Smale-Williams Attractor
Dr. Robert F. Williams accepted the invitation on 29 January 2007 (self-imposed deadline: 29 February 2007).
This article will briefly cover: Attractors with a hyperbolic structure in the sense of Smale. (possible working name: SW attractors)Originated with the 1967 paper of Smale and an article by me, also in 1967, entitled 'One dimensional non-wandering sets' in which 1-dimensional sw attractors were characterized. This was extended to higher dimensions with a paper 'Expanding Attractors'. Where this concept was introduced in 1974. It has recently (1998, Anderson-Putnam) to include 'tiling spaces.' msmath,Amssymb} \Usepackage{Epsf} %\Usepackage{Dvips} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \Newtheorem{Cor}{Corollary} \Newtheorem{Con}{Conjecture} \newtheorem{defn}{Definition} \newtheorem{lem}{Lemma} \newtheorem{prop}{Proposition} \newtheorem{rem}{Remark} \newtheorem{thm}{Theorem} %\newtheorem{stand}[stand]{Assumption} \def \ted{{\mathbb T} } \def \ded{{\mathbb D} } \def \red{{\mathbb R} } \def \zed{{\mathbb Z} } \def \ned{{\mathbb N} } \def \l{\ell} \def \led{\mathcal{L}} \def \ced{\mathcal{C}} \def \bed{\mathcal{B}}
\begin{document} \title{Smale-Williams attractors } \author{R. F. Williams} %\markboth \maketitle \section{Introduction} \section{Contents} \begin{enumerate} \item definition of attractors, and SW attractors \item example dyadic solenoid \item example DA \item show how they collapse to branched manifolds \item branched manifolds \item Theorem 1 dimensional SWA's arise in this manner, where UNIQa5f350079404ab7-MathJax-00000001-QINU satisfies 1,2,3. \item commutative diagram big \item 1-D tiling spaces \item Expanding attractors \item Anderson Putnam theorem \item An unsolved conjecture \end{enumerate} A Smale-Williams attractor SWA for dynamical system is 1) an attractor for the system which 2) has a hyperbolic structure on its chain recurrent set, and 3) is neither a sink nor a periodic orbit. The dyadic solenoid, though long know in many branches of mathematics as a space (even a group), was described by Smale as an attractor in lectures in the mid 60's. \cite[S](The Lorenz attractor, seemingly an attractor by computer experiments, attracted little attention before J. Guckenheimer published ``A strange strange attractor'' in 1975.) Williams, published 1967, found a classification of all SWA which, like the solenoid, are 1 dimensional. Though 'fractal' (a word introduced several years later), SWA are determined by simple map: for the dyadic solenoid, the simple map is the 2-to 1 map UNIQa5f350079404ab7-MathJax-00000002-QINU which raps the circle around itself twice. The role of the circle, a 1-manifold, is enlarged to branched 1-manifolds, or 'train tracks'. Branched 1-manifolds have a (smooth) structure; endomorphisms may be smooth; as usual, an immersion UNIQa5f350079404ab7-MathJax-00000003-QINU is a differentiable map with derivative UNIQa5f350079404ab7-MathJax-00000004-QINU nonsingular at each point; and a map UNIQa5f350079404ab7-MathJax-00000005-QINU is expanding if UNIQa5f350079404ab7-MathJax-00000006-QINU at each point. Some examples and figures.\par For the classification, one has first a map UNIQa5f350079404ab7-MathJax-00000007-QINU where UNIQa5f350079404ab7-MathJax-00000008-QINU is a branched 1-manifold, and UNIQa5f350079404ab7-MathJax-00000009-QINU is an immersion satisfying three conditions: 1) UNIQa5f350079404ab7-MathJax-0000000A-QINU expands (local) distances; 2) the chain recurrent set of UNIQa5f350079404ab7-MathJax-0000000B-QINU is all of UNIQa5f350079404ab7-MathJax-0000000C-QINU and 3) each point of UNIQa5f350079404ab7-MathJax-0000000D-QINU has a neighborhood UNIQa5f350079404ab7-MathJax-0000000E-QINU such that UNIQa5f350079404ab7-MathJax-0000000F-QINU has no branches, for some UNIQa5f350079404ab7-MathJax-00000010-QINU To pass from UNIQa5f350079404ab7-MathJax-00000011-QINU to the fractal attractor, one uses a standard trick in mathematics, sometimes called 'the universal extension', UNIQa5f350079404ab7-MathJax-00000012-QINU Here the space UNIQa5f350079404ab7-MathJax-00000013-QINU is the inverse limit of the sequence UNIQa5f350079404ab7-MathJax-00000014-QINU and UNIQa5f350079404ab7-MathJax-00000015-QINU \begin{thm} An attractor UNIQa5f350079404ab7-MathJax-00000016-QINU of a diffeomorphism UNIQa5f350079404ab7-MathJax-00000017-QINU is a S-W attractor if and only if there exists a branched 1-manifold UNIQa5f350079404ab7-MathJax-00000018-QINU and and immersion UNIQa5f350079404ab7-MathJax-00000019-QINU satisfying the conditions 1-3 above, such that UNIQa5f350079404ab7-MathJax-0000001A-QINU is topologically conjugate to UNIQa5f350079404ab7-MathJax-0000001B-QINU \end{thm} All there is at this date: 3-5-2010. \begin{thebibliography}{99} \bibitem [AP]{AP} \bibitem[FJ]{FJ} New attractors in hyperbolic dynamics, Journal of Differential Geometry, vol. 15, no. 1, 1980, 107-133. \bibitem[G]{G} Guckenheimer, A strange strange attractor. \bibitem[R]{R} Dynamical systems, stability, symbolic dynamics, and chaos, 1995, CRC press \bibitem[Sm]{Sm} Smale, Steve. Differentiable dynamical Systems, Bull. Amer. Math. Soc.,73, 747-817 \bibitem[Wen]{Wen} Wen, L \bibitem[W1] One dimensional nonwandering sets, Topology 6,(1967)473-487. \bibitem[W3]{W3} Williams, R.F.Expanding attractors, Institute des Hautes \'Etudes Scientifique Publ. Math. no. 43(1973)169-203. %\bibitem[W4]{W4} Williams, R.F. The ``DA'' maps of Smale and structural % stability, Global Analysis, vol. XIV of Amer. Math. Soc. Proc. of % Symposia in Pure and Appl. Math., 14(1970) 239-334. \end{thebibliography} \noindent \author{ R.F. Williams\\ Department of Mathematics\\ The University of Texas at Austin\\ Austin, TX 78712 U.S.A.} \end{document}
