For each d ≥ 2 there exists a polynomial p with real coefficients such that the associated Newton function z–[p(z)/p′(z)] has 2d–2 distinct attracting periodic orbits in the complex plane. According to a theorem of G. Julia, this is the maximal number of attracting orbits that any rational function of degree d can possess.

For a polynomial map the measure of maximal entropy is the equilibrium measure for the logarithm potential in the Julia set [1], [4].

Here we will show that in the case where f is a rational map such that f(∞) = ∞ and the Julia set is bounded, then the two measures mentioned above are equal if and only if f is a polynomial.

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.

Let f:Tn→Tn (n ≥ 3) be a hyperbolic toral automorphism. Let A be the set of α > 0 such that there is a Hölder continuous path of index α in Tn with 1-dimensional orbit-closure under f We prove that α0 = sup A can be expressed in terms of the eigenvalues of f and that α0 ∈ A if and only if α0 < 1.

It is shown that a smooth area-preserving monotone twist mapping ϕ of an annulus A can be interpolated by a flow ϕt which is generated by a t-dependent Hamiltonian in ℝ × A having the period 1 in t and satisfying a Legendre condition. In other words, any such monotone twist mapping can be viewed as a section mapping for the extremals of variational problem on a torus:

where F has period 1 in t and x and satisfies the Legendre condition Fẋẋ>0.

For every Perron number λ we construct an infinite collection of topological Markov shifts with entropy log λ whose spectra are disjoint except for the necessary conjugates of λ. This is used to show that Marcus' theorem about every Markov shift of entropy log n factoring onto the full n-shift does not extend to certain entropy values.

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.

In this paper we give an extension to C2 endomorphisms of the circle of the well known Denjoy's theorem on C2 diffeomorphisms. We also give a simple proof of an extension of Block and Franke's theorem on the existence of periodic points for maps of the circle.

We prove that any topological conjugacy between subshifts is decomposed into the product of‘bipartite codes’, and obtain a natural generalization of Williams' theorem to sofic systems: two sofic systems are topologically conjugate iff the ‘representation matrices’ of the right [left] Krieger covers for them are ‘strong shift equivalent’ within right [left] Krieger covers; a similar result with respect to Fischer covers holds for transitive sofic systems.

We present here an investigation of the degree to which mixing properties can be shown to lift to cocycle extensions of an ergodic map by ℤn and ℝn weakly mixing actions. A number of other results on such extensions are also included.

Various definitions of the entropy for countable-state topological Markov chains are considered. Concrete examples show that these quantities do not coincide in general and can behave badly under nice maps. Certain restricted random walks which arise in a problem in magnetic recording provide interesting examples of chains. Factors of some of these chains have entropy equal to the growth rate of the number of periodic orbits, even though they contain no subshifts of finite type with positive entropy; others are almost sofic – they contain subshifts of finite type with entropy arbitrarily close to their own. Attempting to find the entropies of such subshifts of finite type motivates the method of entropy computation by loop analysis, in which it is not necessary to write down any matrices or evaluate any determinants. A method for variable-length encoding into these systems is proposed, and some of the smaller subshifts of finite type inside these systems are displayed.

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.

A natural geometrical representation of the geodesic flow on a surface M of constant negative curvature is given in which the base transformation is the shift on a (finite type) space of shortest words relative to a fixed generating set for π1(M) and the height function is the hyperbolic distance across a fundamental region for π1(M). This representation is obtained by comparing cutting sequences on M with generalised continued fraction expansions of endpoints on ℝ

Extending a result of Livsic [10] it is proved that the coboundary equation f(Tx)−f(x) = g(x) admits a C∞ solution f for C∞g when T is an ergodic toral endomorphism and g sums to zero over every periodic orbit. The same statement is false with C1 in place of C∞, in contrast to the Livsic (hyperbolic) theorem. In one dimension the ‘Lip α’ case leads to questions relating to the generalized Riemann hypothesis.

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.

In this paper, we prove an extension of a theorem of Marcus, which says that every subshift of finite type of entropy log n, n an integer, factors onto the full n-shift. Let p(x) be a monic polynomial, irreducible over ℚ, whose coefficients (except for the leading coefficient) are non-positive integers. Suppose C(λ) is the companion matrix of p(x), where λ is the largest real root of p(x) (λ exists, by the Perron-Frobenius theorem). Then for any aperiodic, non-negative, integral matrix A, with Perron value λ, we give necessary and sufficient conditions for the existence of a positive integer n and a right-closing map f: ΣA→ΣC(λ) satisfying fσn = σnf (where σ is the shift map).

Given an orientation preserving C2 expanding mapping g: S1 → Sl of a circle we consider the family of closed invariant sets Kg(ε) defined as those points whose forward trajectory avoids the interval (0, ε). We prove that topological entropy of g|Kg(ε) is a Cantor function of ε. If we consider the map g(z) = zq then the Hausdorff dimension of the corresponding Cantor set around a parameter ε in the space of parameters is equal to the Hausdorff dimension of Kg(ε). In § 3 we establish some relationships between the mappings g|Kg(ε) and the theory of β-transformations, and in the last section we consider DE-bifurcations related to the sets Kg(ε).

Lyapunov exponents for two-dimensional matrix cocycles are shown to be stable under certain stochastic perturbations.

The crossed product of an AF-algebra by an automorphism, a power of which is approximately inner, is shown to be embeddable into an AF-algebra. The proof uses almost inductive limit automorphisms, i.e. automorphisms possessing a sequence of almost invariant finite-dimensional C*-subalgebras converging to the given AF-algebra.

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.