We study the non-existence of KAM tori for quasi-integrable, analytic Lagrangians. Let ${\cal L}\colon {\bf T}^m\times{\bf R}^m \to {\bf R}$, ${\cal L} (Q,\dot{Q})=\tfrac{1}{2}\vert\dot{Q}\vert^2+h(Q)$ and let $\bar\omega\in{\bf R}^m$ be a frequency exponentially close to resonances. We find $h$ analytic of norm arbitrarily small such that ${\cal L}$ has no invariant torus of frequency $\bar\omega$ projecting diffeomorphically on $\T^m$

Let $G$ be a compact connected Lie group and ${\rm Aut}(G)$ be the group of Lie automorphisms of $G$. We describe a condition on $G$ under which topological conjugacy of the flows on $G$ induced by two one-parameter subgroups $\phi$ and $\psi$ of ${\rm Aut}(G)$ implies conjugacy of $\phi$ and $\psi$ in ${\rm Aut}(G)$. The condition is verified for $Sp(n)$, $SO(2n+1)$ and ${\rm Spin}(2n+1)$.

On étudie une large classe d'immeubles à courbure strictement négative. Deux aspects sont surtout développés : la construction de réseaux dans leur groupe d'automor-phismes, et l'étude de leur type de quasi-isométrie à travers la dimension conforme de Pansu de leur bord.

Given a continuous map on a locally compact metric space and an isolating neighborhood which is decomposed into two disjoint isolating neighborhoods, it is shown that the spectral information of the associated Conley indices is sufficient to conclude the existence of a semi-conjugacy onto the full shift dynamics on two symbols.

Flow-type suspension and homotopy suspension agree for attractor–repellor homotopy data. The connection maps associated in Conley index theory to an attractor–repellor decomposition with respect to the direct flow and its inverse are Spanier–Whitehead duals in the stably parallelizable context and are duals modulo a certain Thom construction in general.

For families of conservative maps near the identity we prove the existence of open sets of parameters with persistence of homoclinic tangencies between stable and unstable leaves of ‘thick’ horse-shoes. Such families are obtained, for instance, by perturbing integrable Hamiltonian systems in $\mathbb{R}^2$ with a rapidly periodic oscillatory term and then performing a slowing change in the time variable.

This paper contains a generalization to the $d$-torus $\mathbb{T}^d$ of earlier results (B. Host. Nombres normaux, entropie, translations. Israel J. Math. 91 (1995), 419–428). Given a probability measure $\mu$ on and an endomorphism $T$ of $\mathbb{T}^d$, we explore the relations between three properties: the uniform distribution of the sequence $(T^n\mathbf{t})$ for $\mu$-almost all $\mathbf{t}$; the behaviour of $\mu$ relative to the translations by some rational subgroups of $\mathbb{T}^d$; and the entropy of $\mu$ for another endomorphism $S$ of $\mathbb{T}^d$.

On semiflows on Banach manifolds, Quandt (1988 and 1989) stated that some special semiflows were inverse limit stable. However, the concepts and proofs used to show inverse limit stability of semiflows are rough and ambiguous. We consider semiflows on finite-dimensional compact manifolds or on finite-dimensional compact non-singular branched manifolds. In this paper we show that more special semiflows are inverse limit stable. That is, Anosov semiflows are inverse limit stable.

We study asymptotic physical measures in certain dynamical systems formed by a skew product. The skew product is formed by two ‘nice’ systems via a skewing function which is asymptotically Brownian. In this setting, no analogue of the Sinai–Ruelle–Bowen measure exists.

Let $G$ and $H$ be locally compact groups. Assume that $G$ acts on a standard probability space $(S,\mu)$, $\mu$ being $G$-invariant. We prove that if there exists a Borel cocycle $\alpha:S\times G\longrightarrow H$ which is proper in an appropriate sense, then $G$ inherits some approximation properties of $H$, for instance amenability or the so-called Haagerup Approximation Property. On the other hand, if $G_{0}$ is a closed subgroup of $G$, if the pair $(G,G_{0})$ has the relative property (T) of Margulis [19] and if either $H$ has Haagerup Approximation Property, or if it is the unitary group of a finite von Neumann algebra with a similar property, then we give rigidity results analogous to that in [23] and [1].

A partially ordered set that is invariantly associated to a subshift is constructed. A property of subshifts, also an invariant of topological conjugacy, is described. If this property is present in a subshift then the constructed partially ordered set is a partially ordered semigroup (with zero). In the description of these invariants the notion of context is instrumental.

Suppose that $X^1,X^2,\dotsc$ is a stationary stochastic process of positive $k\times k$ matrices, and let ${}^nY^1=X^nX^{n-1}\dots X^1$ be the corresponding product matrices. For a special case, Bellman showed that the elements $[{}^nY^1]_{ij}$ converge in the sense that $n^{-1}\mathrm{E}\{\log[{}^nY^1]_{ij}\}\rightarrow a$ as $n\rightarrow\infty$. The constant $a$ is independent of $i$ and $j$. Bellman also conjectured that, asymptotically, the $n^{-1/2}\{\log[{}^nY^1]_{ij}-na\}$ terms are distributed according to a normal distribution with a common variance, independent of $ij$. Later Furstenberg and Kesten generalized and strengthened Bellman's result and established the validity of his conjecture.

This paper extends these results to the case of nonlinear mappings that are monotonic and homogeneous of degree one on $R^k_+$. Specifically, given a stationary process $H^1,H^2,\dots$ of such mappings, we define the composite mappings ${}^nF^1(\cdot)=H^n(H^{n-1}(\dots (H^1(\cdot))\dots)$. Under appropriate conditions, the components $[{}^nF^1(x^0)]_i$ have the property that, almost surely, $n^{-1}\log[{}^nF^1(x^0)]_i\rightarrow a$ independent of $x^0$ and $i$. Furthermore the components $n^{-1/2}\{\log[{}^nF^1(x^0)]_i-na\}$ are asymptotically distributed according to a normal distribution with a common variance.

We analyse the convergence conditions of the Eckmann and Ruelle algorithm (ERA) used to estimate the Liapunov exponents, for the tangent map, of an ergodic measure, invariant under a smooth dynamical system. We find sufficient conditions for this convergence that are related to those ensuring the convergence to the tangent map of the best linear $L^p$-fittings of the action of a mapping $f$ on small balls. Under such conditions, we show how to use ERA to obtain estimates of the Liapunov exponents, up to an arbitrary degree of accuracy. We propose an adaptation of ERA for the computation of Liapunov exponents in smooth manifolds, which allows us to avoid the problem of detecting the spurious exponents.

We prove, for a Borel measurable dynamics $f$, the existence of Liapunov exponents for the function $S_r(x)$, mapping each point $x$ to the matrix of the best linear $L^p$-fitting of the action of $f$ on the closed ball of radius $r$ centred at $x$, and we show how to use ERA to get reliable estimates of the Liapunov exponents of $S_r$. We also propose a test for checking the differentiability of an empirically observed dynamics.

This paper studies a special system of $N$-coupled oscillators under white noise. It is shown that we can get a positive recurrent diffusion process by winding the solution of this system on a cylinder along a special direction. Thus defining the rotation number as the limit of the frequency of the oscillator, we can prove that the rotation number exists and the frequency locking is successful no matter how large the white noise is.

Iosifescu pointed out that earlier attempts to generalize Kuzmin's theorem to fibred systems have a gap. In this paper a new proof is given. The convergence rate is worse, but the result applies to multi-dimensional continued fractions and complex continued fractions.

For an inclusion $F < G < L$ of connected real algebraic groups such that $F$ is epimorphic in $G$, we show that any closed $F$-invariant subset of $L/\Lambda$ is $G$-invariant, where $\Lambda$ is a lattice in $L$. This is a topological analogue of a result due to S. Mozes, that any finite $F$-invariant measure on $L/\Lambda$ is $G$-invariant.

This result is established by proving the following result. If in addition $G$ is generated by unipotent elements, then there exists $a\in F$ such that the following holds. Let $U\subset F$ be the subgroup generated by all unipotent elements of $F$, $x\in L/\Lambda$, and $\lambda$ and $\mu$ denote the Haar probability measures on the homogeneous spaces $\overline{Ux}$ and $\overline{Gx}$, respectively (cf. Ratner's theorem). Then $a^n\lambda\to\mu$ weakly as $n\to\infty$.

We also give an algebraic characterization of algebraic subgroups $F<{\rm SL}_n(\mathbb{R})$ for which all orbit closures on ${\rm SL}_n(\mathbb{R})/{\rm SL}_n(\Z)$ are finite-volume almost homogeneous, namely the smallest observable subgroup of ${\rm SL}_n(\mathbb{R})$ containing $F$ should have no non-trivial algebraic characters defined over $\mathbb{R}$.

A conjecture is formulated and discussed which provides a necessary and sufficient condition for the ergodicity of cylindric billiards (this conjecture improves a previous one of the second author). This condition requires that the action of a Lie-subgroup ${\cal G}$ of the orthogonal group $SO(d)$ ($d$ being the dimension of the billiard in question) be transitive on the unit sphere $S^{d-1}$. If $C_1, \dots, C_k$ are the cylindric scatterers of the billiard, then ${\cal G}$ is generated by the embedded Lie subgroups ${\cal G}_i$ of $SO(d)$, where ${\cal G}_i$ consists of all transformations $g\in SO(d)$ of ${\Bbb R}^d$ that leave the points of the generator subspace of $C_i$ fixed ($1 \le i \le k$). In this paper we can prove the necessity of our conjecture and we also formulate some notions related to transitivity. For hard ball systems, we can also show that the transitivity holds in general: for an arbitrary number $N\ge 2$ of balls, arbitrary masses $m_1, \dots, m_N$ and in arbitrary dimension $\nu \ge 2$. This result implies that our conjecture is stronger than the Boltzmann–Sinai ergodic hypothesis for hard ball systems. We also note a somewhat surprising characterization of the positive subspace of the second fundamental form for the evolution of a special orthogonal manifold (wavefront), namely for the parallel beam of light. Thus we obtain a new characterization of sufficiency of an orbit segment.

Weak equivalence of primitive matrices is a known invariant arising naturally from the study of inverse limit spaces. Several new invariants for weak equivalence are described. It is proved that a positive dimension group isomorphism is a complete invariant for weak equivalence. For the transition matrices corresponding to periodic kneading sequences, the discriminant is proved to be an invariant when the characteristic polynomial is irreducible. The results have direct application to the topological classification of one-dimensional inverse limit spaces.

These two books look like they might form a two-volume set, but this is not so. The first is an introduction to ergodic theory suitable as a text for a first course in the subject, with some of the standard results viewed in an unusual light (that of descriptive set theory) and with detailed discussion of several topics of special interest to the author. The second is a survey at an advanced level of parts of the interaction of measure-preserving and nonsingular ergodic theory with spectral theory and harmonic analysis. It assumes considerable background in all three of these subjects and is written as a guide for researchers, organizing many topics of recent and current interest and highlighting some major open questions on which further progress might be possible at this time. Selected topics could be worked through individually or as part of a seminar or advanced course. Interestingly, the second does not even refer to the first for needed results from basic ergodic theory.