The notion ofdynamical entropy for actions of a countable free abelian group $G$ byautomorphisms of $C^*$-algebras is studied. These results are applied toBogoliubov actions of $G$ on the CAR-algebra. It is shown that the dynamicalentropy of Bogoliubov actions is computed by a formula analogous to thatfound by Størmer and Voiculescu in the case $G={\bf Z}$, and also it isproved that the part of the action corresponding to a singular spectrum giveszerocontribution to the entropy. The case of an infinite number of generators hassome essential differences and requires new arguments.

Using a very simple blow-up, we show how to extend the stable andpseudo-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 continuoussemigroups. We give necessary and sufficient conditions on the semigroup tobehypercyclic, and sufficient conditions on the spectrum of an operator togenerate a hypercyclic semigroup. A variety of examples is provided.

The closing lemma is proved for various classes of holomorphicself-maps on ${\Bbb C}^k$. These classes include thoseconsisting of biholomorphic maps, endomorphisms andsymplectomorphisms.

Let $S(N)$ be a random walk on a countable abelian group$G$ which acts on a probability space $E$ by measure-preservingtransformations$(T_v)_{v\in G}$. For any $\Lambda \subset E$ we consider the random return time$\tau$ at which $T_{S(\tau)}\in\Lambda$. We show that the corresponding inducedskew 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 mildsingularities: the generalized polygon exchanges.This class contains, in particular, polygonal billiards. Our main result is ageometric estimate, from above, on the topologicalentropy of generalized polygon exchanges. One of the applications of ourestimate is that the topological entropy of polygonalbilliards is zero. This implies the subexponential growth of variousgeometric quantities associated with a polygon.Other applications are to the piecewise isometries in two dimensions, and tobilliards in rational polyhedra.

Let $T$ be a locally finite simplicial tree andlet $\Gamma\subset{\rm Aut}(T)$ be a finitely generated discrete subgroup. Weobtain an explicit formula for the critical exponent of the Poincaréseriesassociated with $\Gamma$, which is also the Hausdorff dimension of the limitset of $\Gamma$; this uses a description due to Lubotzky of anappropriate fundamental domain for finite index torsion-free subgroupsof $\Gamma$.Coornaert, generalizing work of Sullivan, showed thatthe limit set is of finite positive measure in its dimension; we give a newproof of this result. Finally, we show that the critical exponent is locallyconstant on the space of deformations of$\Gamma$.

Let $(f,\alpha)$ be the process given by an endomorphism $f$ andby a finite partition $\alpha=\{A\}_{i=1}^{s}$ of a Lebesgue space.Let $E(f,\alpha)$ be the set of densities of absolutely continuousinvariant measures for skew products with the base $(f,\alpha)$.We prove that a process $(f,\alpha)$ is Markovian if and only if$E(f,\alpha)\subset \{g: g=\sum_{i=1}^{s} 1_{A_{i}}\otimesg_{i}\}$. If $f$ is the Lasota–Yorke or Misiurewicz type map withthe 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 ofquasi-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 conjectureby 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 versallyin a neighborhood. We give an unfolding of the codimension-three singularityfor $b=1$, $\varphi=3\pi/2$ and $\alpha=0$ in the system parameters and usenumerical methods to study global phenomena to complete the description ofthe behavior near $\infty$. Our results are evidence in support of theconjecture.

It is proved that every secondcountable locally Hausdorff and locally compactcontinuous groupoid has a Borel set of units that meetsevery orbit and is what is called ‘lacunary’, a propertythat implies that the intersection with every orbit iscountable.

We examine the question whether the dimension $D$ of a setor probability measure is the same as the dimension of itsimage under a typical smooth function, if the range spaceis at least $D$-dimensional. If $\mu$ is a Borelprobability measure of bounded support in ${\Bbb R}^n$with correlation dimension $D$, and if $m\geq D$, thenunder almost every continuously differentiable function(‘almost every’ in the sense of prevalence) from ${\BbbR}^n$ to ${\Bbb R}^m$, the correlation dimension of theimage of $\mu$ is also $D$. If $\mu$ is the invariantmeasure of a dynamical system, the same is true for almostevery delay coordinate map. That is, if $m\geq D$, then$m$ time delays are sufficient to find the correlationdimension using a typical measurement function. Further,it is shown that finite impulse response (FIR) filters donot change the correlation dimension. Analogous theoremshold for Hausdorff, pointwise, and information dimensions.We show by example that the conclusion fails forbox-counting dimension.

We introduce and study a new class of dynamical systems, theprojective billiards, associated with a smooth closed convex planecurve 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 forcertain non-hyperbolic piecewise invertible systems.The multi-dimensional maps we shall considerdo not satisfy Renyi's condition (uniformlybounded distortion for any iterates) and do not necessarilysatisfy the Markov property.The failure of both conditions may cause singularities ofdensities of the invariant measures, even if they arefinite, and causes a crucial difficulty in applying thestandard technique of the Perron–Frobeniusoperator. Typical examples of maps we consideradmit indifferent periodic orbits and arise in many contexts.For the convergence of iterates ofthe Perron–Frobenius operator,we study continuity of the invariant density.