We consider a Ruelle—Perron—Frobenius type of selection procedure for probability measures that are invariant under random subshifts of finite type. In particular we prove that for a class of random functions this method leads to a unique probability exhibiting properties that justify the names Gibbs measure and equilibrium states. In order to do this we introduce the notion of bundle random dynamical systems and provide a theory for their invariant measures as well as give a precise definition of Gibbs measures.

We define entropy pairs for an invariant measure µ on a topological dynamical system (X, T), and show they allow one to construct the maximal topological factorwith entropy 0 for µ. Then we prove that for any µ, a µ-entropy pair is always topologically so, and the reverse is true when (X, T) is uniquely ergodic.

We introduce rotation numbers and pairs characterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.

Broadly speaking, we prove that an action of a group with very little commutativity cannot be stably orbit equivalent to an action of a group with enough commutativity, assuming both actions are free and finite measure preserving. For example, one group may be SL2(ℝ) and the other a group with infinite discrete center (e.g., the universal cover of SL2(ℝ)); I believe this is the first rigidity result of this type for a pair of simpleLie groups both of split rank one. Another example: one group may be any nonelementary word hyperbolic group, the other any group with infinite discrete center.

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 rank 1 locally symmetric space of finite Riemannian volume. It is proved that the set of unit vectors on a non-constant C1 curve in the unit tangent sphere at a point p ∈ M for which the corresponding geodesic is bounded (relatively compact) in M, is a set of Hausdorff dimension 1.

Let V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.

For the solid torus V = S1 × and a C1 embedding f: V → V given by with dϕ/dt > 1, 0 < λi(t) < 1 the attractor Λ = ∩i = 0∞fi(V) is a solenoid, and for each disk D(t) = {t} × (t ∈ S1) the intersection Λ(t) = Λ ∩ D(t) is a Cantor set. It is the aim of the paper to find conditions under which the Hausdorff dimension of Λ(t) is independent of t and determined by where the real numbers pi are characterized by the condition that the pressure of the function log : S1 → ℝ with respect to the expanding mapping ϕ: S1 → S1 becomes zero. (There are two further characterizations of these numbers.)

It is proved that (0.1) holds provided λ1, λ2 are sufficiently small and Λ satisfies a condition called intrinsic transverseness. Then it is shown that in the C1 space of all embeddings f with sup λi > Θ−2 (Θ the mapping degree of ϕ: S1 → S1) all those f which have an intrinsically transverse attractor Λ form an open and dense subset.

We extend the notion of no division for star maps to tree maps and prove that the set of periods of a tree map is cofinite if there exists some periodic orbit of the given map with period larger than one having no division. Using this result we obtain somesimple proofs of known results about the set of periods of a tree map. Also we show that this set is a union of initial segments of a finite number of linear orderings which depend only on the given tree minus a finite subset of ℕ.

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.

Rips' theorem about free actions on ℝ-trees relies on a careful analysis of finite systems of partial isometries of ℝ. We associate a free action on an ℝ-tree to any finite system of isometries without reflection. Any free action may be approximated (strongly in the sense of Gillet-Shalen) by actions arising in this way. Proofs involve the use, in an essential way, of separation properties of systems of isometries. We also interpret these finite systems of isometries as generating sets of pseudogroups of partial isometries between closed intervals of ℝ.

A class of locally isometric, but not necessarily invertible mappings of an interval is considered. We show that under some conditions the study of the dynamical properties of these mappings can be reduced to interval exchange transformations. On the other hand, there are examples of mappings in this class with ergodic invariant measures supported by Cantor sets. The so-called μβ -sets studied by Y. Katznelson appear naturally in such examples.

We first consider the rigidity of the marked length spectrum for non-compact surfaces of finite area.

We consider the following dynamical system: take a d-dimensional real vectorwith positive coordinates. Now keep the smallest coordinate and subtract this one from the others, and iterate this process. When the starting vector is x we denote by xn the result after n iterations. It is shown that for almost all x, limn→∞xn ≠ 0 (the null vector). This is shown to be equivalent to the conjectured finiteness of an algorithm which produces the critical probability in a certain dependent percolation model.

We characterize all possible sets of periods for all continuous self—mapson a tree having all branching points fixed. This result solves a problem which was originally posed by Alsedà, Llibre and Misiurewicz.

We consider the convergence of solutions to various conjugacy problems occurring in Hamiltonian dynamics. These give rise to nonlinear functional equations which we solve by using a norm well adapted to the problem. The approach provides an alternative to the classical majorization method.

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.

A degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

The classification of dynamical systems by the existence of certain common extensions has been carried out very successfully in the class of shifts of finite type (‘finite equivalence’, ‘almost topological conjugacy‘). We consider generalizations of these notions in the class of coded systems. Topological entropy is shown to be a complete invariant for the existence of a common coded entropy preserving extension. In contrast to the shift of finite type setting, this extension cannot be made bounded-to-1 in general. Common extensions with hyperbolic factor maps lead to a version of almost topological conjugacy for coded systems.

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).