In this thesis we investigate the topological nature of the
Collet-Eckmann condition for S-unimodal maps of the interval.
The Collet-Eckmann condition holds, by definition, if the
Lyapunov exponent is positive along the orbit of the critical
point; this is a metric condition.  It implies that the
Lyapunov exponent is positive Lebesgue almost everywhere and
that the map is chaotic.  We relate topological and metric
properties using an extended form of Hofbauer's tower
construction.  The main results are as follows:

We prove that the Collet-Eckmann condition is invariant under
quasi-symmetric conjugacy between S-unimodal maps.

We show that any S-unimodal map whose kneading invariant satisfies
certain simple conditions satisfies the Collet-Eckmann condition.
The conditions are topological analogues of those used by Benedicks
and Carleson in their proof of Jakobson's theorem.  We also give
examples of kneading invariants corresponding to failure of the
Collet-Eckmann condition.  This falls short of a complete
classification of kneading invariants into Collet-Eckmann or
non-Collet-Eckmann.

We prove that Lebesgue almost every value of the topological entropy
has the property that any S-unimodal map with that topological entropy
must satisfy the Collet-Eckmann condition.  This is analogous to
Lebesgue almost every rotation number having the property that any
smooth circle map with that rotation number must possess an absolutely
continuous invariant measure.

These results form part of a topological proof of Jakobson's theorem;
it remains only to show that Collet-Eckmann kneading invariants are
taken on for a positive measure set of parameter values in the
quadratic family.