It is a well-known result in one-dimensional dynamics that if a continuous map of the interval has positive topological entropy, then it has a periodic orbit of period 2i for each integer i ≥ 0 [15] (see also [12]). In fact, one can say rather more: such a map has a sequence of periodic orbits (P)i ≥ 0 with per (Pi) = 2i which form a period-doubling cascade (that is, whose points are ordered and permuted in the way which would occur had the orbits been created in a sequence of period-doubling bifurcations starting from a single fixed point). This result reflects the central role played by period-doubling in transitions to positive entropy in a one-dimensional setting. In this paper we prove an analogous result for positive-entropy orientation-preserving diffeomorphisms of the disc. Using the notion [9] of a two-dimensional cascade, we shall show that such diffeomorphisms always have infinitely many ‘zero-entropy’ cascades of periodic orbits (including a period-doubling cascade, though this need not begin from a fixed point).

This paper is a supplement to [2]. There, solutions ψ., (μ) of certain free-boundary problems, depending on a small parameter μ, are related to the Green function G., (a(μ)) of a second-order, elliptic, partial differential equation defined on a bounded set Ω ⊂ ℝ2. The free boundary is the boundary of a set A(μ) ⊂ Ω that is unknown a priori and corresponds to the cross-section of a steady vortex ring or of a confined plasma in equilibrium, for example.

Let Σ be a closed surface of genus ≥ 1, G a compact Lie group, not necessarily connected with Lie algebra g, ξ,: P → Σ a principal G-bundle, and suppose Σ equipped with a Riemannian metric and g with an invariant scalar product so that the Yang—Mills equations on ξ are defined. Further, let

be the universal central extension of the fundamental group π of Σ and ΓR the group obtained from Γ when its centre Z is extended to the additive group R of the reals. We show that there are bijective correspondences between various spaces of classes of Yang—Mills connections on ξ and spaces of representations of Γ and ΓR (as appropriate) in G. In particular, we show that the holonomy establishes a homeomorphism between the moduli space N(ξ) of central Yang–Mills connections on ξ and the space Repξ(Γ, G) of representations of Γ in G determined by ξ. Our results rely on a detailed study of the holonomy of a central Yang–Mills connection and extend corresponding ones of Atiyah and Bott for the case where G is connected.

K. Motegi has kindly pointed out that the Lemma in Section 4 of the paper is incorrect. (The assertion in Case 1 on p. 95 that the sewing map of E(K) is periodic is false.) Hence Theorem 6 is unproven. However, we have a weaker result.

Some self-dual local rings of integers not free over their associated orders

By N. P. Byott

(Volume 110 (1991), 5–10)

Department of Mathematics, University of Exeter, Exeter EX4 4QE

(Received 19 August 1993; Revised 21 March 1994)