Mathematics – Dynamical Systems
Scientific paper
1998-09-17
Mathematics
Dynamical Systems
57 pages, 33 PostScript figures
Scientific paper
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).
Milnor John
Tresser Charles
No associations
LandOfFree
On Entropy and Monotonicity for Real Cubic Maps does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On Entropy and Monotonicity for Real Cubic Maps, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On Entropy and Monotonicity for Real Cubic Maps will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-536489