Nonlinearity 9 (1996) 999–1014. Printed in the UK
Zeros of the kneading invariant and topological entropy
for Lorenz maps
Paul Glendinningyx and Toby Hallzk
y DAMTP, Silver Street, Cambridge CB3 9EW, UK
z Centre de Recerca Matema`tica, Institut d’Estudis Catalans, Apartat 50, E-08193 Bellaterra,
Spain
Received 9 January 1995, in final form 16 February 1996
Recommended by R S Mackay
Abstract. If f : [0; 1] ! [0; 1] is a unimodal map, then its topological entropy is related
to the smallest positive zero s of a certain power series (the kneading invariant of f ) by
h.f / D log.1=s/ [14]. Moreover, it is implicit in the results of Jonker and Rand [12] that for
each positive entropy basic set i in the renormalization decomposition of the non-wandering
set of f , there is a real zero si of the kneading invariant such that h.f ji / D log.1=si /. Here
we prove a similar result for Lorenz maps. In contrast to the unimodal case, it is possible for
two basic sets in the renormalization decomposition of the non-wandering set of a Lorenz map
to have the same entropy, and we show that in this case there is a corresponding double zero
of the kneading invariant.
AMS classification scheme numbers: 58F13, 58F14
1. Renormalization and kneading theory for unimodal maps
Although the main emphasis in this paper is on Lorenz maps, we begin with a brief review
of some standard results for unimodal maps (that is, continuous maps, f of the interval,
I D [0; 1] with f .0/ D f .1/ D 0 which are strictly increasing on [0; c] and strictly
decreasing on [c; 1] for some c 2 .0; 1/). In fact, since there is a simple correspondence
between the dynamics of unimodal maps and symmetric Lorenz maps [5], proofs of the
results stated at the end of this section can be derived from our main results as simple
special cases. Throughout this section we assume that f .c/ > c, since otherwise the
dynamics is trivial.
The theory of renormalization for unimodal maps is well-documented (see for
example [7, 17, 18]): a unimodal map f : I ! I is renormalizable if there is a proper
subinterval J of I and an integer n > 1 such that the restriction of f n to J is
itself (topologically conjugate to) a unimodal map. This unimodal map may itself be
renormalizable: consideration of the longest possible sequence of renormalizations (possibly
infinite) gives rise to a decomposition, .f / D SpiD0 i of the non-wandering set of f into
p C 1 f -invariant basic sets, for some p 2 N [ f1g which is given by the number of
essential renormalizations in the sequence (those with n > 2). Under additional conditions
(for example, if f has negative Schwarzian derivative), i is the union of a Cantor set and
x Current address: School of Mathematical Sciences, Queen Mary & Westfield College, London E1 4NS, UK.
k Current address: Department of Pure Mathematics, University of Liverpool, Liverpool L69 3BX, UK.
0951-7715/96/040999+16$19.50 c© 1996 IOP Publishing Ltd and LMS Publishing Ltd 999