We study the rectifiable invariant subsets of algebraic dynamical systems determined by ℝ-semisimple one parameter groups. We show that their ergodic components are algebraic. A more precise geometric description of these components is possible in some cases of geodesic flows of locally symmetric spaces with non-positive curvature.

It is shown that diffeomorphisms analogous to a classical example on the circle due to Denjoy can be constructed on the general k-torus. Such a diffeomorphism has the property that it is semiconjugate to an ergodic translation but has a wandering domain with dense orbit. The construction on the k-torus can be made Cr, and by a Cr small perturbation of a translation, for any r < k + 1.

It is shown that a probability measure on a homogeneous space Γ\G which is invariant under a subgroup H < G which is epimorphic in a subgroup L < G is invariant under L. When L = G we obtain a subgroup H such that for any lattice Γ < G its action on Γ\G is uniquely ergodic.

Let {Xθ} be a family of rotated singular real foliations in the Riemann sphere which is the result of the rotation of a meromorphic vector field X with zeros and poles of multiplicity one. We prove that the set of bifurcation values, in the circle {θ}, is for each family a set with at most a finite number of accumulation points. A condition which implies a finite number of bifurcation values is given. We also show that the property of having an infinite set of bifurcation values defines open but not dense sets in the space of meromorphic vector fields with fixed degree.

We prove that the diffeomorphisms with strange attractors shown to exist elsewhere are approximated by diffeomorphisms exhibiting homoclinic tangencies. As a consequence for the Hénon family these diffeomorphisms are approximated by diffeomorphisms exhibiting periodic attractors. This answers a question posed by Benedicks and Carleson.

Let Rw(z): W × C∞ → C∞ be an analytic family of rational functions, J(w) the Julia set of Rw and J*(w) the upper semicontinuous regularization of J(w). We shall discuss the relationship between J(w) and J*(w) as well as some related problems.