The notion of dynamical entropy for actions of a countable free abelian group $G$ by automorphisms of $C^*$-algebras is studied. These results are applied to Bogoliubov actions of $G$ on the CAR-algebra. It is shown that the dynamical entropy of Bogoliubov actions is computed by a formula analogous to that found by Størmer and Voiculescu in the case $G={\bf Z}$, and also it is proved that the part of the action corresponding to a singular spectrum gives zero contribution to the entropy. The case of an infinite number of generators has some essential differences and requires new arguments.

Using a very simple blow-up, we show how to extend the stable and pseudo-stable manifold theorems so as to get (a Banach version of) Sternberg's local conjugacy results for hyperbolic diffeomorphisms [9, 10].

We investigate hypercyclic and chaotic behavior of linear strongly continuous semigroups. We give necessary and sufficient conditions on the semigroup to be hypercyclic, and sufficient conditions on the spectrum of an operator to generate a hypercyclic semigroup. A variety of examples is provided.

The closing lemma is proved for various classes of holomorphic self-maps on ${\Bbb C}^k$. These classes include those consisting of biholomorphic maps, endomorphisms and symplectomorphisms.

Let $S(N)$ be a random walk on a countable abelian group $G$ which acts on a probability space $E$ by measure-preserving transformations $(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time $\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding induced skew product transformation is K-mixing whenever a natural subgroup of $G$ acts ergodically on $E$.

We study the topological entropy of a class of transformations with mild singularities: the generalized polygon exchanges. This class contains, in particular, polygonal billiards. Our main result is a geometric estimate, from above, on the topological entropy of generalized polygon exchanges. One of the applications of our estimate is that the topological entropy of polygonal billiards is zero. This implies the subexponential growth of various geometric quantities associated with a polygon. Other applications are to the piecewise isometries in two dimensions, and to billiards in rational polyhedra.

Let $T$ be a locally finite simplicial tree and let $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. We obtain an explicit formula for the critical exponent of the Poincaré series associated with $\Gamma$, which is also the Hausdorff dimension of the limit set of $\Gamma$; this uses a description due to Lubotzky of an appropriate fundamental domain for finite index torsion-free subgroups of $\Gamma$. Coornaert, generalizing work of Sullivan, showed that the limit set is of finite positive measure in its dimension; we give a new proof of this result. Finally, we show that the critical exponent is locally constant on the space of deformations of $\Gamma$.

Let $(f,\alpha)$ be the process given by an endomorphism $f$ and by a finite partition $\alpha=\{A\}_{i=1}^{s}$ of a Lebesgue space. Let $E(f,\alpha)$ be the set of densities of absolutely continuous invariant measures for skew products with the base $(f,\alpha)$. We prove that a process $(f,\alpha)$ is Markovian if and only if $E(f,\alpha)\subset \{g: g=\sum_{i=1}^{s} 1_{A_{i}}\otimes g_{i}\}$. If $f$ is the Lasota–Yorke or Misiurewicz type map with the Markovian partition $\alpha$, then $(f,\alpha)$ is quasi-Markovian, i.e. $E(f,\alpha)\subset \{g:\supp g=\bigcup_{i=1}^{s} A_{i}\times B_{i}\}$. Moreover, we give the characterization of quasi-Markovian processes.

The equation $\dot{z} = e^{i \alpha} z + e^{i \varphi} z |z|^2 + b \bar{z}^3$ models a map near a Hopf bifurcation with 1:4 resonance. It is a conjecture by V. I. Arnol'd that this equation contains all versal unfoldings of ${\Bbb Z}_4$-equivariant planar vector fields. We study its bifurcations at $\infty$ and show that the singularities of codimension two unfold versally in a neighborhood. We give an unfolding of the codimension-three singularity for $b=1$, $\varphi=3\pi/2$ and $\alpha=0$ in the system parameters and use numerical methods to study global phenomena to complete the description of the behavior near $\infty$. Our results are evidence in support of the conjecture.

It is proved that every second countable locally Hausdorff and locally compact continuous groupoid has a Borel set of units that meets every orbit and is what is called ‘lacunary’, a property that implies that the intersection with every orbit is countable.

We examine the question whether the dimension $D$ of a set or probability measure is the same as the dimension of its image under a typical smooth function, if the range space is at least $D$-dimensional. If $\mu$ is a Borel probability measure of bounded support in ${\Bbb R}^n$ with correlation dimension $D$, and if $m\geq D$, then under almost every continuously differentiable function (‘almost every’ in the sense of prevalence) from ${\Bbb R}^n$ to ${\Bbb R}^m$, the correlation dimension of the image of $\mu$ is also $D$. If $\mu$ is the invariant measure of a dynamical system, the same is true for almost every delay coordinate map. That is, if $m\geq D$, then $m$ time delays are sufficient to find the correlation dimension using a typical measurement function. Further, it is shown that finite impulse response (FIR) filters do not change the correlation dimension. Analogous theorems hold for Hausdorff, pointwise, and information dimensions. We show by example that the conclusion fails for box-counting dimension.

We introduce and study a new class of dynamical systems, the projective billiards, associated with a smooth closed convex plane curve equipped with a smooth field of transverse directions. Projective billiards include the usual billiards along with the dual, or outer, billiards.

We study the convergence to equilibrium states for certain non-hyperbolic piecewise invertible systems. The multi-dimensional maps we shall consider do not satisfy Renyi's condition (uniformly bounded distortion for any iterates) and do not necessarily satisfy the Markov property. The failure of both conditions may cause singularities of densities of the invariant measures, even if they are finite, and causes a crucial difficulty in applying the standard technique of the Perron–Frobenius operator. Typical examples of maps we consider admit indifferent periodic orbits and arise in many contexts. For the convergence of iterates of the Perron–Frobenius operator, we study continuity of the invariant density.