We show that if G is a connected semisimple Lie group with finite center, and if G admits a locally faithful, non-tame action by isometries of a connected Lorentz manifold, then $\mathfrak{g}$ has an ideal which is Lie algebra isomorphic to $\mathfrak{sl}_2(\mathbb{R})$. We also analyze the collection of connected Lie groups G admitting a free action by isometries of a connected Lorentz manifold such that the action is properly ergodic with respect to the Lorentz volume form.

Given a unitary representation u of the locally compact group G on the Hilbert space H, we investigate the notion of infinite tensor power of u. Then we apply the results to study covariance of the associated canonical action of G on the generalized Cuntz algebra $\mathcal{O}_H$ in the GNS representation of (the gauge-invariant extension of) some pure quasi-free state. We examine in detail the case of a measure-preserving action of G on a measure space X. In this case, covariance is (almost) equivalent to the existence of a G-invariant state on $L^\infty(X)$.

Nous donnons des invariants complets pour l'équivalence topologique de champs de vecteurs, en dimension 3, au voisinage d'une connexion (par des variétés invariantes de dimension 1) entre des selles possédant des valeurs propres complexes.

We prove that potentials with summable variations on topologically transitive countable Markov shifts have at most one equilibrium measure. We apply this to multidimensional piecewise expanding maps using their Markov diagrams.

We consider quadratic skew-products over angle-doubling of the circle and prove that they admit positive Lyapunov exponents almost everywhere and an absolutely continuous invariant probability measure. This extends corresponding results of M. Viana and J. F. Alvès for skew-products over the linear strongly expanding map of the circle.

Let $\mathcal{N}$ be a 2n-dimensional manifold equipped with a symplectic structure $\omega$ and $\Lambda(\mathcal{N})$ be the Lagrangian Grassmann bundle over $\mathcal{N}$. Consider a flow $\phi^t$ on $\mathcal{N}$ that preserves the symplectic structure and a $\phi^t$-invariant connected submanifold $\Sigma$. Given a continuous section $\Sigma\to\Lambda(\mathcal{N})$, we can associate to any finite $\phi^t$-invariant measure with support in $\Sigma$, a quantity, The asymptotic Maslov index, which describes the way Lagrangian planes are asymptotically wrapped in average around the Lagrangian Grassmann bundle. We pay particular attention to the case when the flow is derived from an optical Hamiltonian and when the invariant measure is the Liouville measure on compact energy levels. The situation when the energy levels are not compact is discussed.

We classify all factors of the Cartesian product of any two non-singular type III$_\lambda$, $0<\lambda \leq 1$, or type II1 Chacón transformations, as well as the centralizer of finite Cartesian products of such transformations.

This self-contained paper is part of a series (Groups of homeomorphisms of one-manifolds I: actions of nonlinear groups. Preprint, 2001; Group actions on one-manifolds II: extensions of Hölder's Theorem. To appear in Trans. Amer. Math. Soc.) seeking to understand groups of homeomorphisms of manifolds in analogy with the theory of Lie groups and their discrete subgroups. Plante and Thurston proved that every nilpotent subgroup of Diff2(S1) is abelian. One of our main results is a sharp converse: Diff1(S1) contains every finitely generated, torsion-free nilpotent group.

We use matrices over formal power series to represent irreducible and positive recurrent Markov chains and identify a natural class of good finitary isomorphisms (magic word isomorphisms) as those arising from elementary matrix operations. This extends the positive K-theory framework which has been used for other classification problems in symbolic dynamics.

A measure-preserving transformation (respectively a topological system) is null if the metric (respectively topological) sequence entropy is zero for any sequence. Kushnirenko has shown that an ergodic measure-preserving transformation has a discrete spectrum if and only if it is null. We prove that for a minimal system this statement remains true modulo an almost one-to-one extension. It allows us to show that a scattering system is disjoint from any null minimal system. Moreover, some necessary conditions for a transitive non-minimal system to be null are obtained.

Localizing the notion of sequence entropy, we define sequence entropy pairs and show that there is a maximal null factor for any system. Meanwhile, we define a weaker notion, namely weak mixing pairs. It turns out that a system is weakly mixing if and only if any pair not in the diagonal is a sequence entropy pair if and only if the same holds for a weak mixing pair, answering a question in Blanchard et al (F. Blanchard, B. Host and A. Maass, Topological complexity. Ergod. Th. & Dynam. Sys., 20 (2000), 641–662). For a group action we give a direct proof of the fact that the factor induced by the smallest invariant equivalence relation containing the regionally proximal relation is equicontinuous. Furthermore, we show that a non-equicontinuous minimal distal system is not null.

Assuming that two ergodic automorphisms T and R are disjoint, we prove that an ergodic extension $\tilde{T}$ of T remains disjoint with R when, in a representation of $\tilde{T}$ as a skew product over T, the fiber automorphisms belong to a mildly mixing action of a second countable locally countable abelian (LCA) group and the corresponding cocycle with values in this group is ergodic. This allows us to show that the class of ergodic automorphisms which are disjoint from all weakly mixing automorphisms is not closed under taking ergodic joinings, thereby solving a problem stated by E. Glasner and B. Weiss.

A symbolic matrix system and a $\lambda$-graph system for a subshift are generalizations of a symbolic matrix and a $\lambda$-graph (= finite symbolic matrix); for a sofic shift respectively. In [K. Matsumoto. (Presentations of subshifts and their topological conjugacy invariants. Doc. Math. 4 (1999), 285–340)], two equivalence relations between symbolic matrix systems were formulated, properly strong shift equivalence and strong shift equivalence, which are related to topological conjugacy of subshifts. In this paper, we first prove that these two equivalence relations are equivalent to each other for a large class of symbolic matrix systems. State splitting of a $\lambda$-graph system is introduced as a generalization of state splitting of a finite directed graph. State splitting gives rise to a strong shift equivalence for the corresponding symbolic matrix systems. We prove that two subshifts are topologically conjugate if and only if the canonical $\lambda$-graph system for one subshift is obtained by a finite sequence of merged state splittings and merged state amalgamations from the canonical $\lambda$-graph system for the other subshift.

We prove that a generic C1 vector field on a closed 3-manifold either has infinitely many sinks or sources or else is singular Axiom A without cycles. Singular Axiom A means that the non-wandering set of the vector field has a decomposition into compact invariant sets, each being either a hyperbolic basic set or a singular hyperbolic attractor (like the Lorenz-like ones) or a singular hyperbolic repeller. An attractor is a transitive set which attracts all nearby future orbits, and a repeller is an attractor for the time-reversed flow. Our result implies that generic C1 vector fields on closed 3-manifolds do exhibit either attractors or repellers.

In this paper we introduce two notions of measure-theoretical entropy for covers and one notion of topological entropy for open covers. These notions extend the classical concept of measure-theoretical entropy of measurable partitions to measurable covers. Using these notions we prove a local variational principle for any given open cover.

Let $\Sigma = (X, \sigma)$ be a topological dynamical system where $\sigma$ is a homeomorphism in a compact metric space X. Denote by $A(\Sigma)$ the transformation group C*-algebra associated with this system. We describe the shrinking steps of the non-wandering set $\Omega(\sigma)$ down to the Birkhoff center (depth of the center) in terms of a composition series of the particular ideal of type 1 in $A(\Sigma)$, which corresponds to the center. The result implies C*-algebraic characterizations of the cases where the depth is 0 and 1. We also give the structure of dynamical systems for which the associated C*-algebras become algebras with continuous traces.