R. F. Williams

Structural stability of Lorenz attractors

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

The structure of Lorenz attractors

© Publications mathématiques de l’I.H.É.S., 1979, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Knotted periodic orbits in dynamical systems—I: Lorenz's equation

THIS PAPER is the first in a series which will study the following problem. We investigate a system of ordinary differential equations which determines a flow on the 3-sphere S3 (or R3 or ultimately on other 3-manifolds), and which has one or perhaps many periodic orbits. We ask: can these orbits be knotted? .What types of knots can occur? What are the implications? Knotted periodic orbits in dynamical systems do not appear to have been systematically studied, although there is one very well known example. Let (x,, x2, x3, x4) be rectangular coordinates in R4 and let S3 C R4 be the subset of points satisfying Ej=, c? = 1. Let (p, 4) be a pair of coprime integers, and consider the system of ordinary differential equations:

Braids and the Jones polynomial

An important new invariant of knots and links is the Jones polynomia}, and the subsequent generalized Jones polynomial or two-variable polynomial. We prove inequalities relating the number of strands and the crossing number of a braid with the exponents of the variables in the generalized Jones polynomial which is associated to the link formed from the braid by connecting the bottom ends to the top ends. We also relate an exponent in the polynomial to the number of components of this link. In [J] Vaughan Jones introduced a new polynomial invariant of oriented knots and links in 3-space. Subsequently a number of researchers (Freyd and Yetter, Hoste, Lickorish and Millett, and Ocneanu; see [FYHLMO]) independently realized that thlis could be generalized to produce an invariant which is a Laurent polynomial of two variables and which specializes to give both the invariant of [J] and the classical Alexander polynomial (see [R]). More precisely, there is a function P from isotopy classes of oriented links to Z[x,x-1,y,y-l]. If L iS an oriented link we will write P(L) = j(x,y). By abuse of notation we will also write P(b) for b an oriented braid, meaning, of course, the invariant associated to the oriented link determined by closing up b in the usual way. Several equivalent ways of codifying this invariant are described in [FYHLMO]. We have chosen the approach of Lickorish and Millett. Hence our P(K) = j(X, Y) iS precisely their PK with x and y substituted for I and m respectively (see [FYHLMO] and [L-M]). There appears as yet to be no consensus on a name for this two-variable invariant. In order to avoid awkward circumlocutions and being unable to amalgamate the names of the six or eight individuals who actually discovered it, we have chosen to refer to it as the generalized Jones polynomial, or Jones polynomial for short, when there is no likelihood of confusion with the original Jones polynomial. This polynomial, which we will denote j(x, y), is characterized by Figure 1. The interpretation of this is as follows: Given a regular projection of a link K+ with a crossing pictured as below, one can form two new links K_ and KO which are identical to K+ except the one crossing is changed as shown. When this is done the Jones polynomials of the three links are related as in the formula. It is a remarkable fact that this together with the normalization that P (unknot) = 1, un+quely determines a Laurent polynomial j(x, y) and that this polynomial is a link invariant, i.e., is independent of the projection. This result can be found in [L-M]. This formula, in fact, gives a good method for recursively computing j(x, y). One Received by the editors January 13, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 57M25. Research supported in part by NSF Grant MCS 83

The qualitative analysis of a difference equation of population growth

SummaryThe difference equation fb:[0,1]−[0,1] defined by fb(x)=b x(1−x) is studied. In particular complete qualitative information is obtained for the parameter value b=3.83. For example the number of fixed points of (fb)i is given by

A useful functor and three famous examples in topology

N_i = 1 + \left( {\frac{{1 + \sqrt 5 }}{2}} \right)^i + \left( {\frac{{1 - \sqrt 5 }}{2}} \right)^i

The Newtonian graph of a complex polynomial

WE PROVE two theorems, the basic one giving an inequality, the second an equality. THEOREM 1. Suppose f: M” * M” is a diffeomorphism satisfying Axiom A and the no cycle condit~an. Then h cf) 2 log s (f*). Here hcf) is the topological entropy of f and scf*) is the spectral radius of f,: H,(M ; R)+ H&M ; It). That is, s cf,) = max IA 1 where the max is taken over all eigenvalues of fei : El, (M ; R) + Hi f M ; R) and all dimensions i. This theorem is already known in the Morse-Smale casell4, 161 and more generally in the Axiom A and no cycle case with the added hypothesis that the non-wandering set is zero dimensional f4]. Also, Manning proves that h tf) 2 log s(f**), where fe3 is the map in l-dimensional homology and f is a continuous map of quite general spaces. For further discussion of entropy see [lo] and [ Ill. This theorem was conjectured in 1151 and [I61 and has its most striking application in a significant sharpening of the Lefschetz trace formula when the periodic points of f are counted asymptotically. This follows from the work of Bowen[l] where the following was first proved: THEOREM [Bowen]. If f: M + M is a di~eomo~hism sutisfyi~g Axiom A then htf) = lim sup l/n log N,cf). Here N, cf) is the number of periodic points off of period n, that is, the number of fixed points of f”. Since