Herein the authors show that discrete groups of motions on Hn+1 may be conservative on Sn but have no positive measure ergodic components for this boundary action. An explicit example of such a group is given for H3 using the Apollonian circle packing of R2.

In dynamical systems examples are common in which there are regions containing chaotic sets that are not attractors, e.g. systems with horseshoes have such regions. In such dynamical systems one will observe chaotic transients. An important problem is the ‘Dynamical Restraint Problem’: given a region that contains a chaotic set but contains no attractor, find a chaotic trajectory numerically that remains in the region for an arbitrarily long period of time.

We present two procedures (‘PIM triple procedures’) for finding trajectories which stay extremely close to such chaotic sets for arbitrarily long periods of time.

In the papers [9, 10, 3, 11] on perturbations of Hilbert space operators, we studied an invariant (τ) where is a normed ideal of compact operators and τ a family of operators. The size of an ideal for which (τ) vanishes or does not vanish is an upper, respectively lower, bound for a kind of dimension of τ. In the case of systems of commuting self-adjoint operators τ, the results of [9,3] relate (τ) with (an ideal slightly smaller than the Schatten von Neumann class ) to the way the spectral measure of τ compares to p-dimensional Hausdorff measure.

A slightly improved version of the (idealized) C1 closing lemma is proved. This turns out to generalize the C1 closing lemma from diffeomorphisms to nonsingular endomorphisms.

A differentiable G-space is introduced, for a Lie group G, into which every countably separated Borel G-space can be imbedded. The imbedding can be a continuous map if the space is a separable metric space. Such a G-space is called a universal G-space. This universal G-space has a local product structure for the action of G. That structure is inherited by invariant subspaces, giving a local product structure on general G-spaces. This information is used to prove that G-spaces are stratified by the subsets consisting of points whose orbits have the same dimension, to prove that G-spaces with stabilizers of constant dimension are foliated, to give a short proof that closed subgroups of Lie groups are Lie groups, to give a new proof and a stronger version of the Ambrose—Kakutani Theorem and to give a new proof of the existence of near-slices at points having compact stabilizers and hence of the existence of slices for Cartan G-spaces.

We consider some ways in which regularity of a mapping influences dynamics of its iterations and growth of various complexity-type invariants.

To a factor map φ from an irreducible shift of finite type ΣA to a sofic shift S, we associate a subgroup of the dimension group (GA, Â) which is an invariant of eventual conjugacy for φ. This invariant yields new necessary conditions for the existence of factor maps between equal entropy sofic shifts.

We obtain a class of integral formulas for the Lapacian along unstable leaves of the geodesic flow of compact manifolds of negative curvature. Using these formulas, we give two functional descriptions of those manifolds with horospheres having constant mean curvature. More rigidity problems are discussed, including a simple proof of two important Lemmas by Hamenstadt which avoids her use of stochastic process.