## Institute for Mathematical Sciences

## Preprint ims98-9

** J. Milnor and C. Tresser**
* On Entropy and Monotonicity for Real Cubic Maps*

Abstract: It has been known for some time that the topological entropy is a nondecreasing function of the parameter in the real quadratic
family, which corresponds to the intuitive idea that more
nonlinearity induces more complex dynamical behavior.
Polynomial families of higher degree depend on several
parameters, so that the very question of monotonicity needs to
be reformulated. For instance, one can say the entropy is
monotone in a multiparameter family if the isentropes, or sets
of maps with the same topological entropy, are connected. Here
we reduce the problem of the connectivity of the isentropes in
the real cubic families to a weak form of the Fatou conjecture
on generic hyperbolicity, which was proved to hold true by
C. Heckman. We also develop some tools which may prove to be
useful in the study of other parameterized families, in
particular a general monotonicity result for stunted sawtooth
maps: the stunted sawtooth family of a given shape can be
understood as a simple family which realizes all the possible
combinatorial structures one can expect with a map of this
shape on the basis of kneading theory. Roughly speaking, our
main result about real cubic families is that they are as
monotone as the stunted sawtooth families with the same shapes
because of Heckman's result (there are two posible shapes for
cubic maps, depending on the behavior at infinity).

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