In Part 1 we describe a duality method for calculating twisted cocycles. In Part 2 we use our method to prove various results on cohomological rigidity of higher-rank cocompact lattice actions. In Part 3 we use the results of Parts 1 and 2 to prove infinitesimal rigidity of the actions of cocompact lattices on the maximal boundaries of some non-compact type symmetric spaces.

The main purpose of this paper is to study the implications that the homology index of critical sets of smooth flows on closed manifolds M have on both the homology of level sets of M and the homology of M itself. The bookkeeping of the data containing the critical set information of the flow and topological information of M is done through the use of Lyapunov graphs. Our main result characterizes the necessary conditions that a Lyapunov graph must possess in order to be associated to a Morse–Smale flow. With additional restrictions on an abstract Lyapunov graph L we determine sufficient conditions for L to be associated to a flow on M.

Consider a class of Polish groups arising from the subclass of amenable locally compact ones via operations of countable projective limit and group extensions. We show that for each group from this class there exists a cocycle of an ergodic transformation with dense range in it. This is applied to extend and provide a short (orbital) proof for one of the main results from [ALV] on non-coalescence of some ergodic skew product extensions.

We study transfer operators over general subshifts of sequences of an infinite alphabet. We introduce a family of Banach spaces of functions satisfying a regularity condition and a decreasing condition. Under some assumptions on the transfer operator, we prove its continuity and quasi-compactness on these spaces. Under additional assumptions—existence of a conformal measure and topological mixing—we prove that its peripheral spectrum is reduced to one and that this eigenvalue is simple. We describe the consequences of these results in terms of existence and properties of invariant measures absolutely continuous with respect to the conformal measure. We also give some examples of contexts in which this setting can be used—expansive maps of the interval, statistical mechanics.

We investigate a natural connection between inductive limits of $C^*$-algebras, ergodic theory and the theory of subfactors.

The numerical determination of the bottom of the spectrum of a discrete almost periodic Schrödinger operator involves finding the transition to chaotic behaviour of a certain dynamical map on the torus.

We explore substitution minimal systems and their relation to stationary Bratteli diagrams and stationary dimension groups. The constructions involved are algorithmic and explicit and render an effective method to compute an invariant of (ordered) $K$-theoretic nature for these systems. This new invariant is independent of spectral invariants which have previously been extensively studied. Before we state the main results we give some background.

We consider an Abel equation $(*)$ $y^{\prime}=p(x)y^2+q(x)y^3$ with $p(x)$, $q(x)$ polynomials in $x$. A center condition for ($*$) (closely related to the classical center condition for polynomial vector fields on the plane) is that $y_0=y(0)\equiv y(1)$ for any solution $y(x)$ of ($*$). This condition is given by the vanishing of all the Taylor coefficients $v_k(1)$ in the development $y(x)=y_0+\sum^{\infty}_{k=2}v_k(x)y^k_0$. A new basis for the ideals $I_k=\{v_2,\dots,v_k\}$ has recently been produced, defined by a linear recurrence relation. Studying this recurrence relation, we connect center conditions with a representability of $P=\int p$ and $Q=\int q$ in a certain composition form (developing further some results of Alwash and Lloyd), and with a behavior of the moments $\int P^kq$. On this base, explicit center equations are obtained for small degrees of $p$ and $q$.

We show that the Patterson–Sullivan measure on the limit set of a geometrically finite Kleinian group with cusps can be recovered as a weak limit of sums of Dirac masses placed on an appropriate orbit of each parabolic fixed point. A corollary is a sharp asymptotic estimate for a natural counting function associated to a cuspidal subgroup. We also discuss the connection between the above counting and the Riemann hypothesis in some examples of arithmetical lattices.

Nous prouvons que dans l'ensemble $SW^0({\bf T}^1,SU(2))$ des difféomorphismes de ${\bf T}^1\times SU(2)$ de la forme $(\theta,y)\to (\theta+\alpha,A(\theta).y)$, $A\in C^{\infty}({\bf T}^1,SU(2))$, $\alpha\in{\bf T}^1$, les systèmes $(\alpha,A)$ réductibles, c'est-à-dire ceux qui admettent une écriture de la forme $A(\theta)= B(\theta+\alpha)A_0B(\theta)^{-1}$, $B\in C^{\infty}({\bf T}^1,SU(2))$, $A_0\in SU(2)$, forment une partie dense pour la $C^0$-topologie. Ce théorème est à comparer à un résultat de Herman [7], qui dit qu'il en est de même des systèmes non réductibles. Ceci démontre (une forme d')une conjecture de Rychlik (cf. [14]).

We study the natural symbolic dynamics associated with piecewise continuous, non-invertible, dynamical systems. Our study is centered primarily on the relationship between the point-set topological properties of the partition of the system and the symbolic coding. We prove that for a class of maps locally preserving distances with regular partition, the associated symbolic dynamics cannot embed subshifts of finite type of positive entropy. Hence, in particular, almost sofic subshifts obtained from the symbolic dynamics have zero entropy. However, there are examples in Euclidean spaces of systems with non-regular partitions for which the coding maps can be surjective, particularly embedding all subshifts. For all such examples, the associated group of isometries is a subgroup of $O(\mathbb{R}, N)$.

Let $f$ be a smooth homeomorphism of the circle having one cubic-exponent critical point and irrational rotation number of bounded combinatorial type. Using certain pull-back and quasiconformal surgical techniques, we prove that the scaling ratios of $f$ about the critical point are asymptotically independent of $f$. This settles in particular the golden mean universality conjecture. We introduce the notion of a holomorphic commuting pair, a complex dynamical system that, in the analytic case, represents an extension of $f$ to the complex plane and behaves somewhat as a quadratic-like mapping. We define a suitable renormalization operator that acts on such objects. Through careful analysis of the family of entire mappings given by $z\mapsto z+\theta -(1/2\pi)\sin{2\pi z}$, $\theta$ real, we construct examples of holomorphic commuting pairs, from which certain necessary limit set pre-rigidity results are extracted. The rigidity problem for $f$ is thereby reduced to one of renormalization convergence. We handle this last problem by means of Teichmüller extremal methods made available through the recent work of Sullivan on Riemann surface laminations and renormalization of unimodal mappings.

1. Let ${\bf r}=(r_1,r_2,\dots)$ be a sequence of integers greater than one. Then every ${\bf r}$-adic reverse filtration has a maximal standard orbit factor.

2. For ‘entropy-free’ ${\bf r}$, $a\geqq 0$, and any free action $A$ of the group $G=(Z/r_1Z)+(Z/r_2Z)+\dotsb$, there is a free ergodic action $A'$ of $G$ with $h(A')=a$ and such that the reverse filtration produced by $A'$ is isomorphic to that produced by $A$.

Let $\bpropto$ be the topological space obtained by identifying the points 1 and 2 of the segment $[0,3]$ to a point. Let $\binfty$ be the topological space obtained by identifying the points 0, 1 and 2 of the segment $[0,2]$ to a point. An $\bpropto$ (respectively $\binfty$) map is a continuous self-map of $\bpropto$ (respectively $\binfty$) having the branching point fixed. Set $E\in\{\bpropto,\binfty\}$. Let $f$ be an $E$ map. We denote by $\Per(f)$ the set of periods of all periodic points of $f$. The set $K \subset{\mathbb N}$ is the full periodicity kernel of $E$ if it satisfies the following two conditions: (1) if $f$ is an $E$ map and $K\subset \Per(f)$, then $\Per(f)={\mathbb N}$; (2) for each $k\in K$ there exists an $E$ map $f$ such that $\Per(f)={\mathbb N}\setminus\{ k\}$. In this paper we compute the full periodicity kernel of $\bpropto$ and $\binfty$.

Given a $\mathbb{Z}^d$ homeomorphic action, $\alpha$, on the Cantor set, $X$, we consider the higher order continuous integer valued dynamical cohomology, $H^*(X,\alpha)$. We also consider the dynamical $K$-theory of the action, the $K$-theory of the crossed product $C^*$-algebra $C(X)\times_{\alpha}\mathbb{Z}^d$. We show that these two invariants are essentially equivalent. We also show that they only take torsion free values. Our work links the two invariants via a third invariant which is based on topological complex $K$-theory evaluated on an associated mapping torus.

For a sequence $(c_n)$ of complex numbers we consider the quadratic polynomials $f_{c_n}(z):=z^2+c_n$ and the sequence $(F_n)$ of iterates $F_n:= f_{c_n} \circ \dotsb \circ f_{c_1}$. The Fatou set $\mathcal{F}_{(c_n)}$ is by definition the set of all $z \in \widehat{\mathbb{C}}$ such that $(F_n)$ is normal in some neighbourhood of $z$, while the complement of $\mathcal{F}_{(c_n)}$ is called the Julia set $\mathcal{J}_{(c_n)}$. The aim of this paper is to study the connectedness of the Julia set $\mathcal{J}_{(c_n)}$ provided that the sequence $(c_n)$ is bounded and randomly chosen. For example, we prove a necessary and sufficient condition for the connectedness of $\mathcal{J}_{(c_n)}$ which implies that $\mathcal{J}_{(c_n)}$ is connected if $|c_n| \le \frac{1}{4}$, while it is almost surely disconnected if $|c_n| \le \delta$ for some $\delta>\frac{1}{4}$.

The spaces of subgroups and Lie subalgebras with the group actions by conjugations are considered for real Lie groups. Our approach allows one to apply the properties of algebraically regular transformation groups to finding the conditions when those actions turn out to be type I. It follows, in particular, that in this case the stability groups for all the ergodic actions of such groups are conjugate (for example when the stability groups are compact). The isomorphism of the stability groups for ergodic actions is also established under some assumptions. A number of examples of non-conjugate and non-isomorphic stability groups are presented.

By considering the way an $n$-tuple of points in the 2-disk are linked together under iteration of an orientation preserving diffeomorphism, we construct a dynamical cocycle with values in the Artin braid group. We study the asymptotic properties of this cocycle and derive a series of topological invariants for the diffeomorphism which enjoy rich properties.

Let $(E,{\cal A},\mu,T)$ be a dynamical system and let $\Phi$ be a function defined on $E$ with values in $\mathbb{R}^2$. We give a criterion, the central limit theorem along subsequences of positive density, for the recurrence of the corresponding ‘stationary walk’ defined as the cocycle $\big(\sum^{n-1}_{j=0}\Phi(T^jx)\big)_{n\geq1}$.

This criterion is satisfied by functions which are homologous to a martingale difference (a property which holds for regular functions in many systems). It can also be applied to the periodic Lorentz gas in the plane and shows recurrence for this model.

We establish bounds for the multipliers of those periodic orbits of $R_\mu(z) = z(z+\mu)/(1+\overline\mu z) $, which have a Poincaré rotation number $ p/q $. The bounds are given in terms of $ p/q $ and the (logarithmic) hororadius of $\mu$ to $e^{2\pi ip/q} $. The principal tool is a new construction denoted a ‘star’ of an immediate attracting basin. The bounds are used to prove properties of the space of Möbius conjugacy classes of quadratic rational maps. These properties are related to the mating and non-mating conjecture for quadratic polynomials lsqb;Ta]. Moreover they are also reminiscent of Chuckrows theorem on the non-existence of elliptic limits of loxodromic elements in quasiconformal deformations of Kleinian groups. We bear this analogy further by proving an analog of Chuckrows theorem for deformations of certain holomorphic maps.