Kneading of Lorenz type for intervals and product spaces

Abstract

The dynamical properties of Lorenz attractors have been studied in a number of recent papers. While the Lorenz dynamical system is a flow in 3-space, a major tool for its study has been the kneading invariant of a piecewise monotonic map of an interval to itself. This is related to a semi-flow on a 2 dimensional branched manifold. The semi-flow can be used to define a flow which admits a geometric realisation in 3-space. The main results can be traced through references (1), (5) and (6). It is known that if the slope of the map always exceeds √2 then the periodic points are dense. That this cannot be true without some restriction on the kneading function is clear from Parry's measure theoretic results for piecewise linear kneading functions (4). In the first part of this paper I prove some analogous topological results concerning the condition that the kneading function is locally eventually onto, and concerning the non-wandering set and set of periodic points of the discrete semi-flow defined by the function. In the second part of the paper I give alternative proofs of results on kneading invariants in a more general context than Rand's (5). The new methods of proof are needed in the third part where kneading functions on intervals are used to define kneading functions on product spaces of intervals with compact metric spaces.