We introduce rotation numbers and pairs characterizing cyclic patterns on an interval and a special order among them; then we prove the theorem which specializes the Sharkovskii theorem in this setting.

Let V be a four-dimensional complex symplectic vector space. This paper classifies those connected linear algebraic subgroups of the symplectic group Sp(V) that admit two independent rational invariants. As an application we show the non integrability of a three degree of freedom Hamiltonian system.

We consider the convergence of solutions to various conjugacy problems occurring in Hamiltonian dynamics. These give rise to nonlinear functional equations which we solve by using a norm well adapted to the problem. The approach provides an alternative to the classical majorization method.

The purpose of this paper is to present some simple examples that are hyperbolic everywhere except at one point, but which do not admit SBR measures. Each example has a fixed point at which the larger eigenvalue is equal to one and the smaller eigenvalue is less than one.

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak if

It is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

We prove that unimodal Fibonacci maps with negative Schwarzian derivative and a critical point of order ℓ have a finite absolutely continuous invariant measure if ℓ ∈ (1 ℓ1) where ℓ1 is some number strictly greater than 2. This extends results of Lyubich and Milnor for the case ℓ = 2.

Systems possessing symmetries often admit heteroclinic cycles that persist under perturbations that respect the symmetry. The asymptotic stability of such cycles has previously been studied on an ad hoc basis by many authors. Sufficient conditions, but usually not necessary conditions, for the stability of these cycles have been obtained via a variety of different techniques.

We begin a systematic investigation into the asymptotic stability of such cycles. A general sufficient condition for asymptotic stability is obtained, together with algebraic criteria for deciding when this condition is also necessary. These criteria are always satisfied in ℝ3 and often satisfied in higher dimensions. We end by applying our results to several higher-dimensional examples that occur in mode interactions with O(2) symmetry.

Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent one-parameter subgroups, then the limit μ of such a sequence is supported on a closed orbit of the subgroup preserving it, and is invariant and ergodic for the action of a unipotent one-parameter subgroup of G.

In this paper the Pesin's entropy formula and ‘large ergodic theorem’ are constructed for Axiom A endomorphisms.

We introduce a notion of autonomous dynamical systems which generalizes algebraic dynamical systems. We show by giving examples and by describing some properties that this generalization is not a trivial one. We apply the methods then developed to algebraic Anosov systems. We prove that a C1-submanifold of finite volume, which is invariant by an algebraic Anosov system is ‘essentially’ algebraic.