We study the dynamics of continuous maps of the circle with periodic points. We show that the centre is the closure of the periodic points and that the depth of the centre is at most two. We also characterize the property that every power is transitive in terms of transitivity of a single power and some periodic data.

The cohomology action of an Anosov diffeomorphism on a nilmanifold resembles that of a Cartesian product map. Corresponding results hold for infranilmanifolds, giving an invariant bigrading of the cohomology and a fourfold symmetry that extends Poincaré duality. Holonomy invariant cocycles are applied to the action on first cohomology.

Let φ: ℝ × M → M be a continuous flow on a compact C∞ two-manifold M. It is proved that there exists a C1 flow ψ on M which is topologically equivalent to φ, and that the following conditions are equivalent:

(a) any minimal set of φ is trivial;

(b) φ is topologically equivalent to a C2 flow;

(c) φ is topologically equivalent to a C∞ flow.

Also proved is a structure and an existence theorem for continuous flows with non-trivial recurrence.

Vector fields of , 1 ≤ r ≤ ∞, with non-trivial recurrent points are classified in two types, one of which we call the constant type inspired by the terminology of continued fractions. Let have finitely many singularities and p ∈ T2 be a non-wandering point of X. With the exception of the case when X is of constant type and, simultaneously, p is non-trivial recurrent, we prove that there exists arbitrarily close to X (in the Cr-topology) having a periodic trajectory through p.

Let P and f be polynomials in several (real) variables, with P having no negative coefficients. We give necessary and sufficient conditions for there to exist a positive integer n with Pnf having no negative coefficients; roughly speaking, the conditions involve the behaviour of f as a function on the positive orthant, together with its behaviour on a boundary constructed from the supporting monomials of P. This completes a series of results due to Poincaré (1883), Meissner (1911), and Polyà (1927). The former discusses the one variable case, the latter two deal with the situation that the Newton polyhedra of both P and f be, respectively, standard hypercubes, standard simplices.

Shift equivalence is the relation between A, B that there exists S, R, n > 0 with RA = BR, AS = SB, SR = An, RS = Bn. Strong shift equivalence is the equivalence relation generated by these equations with n = 1. We prove that for many Boolean matrices strong shift equivalence is characterized by shift equivalence and a trace condition. However, we also show that if A is strongly shift equivalent to B, then there exists a homomorphism from an iterated directed edge graph of A to the graph of B preserving the traces of powers. This yields results on colourings of iterated directed edge graphs and might distinguish new strong equivalence classes.

We consider the action of the subgroup of the mapping class group consisting of homeomorphisms that extend to the handlebody on Thurston's sphere of measured foliations. Properties of the limit set and domain of discontinuity are described and for genus two it is shown the limit set has measure zero.

We study a class of maps of the real line into itself which are degree one liftings of maps of the circle and have discontinuities only of a special type. This class contains liftings of continuous degree one maps of the circle, lifting of increasing mod 1 maps and some maps arising from Newton's method of solving equations. We generalize some results known for the continuous case.

We consider G (Galois) coverings of Axiom A flows (restricted to basic sets) and prove an analogue of Chebotarev's theorem. The theorem provides an asymptotic formula for the number of closed orbits whose Frobenius class is a given conjugacy class in G. An application answers a question raised by J. Plante. The basic method is then extended to compact group extensions and applied to frame bundle flows defined on manifolds of variable negative curvature.

Let G be a connected finite-dimensional Lie group and M a compact surface. We investigate whether, for a given G and M, every continuous action of G on M must have a fixed (stationary) point. It is shown that when G is nilpotent and M has non-zero Euler characteristic that every action of G on M must have a fixed point. On the other hand, it is shown that the non-abelian 2-dimensional Lie group (affine group of the line) acts without fixed points on every compact surface. These results make it possible to complete this investigation for Lie groups of dimension at most 3.