scavo@spencer.cs.uoregon.edu (Tom Scavo) (08/22/90)
Can anyone give some references on the dynamics of piecewise linear maps of the interval? Many textbooks use these functions to introduce concepts in chaotic dynamics, but in an overly general way. I'm interested in more detailed and specific results (what are the 2^n periodic points of period n of the tent map, for instance) which are probably buried in some papers somewhere. I sure would appre- ciate relevant pointers into the literature. Thanks, Tom Scavo <scavo@cs.uoregon.edu> ---------
scavo@spencer.cs.uoregon.edu (Tom Scavo) (09/05/90)
Recently, I asked for references concerning piecewise linear
maps of the interval. Here's a chronological summary of the
response (bracketed [] comments are my own):
*******************************************************************
Gerald Edgar <edgar@shape.mps.ohio-state.edu>:
Hao, Bai-lin. _Elementary_Symbolic_Dynamics_and_Chaos_in_
_Dissipative_Systems_. World Scientific, 1989. [A very
entertaining book written by and for physicists; has one
of the most exhaustive bibliographies around.]
Ken Olstad <olstad@uh.msc.umn.edu>:
Brucks, K.M. "Uniqueness of aperiodic kneading sequences."
_Proceedings_of_the_AMS_ (to appear). [Haven't seen it---
apparently deals with a class of functions called
trapezoidal maps (see below).]
Evans, Michael, Paul Humke, Cheng-Ming Lee, and Richard J.
O'Malley. "Characterizations of turbulent one dimensional
mappings via omega-limit sets."
Brucks, K., M. Misiurewicz, and C. Tresser. "Monotonicity
properties of the family of trapezoidal maps." Preprint,
1989.
Gambaudo, Jean Marc and Charles Tresser. "A monotonicity
property in one dimensional dynamics."
chopin@ucscc.UCSC.EDU (Toshiro Kendrick Ohsumi):
Collet, Pierre and Jean-Pierre Eckmann. _Iterated_Maps_on_
_the_Interval_as_Dynamical_Systems_. Birkhauser, 1980.
[One of the first to treat maps as bona fide dynamical sys-
tems; poorly typeset, but nevertheless a classic.]
Devaney, R.L. _An_Introduction_to_Chaotic_Dynamical_Systems_
(second edition). Addison-Wesley, 1989. [Certainly one of
the most authoritative sources of material on iterated maps.]
Lichtenberg, A.J. and M.A. Liberman. _Regular_and_Stochastic_
_Motion_. Springer-Verlag, 1983.
pmd@axiom.maths.uq.oz.au (phil diamond):
Baldwin, S. "A complete classification of piecewise mono-
tone functions on the interval." _Trans_Amer_Math_Soc_
319(1) 1990, pp.155-178.
Parker, Thomas R. and Leon O. Chua. _Practical_Numerical_
Algorithms_for_Chaotic_Systems_. Springer-Verlag, 1989.
[Indispensible source of algorithms for exploring continuous
dynamical systems; implemented and available as part of the
INSITE software package.]
Guckenheimer, J. and S. Johnson. "Distortion of S-unimodal
maps." _Annals_Math_ 132 (1990), pp.71-130.
Diamond, Phil. "Iterated maps on discretized meshes of
the unit interval." _Computers_Math_Applic_ (to appear).
[Available from the author upon request.]
*******************************************************************
I still haven't found what I'm looking for (but then maybe I don't
know what I'm looking for 8-). Something like "Everything you ever
wanted to know about 'x' but were afraid to ask..." where 'x' might
be the tent map, the doubling map, etc. Surely somebody must have
computed the periodic points of the tent map explicitly! Not
approximations (which would be difficult anyway, since these points
are repelling) but the general form of the rational numbers com-
prising the 2^n periodic points of period n under iteration of the
tent map, for instance.
Tom Scavo <scavo@cs.uoregon.edu>
---------