We show that given some natural conditions on a 3 × 3 hyperbolic matrix of integers A(det A = 1) there exists a Markov partition for the induced map A(x + ℤ3) = A(x)+ℤ3 on T3 whose transition matrix is (A−1)t. For expanding endomorphisms of T2 we construct a Markov partition so that there is a semiconjugacy from a full (one-sided) shift.

Let f denote a continuous map of a compact interval to itself, P(f) the set of periodic points of f and Λ(f) the set of ω-limit points of f. Sarkovskǐi has shown that Λ(f) is closed, and hence ⊆Λ(f), and Nitecki has shown that if f is piecewise monotone, then Λ(f)=. We prove that if x∈Λ(f)−, then the set of ω-limit points of x is an infinite minimal set. This result provides the inspiration for the construction of a map f for which Λ(f)≠.

Kupka-Smale like theorems are proven in the real analytic case, using existing perturbation schemes for the smooth case and the heat operator. As a consequence, a topological proof is obtained of Siegel's theorem on the generic divergence of normal form transformations.

In the class of rank-1 transformations, there is a strong dichotomy. For such a T, the commutant is either irivial, consisting only of the powers of T, or is uncountable. In addition, the commutant semigroup, C(T), is in fact a group. As a consequence, the notion of weak isomorphism between two transformations is equivalent to isomorphism, if at least one of the transformations is rank-1. In § 2, we show that any proper factor of a rank-1 must be rigid. Hence, neither Ornstein's rank-1 mixing nor Chacón's transformation, can be a factor of a rank-1.

We give a new proof of the existence of analytic models of pseudo-Anosov maps. The persistence properties of Thurston's maps ensure that any Co-perturbation of them presents all their dynamical features. Using Lyapunov functions of two variables we are able to choose certain analytic perturbations which do not add any new dynamical behaviour to the original pseudo-Anosov map.

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.

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.

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.

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.

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.