R. F. Williams :
“The ‘\( \mathrm{DA} \) ’ maps of Smale and structural stability ,”
pp. 329–334
in
Global analysis
(Berkeley, CA, 1–26 July 1968 ).
Edited by S.-S. Chern and S. Smale .
Proceedings of Symposia in Pure Mathematics 14 .
American Mathematical Society (Providence, RI ),
1970 .
MR
264705
Zbl
0213.50303
incollection

Abstract
People
BibTeX

The purpose of this paper is to describe in detail the “derived from Anosov” maps of the title (due to S. Smale [1967, p. 789]) and to show how these examples allow one to lower by 1 the dimension in the following theorem of Smale [1966]:

There is an open set \( U \) in the space of \( C^r \) vector fields (\( r > 0 \) ) on a 4-dimensional manifold such that no \( X \in U \) is structurally stable.

The resulting theorem is definitive in that M. Peixoto has shown [1962] that structurally stable systems on compact two manifolds are dense. The present construction is based on a one-dimensional attractor (generalized solenoid [Williams 1967]) whereas Smale’s is based on a two-dimensional attractor (a torus). Otherwise this construction is just like that of Smale’s. The paper of Peixoto and Pugh [1968] contains another variation of Smale’s construction for noncompact 2-manifolds and has a good discussion as to how all of these examples work.

@incollection {key264705m,
AUTHOR = {Williams, R. F.},
TITLE = {The ``\$\mathrm{DA}\$'' maps of {S}male
and structural stability},
BOOKTITLE = {Global analysis},
EDITOR = {Chern, Shiing-Shen and Smale, Stephen},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {14},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1970},
PAGES = {329--334},
DOI = {10.1090/pspum/014/0264705},
NOTE = {(Berkeley, CA, 1--26 July 1968). MR:264705.
Zbl:0213.50303.},
ISSN = {0082-0717},
ISBN = {9780821814147},
}
R. F. Williams :
“Classification of one dimensional attractors ,”
pp. 341–361
in
Global analysis
(Berkeley, CA, 1–26 July 1968 ).
Edited by S.-S. Chern and S. Smale .
Proceedings of Symposia in Pure Mathematics 14 .
American Mathematical Society (Providence, RI ),
1970 .
MR
266227
Zbl
0213.50401
incollection

Abstract
People
BibTeX

This is a sequel to a paper [1967] in which one dimensional attractors are characterized. Some familiarity with [1967] is assumed. Attractor is meant in the sense of S. Smale; for a general introduction to this subject, see the important paper [Smale 1967]. As this is written, the axioms A and B of Smale [1967, pp. 777–778] have been shown to be nongeneric by Abraham and Smale [1970]. Thus the formulation of Smale (sometimes called Anosov–Smale systems or diffeomorphisms) will have to be varied again. But for technical reasons, the example of Abraham and Smale does not directly affect work done on attractors. Moreover, the analysis carried out here and in [1967] does concern an open set of diffeomorphisms, and thus will be pertinent to any formulation of this theory.

@incollection {key266227m,
AUTHOR = {Williams, R. F.},
TITLE = {Classification of one dimensional attractors},
BOOKTITLE = {Global analysis},
EDITOR = {Chern, Shiing-Shen and Smale, Stephen},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {14},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1970},
PAGES = {341--361},
DOI = {10.1090/pspum/014/0266227},
NOTE = {(Berkeley, CA, 1--26 July 1968). MR:266227.
Zbl:0213.50401.},
ISSN = {0082-0717},
ISBN = {9780821814147},
}
R. F. Williams :
“Zeta function in global analysis ,”
pp. 335–339
in
Global analysis
(Berkeley, CA, 1–26 July 1968 ).
Edited by S.-S. Chern and S. Smale .
Proceedings of Symposia in Pure Mathematics 14 .
American Mathematical Society (Providence, RI ),
1970 .
MR
266252
Zbl
0213.50402
incollection

Abstract
People
BibTeX
@incollection {key266252m,
AUTHOR = {Williams, R. F.},
TITLE = {Zeta function in global analysis},
BOOKTITLE = {Global analysis},
EDITOR = {Chern, Shiing-Shen and Smale, Stephen},
SERIES = {Proceedings of Symposia in Pure Mathematics},
NUMBER = {14},
PUBLISHER = {American Mathematical Society},
ADDRESS = {Providence, RI},
YEAR = {1970},
PAGES = {335--339},
DOI = {10.1090/pspum/014/0266252},
NOTE = {(Berkeley, CA, 1--26 July 1968). MR:266252.
Zbl:0213.50402.},
ISSN = {0082-0717},
ISBN = {9780821814147},
}
S. Smale and R. F. Williams :
“The qualitative analysis of a difference equation of population growth ,”
J. Math. Biol.
3 : 1
(1976 ),
pp. 1–4 .
MR
414147
Zbl
0342.92014
article

Abstract
People
BibTeX

The difference equation \( f_b:[0,1]\to [0,1] \) defined by
\[ f_b(x) = bx(1-x) \]
is studied. In particular complete qualitative information is obtained for the parameter value \( b = 3.83 \) . For example the number of fixed points of \( (f_b)^i \) is given by
\[ N_i = 1 + \Bigl(\frac{1+\sqrt{5}}{2}\Bigr)^i + \Bigl(\frac{1-\sqrt{5}}{2}\Bigr)^i. \]

@article {key414147m,
AUTHOR = {Smale, S. and Williams, R. F.},
TITLE = {The qualitative analysis of a difference
equation of population growth},
JOURNAL = {J. Math. Biol.},
FJOURNAL = {Journal of Mathematical Biology},
VOLUME = {3},
NUMBER = {1},
YEAR = {1976},
PAGES = {1--4},
DOI = {10.1007/BF00307853},
NOTE = {MR:414147. Zbl:0342.92014.},
ISSN = {0303-6812},
}
R. F. Williams :
“A new zeta function, natural for links ,”
pp. 270–278
in
From topology to computation: Proceedings of the Smalefest
(Berkeley, CA, 5–9 August 1990 ).
Edited by M. W. Hirsch, J. E. Marsden, and M. Shub .
Springer (New York ),
1993 .
MR
1246126
Zbl
0851.58037
incollection

Abstract
People
BibTeX

We overcome a former frustration: A new “determinant” is defined, not requiring completely abelian matrix entries. In fact, non-abelian invariants do occur in dynamical systems, so that the usual determinant is not available. This “link-determinant” is invariant only under the group of permutations, not the general linear group. But permutations are the only pertinent transformations in the setting of symbolic dynamics. That is to say, permuting the elements of a Markov matrix makes sense; linear combinations of them do not. For our immediate purpose, keeping track of knots and links via their fundamental group words, the zeta function defined in terms of this new determinant seems ideal.

@incollection {key1246126m,
AUTHOR = {Williams, R. F.},
TITLE = {A new zeta function, natural for links},
BOOKTITLE = {From topology to computation: {P}roceedings
of the {S}malefest},
EDITOR = {Hirsch, M. W. and Marsden, J. E. and
Shub, M.},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {270--278},
DOI = {10.1007/978-1-4612-2740-3_27},
NOTE = {(Berkeley, CA, 5--9 August 1990). MR:1246126.
Zbl:0851.58037.},
ISBN = {9781461227403},
}
R. F. Williams, D. Karp, D. Brown, O. Lanford, P. Holmes, R. Thom, E. C. Zeeman, and M. M. Peixoto :
“Final panel ,”
pp. 589–605
in
From topology to computation: Proceedings of the Smalefest
(Berkeley, CA, 5–9 August 1990 ).
Edited by M. W. Hirsch, J. E. Marsden, and M. Shub .
Springer (New York ),
1993 .
MR
1246149
incollection

People
BibTeX
@incollection {key1246149m,
AUTHOR = {Williams, R. F. and Karp, D. and Brown,
D. and Lanford, O. and Holmes, P. and
Thom, R. and Zeeman, E. C. and Peixoto,
M. M.},
TITLE = {Final panel},
BOOKTITLE = {From topology to computation: {P}roceedings
of the {S}malefest},
EDITOR = {Hirsch, M. W. and Marsden, J. E. and
Shub, M.},
PUBLISHER = {Springer},
ADDRESS = {New York},
YEAR = {1993},
PAGES = {589--605},
DOI = {10.1007/978-1-4612-2740-3_52},
NOTE = {(Berkeley, CA, 5--9 August 1990). MR:1246149.},
ISBN = {9781461227403},
}