We use a new method to construct a class of strongly uniformly distributed D-sequences. Such sequences are used to determine ergodic measure-preserving transformations.

The aim of this paper is to prove that any transitive constant-to-one extension of a coded system is still coded. The corresponding statement for sofic systems is false, and it is an open question whether it is true for subshifts of finite type.

Let us consider (Ω, , μ, G) a measure-preserving dynamical system, (Ω, , μ) is a probability space. The group G, which is supposed to be either ℤd or ℝd (d ≥ 1), acts on Ω by measure-preserving transformations. This action is denned by a map

which is jointly measurable, such that Tx+y = TxTy and Txμ = μ

This paper introduces a Riemannian invariant of a compact Riemannian manifold based on the spectral theory for the Jacobi field operator. It is the Floquet exponent for this operator, a purely dynamical quantity computable directly from the asymptotic behavior of Jacobi fields. We show that it is related to certain traces of the Green's function, and we derive further regularity and analyticity properties for the Green's function. In case the geodesic flow is ergodic, the Floquet exponent generalizes the measure entropy, and several entropy estimates follow. An asymptotic expansion of the Floquet exponent gives rise to a sequence of ‘Jacobi invariants’, which are related to the polynomial invariants of the K dV equation.

We consider g-measures for the shift on where S is a finite set. For a certain class of continuous g, two g-measures are identified; they are equal if and only if there is a unique g-measure. We prove that the natural extensions of these measures are Bernoulli. With a further restriction on g when S is a two-point set, we show that there is a unique g-measure. We also consider extensions of these results to the non-continuous case.

We present an existence theorem for certain kinds of orbits of a monotone twist map and use it to obtain a criterion for proving that there are no invariant circles with a certain range of rotation numbers. We have used this criterion to prove (computer assisted) that the standard map has no invariant circles for several parameter values including k = 0.9718.

We prove some results related to the question of the existence of factor maps and eventual factor maps between shifts of finite type. Our main result is that if A and B are integral eventually positive (IEP) matrices, and A eventually factors finite-to-one onto B, then there exists an IEP matrix C such that A eventually factors onto C by left closing maps and C eventually factors onto B by right closing maps. This answers the question of the existence of finite-to-one eventual factor maps when the spectrum of A is simple. As a corollary, if in addition to the above hypothesis, χ*A=χ*B, (where χ*A is the characteristic polynomial of A modulo x), then A is shift equivalent to B.

We show that if a homeomorphism f of the torus, isotopic to the identity, has three or more periodic orbits with non-collinear rotation vectors, then it has positive topological entropy. Furthermore, for each point ρ of the convex hull Δ of their rotation vectors, there is an orbit of rotation vector ρ, for each rational point p/q, p ∈ ℤ2, q ∈ ℕ, in the interior of Δ, there is a periodic orbit of rotation vector p / q, and for every compact connected subset C of Δ there is an orbit whose rotation set is C. Finally, we prove that f has ‘toroidal chaos’.

We study Markov chains via invariants constructed from periodic orbits. Canonical extensions, based on these invariants, are used to establish a constraint on the degree of finite-to-one block homomorphisms from one Markov chain to another. We construct a polytope from the normalized weights of periodic orbits. Using this polytope, we find canonically-defined induced Markov chains inside the original Markov chain. Each of the invariants associated with these Markov chains gives rise to a scaffold of invariants for the original Markov chain. This is used to obtain counterexamples to the finite equivalence conjecture and to a conjecture regarding finitary isomorphism with finite expected coding time. Also included are results related to the problem of minimality (with respect to block homomorphism) of Bernoulli shifts in the class of Markov chains with beta function equal to the beta function of the Bernoulli shift.

We prove an infinite-area lemma for maps which are area non-contracting on the boundaries of certain domains; the maps are required to be smooth to the extent that their Jacobians are twice differentiable (see Main Lemma below). It will follow that a hyperbolic rational map has no wandering simply connected domains. As a more direct corollary, a C3 diffeomorphism f of a compact smooth 2-manifold cannot have a wandering domain Δ if f is area non-contracting on the boundary of each forward image of Δ.

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.

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.