WANDERING INTERVALS FOR LORENZ MAPS WITH BOUNDED NONLINEARITY D. BERRY AND B. D. MESTEL ABSTRACT A Lorenz map satisfying a bounded nonlinearity condition is shown not to admit wandering intervals unless it renormalises to a non-surjective circle map (gap map) with irrational rotation number. 1. Introduction and statement of results The Lorenz equations are a well-known example of a system of differential equations that exhibit chaotic behaviour. We refer the reader to [5] for a detailed study of these equations. Lorenz maps are one-dimensional maps with a single discontinuity that arise as the Poincare return map for the flow on the branched manifold that models the strange attractor of the Lorenz equations. More precisely, let / = [a, b] be an interval containing 0. A Lorenz map F is a map F\I-*I such that (a) F\[a,0) and F\(0,b] are increasing homeomorphisms onto their images; (b) F(0-) = b,F(0 + ) = a. Here F(0 —) and F(0 + ) denote the left and right limits of Fat 0. It is convenient to leave F undefined at 0, because the iterate of 0 plays no real important role for these maps. What are important are the iterates of a = ^(0 + ) and b = F(0 —). Indeed, it is often convenient to regard 0 as two distinct points, 0 -I- and 0 —. We shall do this often in what follows. Figure 1 illustrates three examples of Lorenz maps. In this note we are concerned with the existence of wandering intervals. We say that an interval / is a wandering interval for a map F:/-> / if it satisfies: (a) its forward iterates {F n (J) \ n ^ 0} are all disjoint; (b) J is not contained in the basin of attraction of an attracting periodic orbit; (c) F n | / is a homeomorphism for all n > 0. The dynamics of Lorenz maps are complicated. An important question to ask is whether a Lorenz map has sensitive dependence on initial conditions, that is, whether the orbits of nearby points separate exponentially quickly. This sensitive dependence is a necessary condition for the Lorenz attractor modelled by the Lorenz map to be chaotic. If a Lorenz map has a wandering interval J, then sensitive dependence on initial conditions cannot hold at all points. In particular, iterates of nearby points within J do not separate because they remain within the iterates of /. The question of the existence or non-existence of wandering intervals for a dynamical system is subtle, and the answer depends on the analytical and topological properties of the map F. For circle maps the non-existence of wandering intervals for C ^homeomorphisms of the circle with log-F' of bounded variation is a classical result due to Denjoy [2]. His results have recently been extended to maps with critical points Received 11 October 1989; revised 27 March 1990. 1980 Mathematics Subject Classification 58F12, 58F13. Bull. London Math. Soc. 23 (1991) 183-189