that raises the question of whether a geometrical relation may exist between the Minkowski light cones and characteristics.

for t E T and s E S. This definition can be restated in equivalent ways in terms of particular flows. A flow (X, T) is a transformation group [8] where X is a topological space and Tis the additive group of the reals. We designate the action of t E T on x eX by x x,. A point xeX is said to be almost periodic if for each neighborhood U of x there is a relatively dense set S c T such that x. E U if s E S. We let B be the Banach space of uniformly continuous bounded real valued functions of the reals. The norm of an element of B shall be the supremum of its absolute values. We let (B, T) be the flow defined by x -) x, where x,(s) = x(t + s). Uniformly almost periodic functions may be defined in any one of the three following ways. (1) A function x in B is uniformly almost periodic if x is an almost periodic point of (B, T). (2) A function x in B is uniformly almost periodic if its orbit closure in (B,T) is compact. (3) A function x in B is uniformly almost periodic if it may be extended to a continuous function on the Bohr compactification of the reals. The fact that these three statements are equivalent may be found in [3; 12]. For many notions of functions being recursive under translation the compactopen topology is more natural than the topology of uniform convergence. With this in mind we let C be the set of continuous real valued functions of the reals with the compact open topology. This topology is equivalent to the topology induced by the metric

The role of local actors in transforming the urban fringe

The nonlinear realizations of the chiral group SU(2)⊗SU(2) are studied from a geometric point of view. The three-dimensional nonlinear realization, associated with the pion field, is considered as a group of coordinate transformations in a three-dimensional isospin space of constant curvature, leaving invariant the line element. Spinor realizations in general coordinates are constructed by combined coordinate-spin-space transformations in analogy to Pauli's method for spinors in general relativity. The description of vector mesons and possible chiral-invariant Lagrangians, which yield the various nonlinear models in specific frames of general coordinates, are discussed.

This paper is an expanded and updated version of the preprint arXiv:math/0406499. It includes a more detailed description of the basics of the theory of Cherednik and Hecke algebras of varieties started in arXiv:math/0406499, as well as a new Section 4, which reviews the developments in this theory since 2004 with references to the relevant literature. Let $G$ be a finite group of linear transformations of a finite dimensional complex vector space $V$. To this data one can attach a family of algebras $H_{t,c}(V,G)$, parametrized by complex numbers $t$ and conjugation invariant functions $c$ on the set of complex reflections in $G$, which are called rational Cherednik algebras. These algebras have been studied for over 15 years and revealed a rich structure and deep connections with algebraic geometry, representation theory, and combinatorics. In this paper, we define global analogs of Cherednik algebras, attached to any smooth algebraic or analytic variety $X$ with a finite group $G$ of automorphisms. We show that many interesting properties of Cherednik algebras (such as the PBW theorem, universal deformation property, relation to Calogero-Moser spaces, action on quasiinvariants) still hold in the global case, and give several interesting examples. Then we define the KZ functor for global Cherednik algebras, and use it to define (in the case $\pi_2(X)\otimes \Bbb Q=0$) a flat deformation of the orbifold fundamental group of the orbifold $X/G$, which we call the Hecke algebra of $X/G$. This includes usual, affine, and double affine Hecke algebras for Weyl groups, Hecke algebras of complex reflection groups, as well as many new examples.

Subalgebras of the Lie algebra AC(2, 2) of the group C(2, 2), which is the group of conformal transformations of the pseudo-Euclidean space R 2,2, are studied. All subalgebras of the algebra AC(2, 2) are splitted into three classes, each of those is characterized by the isotropic rank 0, 1, or 3. We present the complete classification of the class 0 subalgebras and also of the class 3 subalgebras which satisfy an additional condition. The results obtained are applied to the reduction problem for the d’Alembert equation □u + λu 3 = 0 in the space R 2,2.

Let G be a group and let k be a field. A K-representation ρ of G is a homomorphism of G into the group of non-singular linear transformations of some finite-dimensional vector space V over k. Let K be the field of fractions of the symmetric algebra S(V) of V, then G acts naturally on K as k-automorphisms. There is a natural inclusion map V→K, so we view V as a k-subvector space of K. Let v1, v2, · · ·, vn be a basis for V, then K is generated by v1, v2, · · ·, vn over k as a field and these are algebraically independent over k, that is, K is a rational field over k with the transcendence degree n. All elements of K fixed by G form a subfield of K. We denote this subfield by KG.

Our concern is with the group of conformal transformations of a finite-dimensional real quadratic space of signature (p,q), that is one that is isomorphic to $ℝ^{p,q}$, the real vector space $ℝ^{p+q}$, furnished with the quadratic form $x^{(2)} = x · x =

This work is about examples of applications of group classifications for continuous (differential equations) and discrete models of flow in porous media. Difference schemes and discrete dynamical systems are considered as discrete models. Group classifications has been carried out on the basis of continuous groups (Lie groups) of transformations in n-dimensional Euclidean spaces. The example of explicit difference scheme for Buckley-Leverett equation has been demonstrated where the correspondence to one of classes of the obtained earlier classifications of discrete dynamical system has been shown. The example of fractal capillary network on the basis of Sierpinski triangle has been demonstrated for the group classification of discrete dynamical systems. The approach of continuous symmetry absence or presence proof has been demonstrated using classifications of discrete dynamical systems. Difference schemes with continuous symmetry groups for partial differential equations of parabolic type, which correspond to the equation of gas flow in porous media and Rapoport-Leas equation (generalized Buckley-Leverett equation), are considered. Examples of difference schemes for these cases are given and related to classes of the obtained earlier classifications. Applications of continuous symmetry groups for numerical solutions generations are demonstrated for these cases.