Let (Y, g) be a compact connected n-dimensional Riemannian manifold and let () be its universal cover endowed with the pulled-back metric. If y ∈ , we define

where B(y, R) denotes the ball of radius R around y in . It is a well known fact that this limit exists and does not depend on y ([Man]). The invariant h(g) is called the volume entropy of the metric g but, for the sake of simplicity, we shall use the term entropy. The idea of recognizing special metrics in terms of this invariant looks at first glance very optimistic. First the entropy, which behaves like the inverse of a distance, is sensitive to changes of scale which makes it a bad invariant: however, this is a minor drawback that can be circumvented by looking at the behaviour of the entropy functional on the space of metrics with fixed volume (equal to one for example). Nevertheless, it seems very unlikely that two numbers, the entropy and the volume, might characterize any metric. The very first person to consider such a possibility was Katok ([Kat1]). In this article the entropy is thought of as a dynamical invariant which actually is suggested by its name. More precisely, let us define this dynamical invariant, which is called the topological entropy: let (M, d) be a compact metric space and ψt, a flow on it, we define

.

A family of quadratic maps of the plane has been found numerically for certain parameter values to have three attractors, in a triangular pattern, with ‘intermingled’ basins. This means that for every open set S, if the basin of attraction of one of the attractors intersects S in a set of positive Lebesgue measure, then so do the other two basins. In this paper we mathematically verify this observation for a particular parameter, and prove that our results hold for a set of parameters with positive Lebesgue measure.

We investigate the relation between the complexity function of a sequence, that is the number p(n) of its factors of length n, and the rank of the associated dynamical system, that is the number of Rokhlin towers required to approximate it. We prove that if the rank is one, then lim , but give examples with lim for any prescribed function G with G (n) = 0(an) for every a > 1. We give exact computations for examples of the ‘staircase’ type, which are strongly mixing systems with quadratic complexity. Conversely, for minimal sequences, if p(n) < an + b for some a ≥ 1, the rank is at most 2[a], with bounded strings of spacers, and the system is generated by a finite number of substitutions.

We show that for a certain family of integrable reversible transformations, the curves of periodic points of a general transformation cross the level curves of its integrals. This leads to the divergence of the normal form for a general reversible transformation with integrals. We also study the integrable holomorphic reversible transformations coming from real analytic surfaces in ℂ2 with non-degenerate complex tangents. We show the existence of real analytic surfaces with hyperbolic complex tangents, which are contained in a real hyperplane, but cannot be transformed into the Moser—Webster normal form through any holomorphic transformation.

Given a connected Lie group G acting ergodically on a space S with finite invariant measure, one can ask when G will contain single elements (or one-parameter subgroups) that still act ergodically. For a compact simple group or the isometry group of the plane, or any group projecting onto such groups, an ergodic action may have no ergodic elements, but for any other connected Lie group ergodic elements will exist. The proof uses the unitary representation theory of Lie groups and Lie group structure theory.

An IP set in ℕ is a subset of ℕ which coincides with the set of finite sums taken from an infinite sequence in ℕ with not necessarily distinct terms. It has been established by H. Furstenberg and others that there is a rich connection between IP sets and idempotents in the enveloping semigroup E(X) of a compact metric dynamical system. Our aim in this paper is to further develop this program by considering special elements of E(X) with various IP properties.

This paper describes a method for obtaining rigorous numerical bounds on time averages for a class of one-dimensional expanding maps. The idea is to directly estimate the absolutely continuous invariant measure for these maps, without computing trajectories. The main theoretical result is a bound on the convergence rate of the Frobenius—Perron operator for such maps. The method is applied to estimate the Lyapunov exponents for a planar map of recent interest.

We investigate invariant ergodic measures for certain partially hyperbolic and Anosov actions of ℝk, ℤk and We show that they are either Haar measures or that every element of the action has zero metric entropy.

Consider the partial differential equation f(x, y(x), dy(x)) = 0, where f is a smooth real function on ℝn × ℝ × (ℝn)*. Near each singularity of the characteristic foliation, a Liouville field is associated to the equation; we classify hyperbolic germs of Liouville fields up to symplectic transformations, hence we deduce normal forms for partial differential equations up to transformations which preserve the standard contact form of ℝ2n+1. For n = 1, a theorem of Davydov enables us to deduce normal forms for such equations up to transformations of the x, y plane.

In this paper we estimate fractal dimensions of almost periodic orbits in terms of two kinds of exponents: the exponent in the inclusion lengths for ε-almost period and the exponent in Hölder conditions. Further, we estimate the inclusion lengths for ε-almost period of quasi-periodic functions by using Diophantine approximations. In the n-frequency quasi-periodic case we can show that the fractal dimension of its orbit is majorized by the value n/δ when it is Hölder continuous with exponent δ, 0 < δ ≤ 1.

We consider a generalized Fredholm determinant d(z) and a generalized Selberg zeta function ζ(ω)−1 for Axiom A diffeomorphisms of a surface and Axiom A flows on three-dimensional manifolds, respectively. We show that d(z) and ζ(ω)−1 extend to entire functions in the complex plane. That the functions are entire and not only meromorphic is proved by a new method, identifying removable singularities by a change of Markov partitions.

We show that a C0 minimal action of a semisimple Lie group without compact factors is either locally free or induced from a minimal action of a proper parabolic subgroup. We describe the orbit structure of the action restricted to a maximal compact subgroup, and then apply this to minimal actions in low dimension. We give some examples, applications, and open problems.

In this paper we shall give examples of rational maps on the Riemann sphere and also of polynomial interval maps which are transitive but not ergodic with respect to Lebesgue measure. In fact, these maps have two disjoint compact attractors whose attractive basins are ‘intermingled’, each having a positive Lebesgue measure in every open set. In addition, we show that there exists a real bimodal polynomial with Fibonacci dynamics (of the type considered by Branner and Hubbard), whose Julia set is totally disconnected and has positive Lebesgue measure. Finally, we show that there exists a rational map associated to the Newton iteration scheme corresponding to a polynomial whose Julia set has positive Lebesgue measure.

The Hausdorff dimension of closed invariant sets under diffeomorphisms is an interesting concept as it is a measure of their complexity. The theory of holomorphic dynamical systems provides us with many examples of fractal sets and, in particular, a theorem of Ruelle [Ru1] shows that the Hausdorff dimension of the Julia set depends real analytically on f if f is a rational function of ℂ and the Julia set J of f is hyperbolic. In this paper we generalize Ruelle's result for complex dimension two and show the real analytic dependence of the Hausdorff dimension of the corresponding Julia sets of hyperbolic Hénon mappings.