In this paper we present a generalization to higher dimensions of the techniques for computation of the entropy of graphs in dimension one. Following these methods, we obtain a lower bound for the topological entropy of a differentiable map F:ℝn→ℝn possessing a snap-back repeller.

We show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.

Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

We define a class Σ of entire functions whose covering properties are similar to those of rational maps. The set Σ is closed under composition of functions, and we show that when regarded as dynamical systems of the plane, the elements of Σ share many properties with rational maps. In particular, they have finite dimensional spaces of quasiconformal deformations, and they contain no wandering domains in their stable sets.

When does a non-singular flow on a 3-manifold have a 2-dimensional foliation everywhere transverse to it? A complete answer is given for a large class of flows, those with 1-dimensional hyperbolic chain recurrent set. We find a simple necessary and sufficient condition on the linking of periodic orbits of the flow.

We investigate the implications of transverse homoclinic orbits to fixed points in dissipative diffeomorphisms of the annulus. We first recover a result due to Aronson et al. [3]: that certain such ‘rotary’ orbits imply the existence of an interval of rotation numbers in the rotation set of the diffeomorphism. Our proof differs from theirs in that we use embeddings of the Smale [61] horseshoe construction, rather than shadowing and pseudo orbits. The symbolic dynamics associated with the non-wandering Cantor set of the horseshoe is then used to prove the existence of uncountably many invariant Cantor sets (Cantori) of each irrational rotation number in the interval, some of which are shown to be ‘dissipative’ analogues of the order preserving Aubry-Mather Cantor sets found by variational methods in area preserving twist maps. We then apply our results to the Josephson junction equation, checking the necessary hypotheses via Melnikov's method, and give a partial characterization of the attracting set of the Poincaré map for this equation. This provides a concrete example of a ‘Birkhoff attractor’ [10].

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.

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.

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).

We study the effects of small random errors on the asymptotic distribution of points in the basin of a hyperbolic attractor.

The author regrets that there are certain errors in [1] and would like to give the following corrections.

All the theorems stated in [2] are true. In particular, theorem B, on the existence of an invariant measure, is true. However, in the proof of theorem B, it is necessary to use the result of G. Keller [1] on the existence of an invariant measure for a C1+α function. The attempt, in § 4 of this paper, to modify the proof of S. Wong so that it applies to this general case of a C1+α function, is faulty. These calculations do not overcome the real difficulties involved in the proof and leave major gaps in the estimates needed.