We consider stochastic processes {xn}n≥0 of the form

where F: ℝm → ℝm is C2, {λi}i≥1 is a sequence of positive numbers decreasing to 0 and {Ui}i≥1 is a sequence of uniformly bounded ℝm-valued random variables forming suitable martingale differences. We show that when the vector field F is Morse-Smale, almost surely every sample path approaches an asymptotically stable periodic orbit of the deterministic dynamical system dy/dt = F(y). In the case of certain generalized urn processes we show that for each such orbit Γ, the probability of sample paths approaching Γ is positive. This gives the generic behavior of three-color urn models.

Let M be a closed n-dimensional manifold, and let U be a n-dimensional isolating block such that U is smoothly embedded in M. Let φ be a smooth semi-flow on U and let Λ contained in U, be isolated and invariant under φ Then there exists a semi-flow φ′ on M which extends φ such that φ′ is Morse-Smale outside of U, and no new recurrence is introduced in U. The theorem is true for any finite number of pairwise-disjoint Ui. Furthermore, if Λ is hyperbolic, topologically transitive and is the closure of periodic orbits, then φ′ is an Axiom A flow and is Ω-stable. In dimensions two and three, we have the stronger result that φ′ is structurally stable. Also, as a corollary, we give sufficient conditions for the flow φ′ to be nonsingular. One application of the corollary permits the formation of allowable knots and links in three-manifolds such that there exists a structurally stable nonsingular Morse-Smale flow φ′ which contains the specified knots and links in Ω(φ′) Moreover, the knots and links can be specified to be any combination of attractors, repellers or saddles.

We show, using elementary methods, that for 0 < a the measure-preserving, orientation-preserving Hénon map, H, has a horseshoe. This improves on the result of Devaney and Nitecki who have shown that a horseshoe exists in this map for a ≥ 8. For a > 0, we also prove the conjecture of Devaney that the first symmetric homoclinic point is transversal.

To obtain our results, we show that for a branch, Cu, of the unstable manifold of a hyperbolic fixed point of H, Cu crosses the line y = − x and that this crossing is a homoclinic point, χc. This has been shown by Devaney, but we obtain the crossing using simpler methods. Next we show that if the crossing of Wu(p) and Ws(p) at χc is degenerate then the slope of Cu at this crossing is one. Following this we show that if χc is a degenerate homoclinic its x-coordinate must be greater than l/(2a). We then derive a contradiction from this by showing that the slope of Cu at H-1(χc) must be both positive and negative, thus we conclude that χc is transversal.

Our approach uses a lemma that gives a recursive formula for the sign of curvature of the unstable manifold. This lemma, referred to as ‘the curvature lemma’, is the key to reducing the proof to elementary methods. A curvature lemma can be derived for a very broad array of maps making the applicability of these methods very general. Further, since curvature is the strongest differentiability feature needed in our proof, the methods work for maps of the plane which are only C2.

We develop a variant of the Taylor approximation approach to the periodic points of systems of contraction mappings [Rl] that does not invoke compactness conditions. Our presentation is simpler, in that certain steps are bypassed and only one basic estimate is used (Lemma 1). We also study the distribution of the discrete spectrum for the relevant transfer operators (Proposition 2).

We give sufficient conditions for a group of homeomorphisms of a compact Hausdorff space to have an invariant probability measure. For a complex projective space CPn we give a necessary condition for a subgroup of Aut(CPn) to have an invariant probability measure. We discuss two approaches to Auslander's conjecture.

We define a new invariant for symbolic ℤ2-actions, the projective fundamental group. This invariant is the limit of an inverse system of groups, each of which is the fundamental group of a space associated with the ℤ2-action. The limit group measures a kind of long-distance order that is manifested along loops in the plane, and roughly speaking bears the same relation to the mixing properties of the ℤ2-action that π1; of a topological space bears to π0. The projective fundamental group is invariant under topological conjugacy. We calculate this invariant for several important examples of ℤ2-actions, and use it to prove non-existence of certain constant-to-one factor maps between two-dimensional subshifts. Subshifts that have the same entropy and periodic point data can have different projective fundamental groups.

The local dimension of invariant and conformal measures for piecewise monotonic transformations on the interval is considered. For ergodic invariant measures m with positive characteristic exponent χm we show that the local dimension exists almost everywhere and equals hm/χm For certain conformal measures we show a relation between a pressure function and the Hausdorff dimension of sets, on which the local dimension is constant.

This paper yields diffusion and moderate deviation type asymptotics for solutions of differential equations of the form dZε(t)/dt = εB(Zε(t), fty) where ft is a suspension flow (in particular, a hyperbolic flow) over a sufficiently fast mixing transformation. Such problems emerge in the study of perturbed Hamiltonian systems. These exhibit a new class of limit theorems for dynamical systems and extend a number of previously known results.

Given a measure-preserving equivalence relation R with countable classes, we study relations between the properties of R and metric invariants. We give applications to pseudogroups of measure-preserving homeomorphisms.

Let Λ be a basic set of an Axiom A diffeomorphism of a compact Riemannian manifold M without boundary. If ε is small enough one can find by local product structure that for x ε Λ there is a neighborhood V(x) in M such that V ∩ Λ is homeomorphic to . The author proves that this homeomorphism can be extended to a homeomorphism of V onto .

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.