Let \Gamma be a non-solvable pseudogroup of holomorphic transformations in one variable fixing zero. Then for any z_0 sufficiently near zero and outside some real analytic set containing zero and depending on \Gamma, for any germ of biholomorphism \phi defined at z_0 with \phi(z_0)-z_0 sufficiently small, there exists a sequence \gamma_n \in \Gamma which tends to \phi uniformly on some neighborhood of zero. If we let \Gamma be the holonomy pseudogroup of a compact leaf, we find that holomorphic codimension-1 foliations with non-solvable holonomy admit no transverse geometric structure in addition to the conformal one. This applies, in particular, to the singular foliation induced on \mathbb{C}\mathbb{P}^2 by the differential equation dw/dz = P_n(z,w)/Q_n(z,w) where P_n and Q_n are the generic polynomials of degree n, hence the title of this paper.

Let BV(I) be the (Banach) space of real functions h on I:=[0,1] of bounded variation \mathop{\rm var}\nolimits h with the norm \Vert h\Vert = \vert h\vert + \mathop{\rm var}\nolimits h, where \vert h\vert :=\sup_{t\in I}\vert h(t)\vert. We introduce a condition (M) on the monotony of the transition probability function (TPF) of the piecewise monotonic transformation \tau associated with an f-expansion. The transfer operator associated with \tau is denoted by U. Under (M) we can apply methods used in the continued fraction expansion case to U acting on BV(I). An expression in terms of the TPF of a contraction constant, \theta, of \mathop{\rm var}\nolimits Uh is thus obtained which is subsequently used to state a Gauss–Kuzmin–Levy theorem. This result yields the answer in a special case to the problem of determining a numerical value of the constant \theta occurring in the Kuzmin-type theorem proved in [1].

Examples of numerical values of \theta in particular cases are given. Possible extensions of the methods used are mentioned.

The notion of isomorphism on stable AF-C^{\ast}-algebras is considered in this paper in the case when the corresponding Bratteli diagram is stationary, i.e. is associated with a single square primitive incidence matrix. A C^{\ast}-isomorphism induces an equivalence relation on these matrices, called C^{\ast}-equivalence. We show that the associated isomorphism equivalence problem is decidable, i.e. there is an algorithm that can be used to check in a finite number of steps whether two given primitive matrices are C^{\ast}-equivalent or not. Special cases of this problem will be considered in a forthcoming paper.

We prove that on any surface there is a C^\infty diffeomorphism exhibiting a wandering domain D with the following ergodic property: for any orbit starting in D the corresponding Birkhoff mean of Dirac measures converges to the invariant measure supported on a hyperbolic horseshoe \Lambda which is equivalent to the unique non-trivial Hausdorff measure in \Lambda. The construction is obtained by perturbation of a diffeomorphism such that the unstable and stable foliations of this horseshoe \Lambda are relatively thick and in tangential position. We describe, in addition, the set of accumulation points of orbits starting in D.

Certain classes of automorphisms of reduced amalgamated free products of C^*-algebras are shown to have Brown–Voiculescu topological entropy zero. Also, for automorphisms of exact C^*-algebras, the Connes–Narnhofer–Thirring entropy is shown to be bounded above by the Brown–Voiculescu entropy. These facts are applied to generalize Størmer's result about the entropy of automorphisms of the {\rm II}_1-factor of a free group.

A general framework for investigating topological actions of \mathbb{Z}^d on compact metric spaces was proposed by Boyle and Lind in terms of expansive behavior along lower-dimensional subspaces of \mathbb{R}^d. Here we completely describe this expansive behavior for the class of algebraic \mathbb{Z}^d-actions given by commuting automorphisms of compact abelian groups. The description uses the logarithmic image of an algebraic variety together with a directional version of Noetherian modules over the ring of Laurent polynomials in several commuting variables.

We introduce two notions of rank for topological \mathbb{Z}^d-actions, and for algebraic \mathbb{Z}^d-actions describe how they are related to each other and to Krull dimension. For a linear subspace of \mathbb{R}^d we define the group of points homoclinic to zero along the subspace, and prove that this group is constant within an expansive component.

To obtain connections between grammars of languages and subshifts, Krieger introduced the idea of associating to a subshift X a certain increasing sequence of shifts of finite type X_{n} sitting inside X. Their union is the set of presynchronizing points. A system is called presynchronized if there is a sequence of irreducible components C_{n} \subset X_{n}, n \in \mathbb{N}, increasing with n and with dense union (a characterization of coded systems shows that presynchronized systems are coded). We investigate the uniqueness (up to eventual equality) of such sequences. For that an equivalence relation on the periodic presynchronizing points captures all the possible choices of increasing sequences of irreducible components. Dense equivalence classes correspond to dense unions. There is a one-sided and a two-sided theory. In the one-sided setting we show the uniqueness of a dense equivalence class and identify the right presynchronized systems as the well-known half-synchronized systems. The two-sided theory turns out to be more complicated (and interesting). There can be more than one dense class. Many examples are discussed in detail. There is a natural ordering on the set of (dense) equivalence classes. For each finite-order structure we find a representative in the countable class of systems which are intersections of Dyck shifts with shifts of finite type. Finally we discuss the relation to other known classes of subshifts, which is more subtle than in the one-sided setting. Almost all systems considered in this work will be coded.

We consider measure-preserving diffeomorphisms of the two-dimensional torus with zero entropy. We prove that every ergodic C^3-diffeomorphism f of the two-dimensional torus with linear growth of the derivative (i.e. the sequence \{n^{-1}Df^n\}_{n\in{\mathbb N}} is uniformly separated from 0 and \infty and it is bounded in the C^2-norm) is algebraically conjugate to a skew product of an irrational rotation on the circle and a circle C^3-cocycle with non-zero topological degree.

We show that every real analytic action of a connected supersoluble Lie group on a compact surface with non-zero Euler characteristic has a fixed point. This implies that Lima's fixed point free C^{\infty} action on S^2 of the affine group of the line cannot be approximated by analytic actions. An example is given of an analytic, fixed point free action on S^2 of a solvable group that is not supersoluble.

C^*-algebras arising from directed graphs and their generalizations, Exel–Laca algebras \mathcal{O}_A of an infinite matrix A, are considered. Characterizations of pure infiniteness (in the sense of [11]) and of stability are given. In particular, these C^*-algebras admit no non-zero trace if and only if they are purely infinite. Stability is characterized by the absence of non-zero unital quotients and a bounded trace. Also, graph-theoretical analogues of pure infiniteness and stability are obtained.

We prove L^2 variation inequalities for operators defined by the convolution powers of probability measures on locally compact Abelian groups. In some cases we also obtain L^p results for 1<p<\infty. These inequalities imply the pointwise convergence of these operators and give an estimate of the number of upcrossings.

It will be proved that some topological conjugacy invariants, called K-groups and Bowen–Franks groups, for subshifts are invariants for flow equivalence by an argument of symbolic dynamical systems. Some examples of non-sofic subshifts that are not flow equivalent will be presented.

We construct, for each irrational number \alpha, a minimal C^1-diffeomorphism of the circle with rotation number \alpha which is not ergodic with respect to the Lebesgue measure.

Let (X, S), (Y, T) be topological dynamical systems and \pi : X \rightarrow Y a factor map. A function F \in C(X) is a compensation function if P (F + \phi \circ \pi) = P (\phi) for all \phi \in C(Y). We present an example of a factor map \pi : X \rightarrow Y between subshifts of finite type X, Y that does not have a saturated compensation function and an example of a non-Markovian factor map with a saturated compensation function. Also we provide a necessary and sufficient condition for a certain type of factor map to have a saturated compensation function.

Let (X,T) be a Cantor minimal system associated with a properly ordered Bratteli diagram with the equal path number property. In this paper we will show that there exists a Toeplitz flow (Y,S) such that (X,T) and (Y,S) are strongly orbit equivalent. This is an affirmative answer to the open problem in R. Gjerde and Ø. Johansen (Bratteli–Vershik models for Cantor minimal systems: applications to Toeplitz flows. Ergod. Th. & Dynam. Sys.20 (2000), 1687–1710).

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author's ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.