A \mathbb{G}-solenoid is a laminated space whose leaves are copies of a single Lie group \mathbb{G} and whose transversals are totally disconnected sets. It inherits a \mathbb{G}-action and can be considered as a dynamical system. Free \mathbb{Z}^d-actions on the Cantor set as well as a large class of tiling spaces possess such a structure of \mathbb{G}-solenoids. For a large class of Lie groups, we show that a \mathbb{G}-solenoid can be seen as a projective limit of branched manifolds modeled on \mathbb{G}. This allows us to give a topological description of the transverse invariant measures associated with a \mathbb{G}-solenoid in terms of a positive cone in the projective limit of the dim(\mathbb{G})-homology groups of these branched manifolds. In particular, we exhibit a simple criterion implying unique ergodicity. Particular attention is paid to the case when the Lie group \mathbb{G} is the group of affine orientation-preserving isometries of the Euclidean space or its subgroup of translations.

In this paper we will consider a bifurcation that occurs in the topology of the Julia set of meromorphic maps as they become entire. For some meromorphic maps the Julia set will be a Cantor set. We will investigate how the Julia set changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet. Other meromorphic maps have a Julia set that is a Jordan curve. Again we will study how this curve changes as these meromorphic maps approach the entire function whose Julia set is a Cantor bouquet.

A combinatorial model for a property of continuous self-maps of a compact interval is a self-map \pi of a finite ordered set such that every continuous \pi-weakly monotone self-map of a compact interval has that property. We identify the minimal combinatorial models for the property ‘the set of periods is a given set’. Here the word minimal refers to the number of points in the domain of the model. We also identify the minimal permutation models and, in appropriate cases, the minimal combinatorial models for properties involving ‘horseshoes’.

For a compact Riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Hölder bicontinuous homeomorphism. However, the Riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, which does depend on the metric. For a surface S, we show that the map which to a hyperbolic metric on S associates its Liouville transverse measure is differentiable, in an appropriate sense. Its tangent map is valued in the space of transverse Hölder distributions for the geodesic foliation.

We investigate a basic system T of two rotations with a common angle of rotation. Using a new idea, we prove that, except for an invertible T, the system T must have a global attractor and a global repeller. We give a specific condition for both cases.

This paper has four main results: (i) it shows that left-invariant geodesic flows on a broad class of two-step nilmanifolds—which are dubbed almost non-singular—are integrable in the non-commutative sense of Nehoros˘ev; (ii) the left-invariant geodesic flows on all Heisenberg–Reiter nilmanifolds are Liouville integrable; (iii) the topological entropy of a left-invariant geodesic flow on a two-step nilmanifold vanishes; (iv) there exist two-step nilmanifolds with non-integrable left-invariant geodesic flows. It is also shown that for each of the integrable Hamiltonians investigated here, there is a C^2-open neighbourhood in C^2(T^* M) such that every integrable Hamiltonian vector field in this neighbourhood must have wild first integrals.

This paper addresses the question of genericity of existence of elliptic islands for the billiard map associated to strictly convex closed curves. More precisely, we study 2-periodic orbits of billiards associated to C^5 closed and strictly convex curves and show that the existence of elliptic islands is a dense property on the subset of those billiards having an elliptic 2-periodic point.

Our main tools are normal perturbations, the Birkhoff Normal Form for elliptic fixed points and Moser's Twist Theorem. In order to perform some of the long computations involved, Maple^{\circledR} software was employed.

We find an explicit example of a Markov odometer action which is not orbit-equivalent to any product odometer action. We make use of some techniques from Krieger's construction of an action of non-product type arising from a flow. However, his example is not realized as an odometer action with an explicit measure, nor does it arise from a Markov measure.

This paper studies the problem of characterizing the simplest aperiodic discrete point sets, using invariants based on topological dynamics. A Delone set of finite type is a Delone set X such that X-X is locally finite. Such sets are characterized by their patch-counting function N_X (T) of radius T being finite for all T. We formulate conjectures relating slow growth of the patch-counting function N_X (T) to the set X having a non-trivial translation symmetry.

A Delone set X of finite type is repetitive if there is a function M_X(T) such that every closed ball of radius M_X(T) + T contains a complete copy of each kind of patch of radius T that occurs in X. This is equivalent to the minimality of an associated topological dynamical system with \mathbb{R}^n-action. There is a lower bound for M_X(T) in terms of N_X(T), namely M_X(T) \ge c( N_X (T))^{1/n} for some positive constant c depending on the Delone set constants r,R, but there is no general upper bound for M_X(T) purely in terms of N_X(T). The complexity of a repetitive Delone set X is measured by the growth rate of its repetitivity function M_X (T). For example, the function M_X (T) is bounded if and only if X is a periodic crystal. A set X is linearly repetitive if M_X (T) = O(T) as T \to \infty and is densely repetitive if M_X (T) = O(N_X(T))^{1/n} as T \to \infty. We show that linearly repetitive sets and densely repetitive sets have strict uniform patch frequencies, i.e. the associated topological dynamical system is strictly ergodic. It follows that such sets are diffractive, in the sense of having a well-defined diffraction measure. In the reverse direction, we construct a repetitive Delone set X in \mathbb{R}^n which has M_X (T) = O(T (\log T)^{2/n} (\log\log\log T )^{4/n}), but does not have uniform patch frequencies. Aperiodic linearly repetitive sets have many claims to be the simplest class of aperiodic sets and we propose considering them as a notion of ‘perfectly ordered quasicrystals’.

We prove that the general (i.e. possibly aperiodic) Lorenz gas in two dimensions, with finite horizon and non-degenerate geometrical features, is ergodic if it is recurrent. We also give examples of aperiodic recurrent gases.

A dimension invariant for isomorphism of non-singular ergodic integer actions is presented. A result analogous to Shannon's Theorem is used to calculate the critical dimension of type III odometer actions. It is shown that the dimension is non-trivial and that it is not an invariant of orbit equivalence.

In dynamical systems examples are common in which two or more attractors coexist and in such cases the basin boundary is non-empty. The purpose of this paper is to describe the structure and properties of basins and their boundaries for two-dimensional diffeomorphisms. If a two-dimensional basin has a basin cell (a trapping region whose boundary consists of pieces of the stable and unstable manifolds of a well-chosen periodic orbit), then the basin consists of a central body (the basin cell) and a finite number of channels attached to it and the basin boundary is fractal. We prove the following surprising property for certain fractal basin boundaries: a basin of attraction B has a basin cell if and only if every diverging path in basin B has the entire basin boundary \partial\bar{B} as its limit set. The latter property reflects a complete entangled basin and its boundary.

We prove that any finitely generated solvable group of zero entropy contains a nilpotent subgroup of finite index. In particular, any finitely generated solvable group of exponential growth is of uniformly exponential growth.

The homogenizability of linear transport equations with periodic data is investigated. It can be interpreted in terms of the dynamical system properties of the associated ODEs. De Giorgi conjectured that all equations of this kind are homogenizable. In this paper, the homogenizability is proved is the two-dimensional case for non-vanishing functions (partial results in this direction have been previously proved by other authors). On the other hand, in the three-dimensional case, an example of a non-homogenizable equation is given.

In this paper we study the behavior of diffeomorphisms, contained in the closure \overline{\mathcal{A}_\alpha} (in the inductive limit topology) of the set \mathcal{A}_\alpha of real-analytic diffeomorphisms of the torus \mathbb T^2, which are conjugated to the rotation R_\alpha:(x,y)\mapsto (x + \alpha , y) by an analytic measure-preserving transformation. We show that for a generic \alpha\in [0,1], \overline{\mathcal{A}_\alpha} contains a dense set of uniquely ergodic diffeomorphisms. We also prove that \overline{\mathcal{A}_\alpha} contains a dense set of diffeomorphisms that are minimal and non-ergodic.

Let f be a real rational function with all critical points on the extended real axis and of even order. Then:

(1) f carries no invariant line field on the Julia set unless it is doubly covered by an integral torus endomorphism (a Lattés example); and

(2) f|J(f) has only finitely many ergodic components.

We draw attention to the following mistakes which appear in the above-mentioned paper (pp. 732–734). We are grateful to J.-F. Mattei who pointed out with an example that the notion of singular holonomy should be enlarged and consequently the Structure Theorem restated. The main result (Theorem 1) remains untouched. We refer to the same notation and terminology of the paper.