#### by Robert Fones Williams

My family always lived in Austin except when forced for a six-year retreat (back) to Alabama; I was born there in 1928. This hard period for my father included plowing with a mule. But he was determined to get us, now five, back to Austin and the University of Texas (there was only one until decades later.) My mother’s parents were Leon Walter and Lucretia Brown of Runge, Texas, and my father’s were Joseph Marble and Mary Michael (Fones) Williams, of Montgomery Alabama. The Great Depression, World War II and rigid Jim Crow defined the young lives of my two brothers and me.

I graduated from high school — neither parent had been to high school — in 1945 and entered the University. All three boys fulfilled our father’s wish and graduated from the University of Texas: me in 1948, BA in Mathematics; Elgin, Jr. with a BA in Economics, 1942, Masters 1943; Joe (after a stint in the Navy), with a BS in Architecture, 1949. My brothers were very important to me.

My curriculum at UT was mostly physical science but included economics, psychology and advanced English courses. By my sophomore year I was grading mathematics homework and had a key to the Mathematics Department office. Calculus under R. L. Moore had only two students during the first semester but the number increased to seven in the second (an effect of World War II). Next was Moore’s (undergraduate) analysis. By this time I was much taken by the “Moore method” of teaching and certain that I wanted to be a mathematician. Along with my close friends, Eldon Dyer and Mary-Elizabeth Hamstrom, I was ready to show the world the Moore method (now universally admired as inquiry-based learning). The math community included — at some stage of their careers — Mary Ellen Estell (later, Rudin), R. D. Anderson, B. J. Ball, R H Bing, Cecil Burgess and Burton Jones. I had a year in graduate school, continuing Moore’s sequence of courses, on which I spent most of my time.

In the fall of 1948 I applied for support at the two universities suggested by Moore; I accepted a position at the University of Virginia where I continued my graduate work under Gordon Whyburn. There are various stories about this move: one was that Moore hoped I would mature, and return for my Ph.D. Another, see ([e7], p. 232), concerned my modern dancing which had become important to me. Could I have been a professional modern dancer? I had a summer fellowship to study dance under Hanya Holm, but instead accepted a summer job teaching math — which was canceled without warning.

Virginia was a major change and at first I resisted the more balanced
program under Whyburn.
With momentum from Moore, I tried to prove some foolish things about
complexes, but did
solve a (dynamics) problem suggested by
Gustav Hedlund
[1]:
I
reinvented the dyadic solenoid and doubling diffeomorphism; later
E. E. Floyd
showed me how inverse limits were good for such examples.
Another scholar suggested a pointwise definition for
local contractions, *pl contraction*: I showed
how poorly this worked; in particular I put a topologically equivalent
metric on the circle
[5]
so
that an irrational rotation was a
pl contraction but not a
pl isometry. I began to have more contact with E. E. Floyd which continued for
many years. Also
David Lowdenslager
and
Bob Plunkett
(fellow graduate students) became important friends.

In 1952 I married Catherine Barnes — whose BA was a double major in painting and art history. She continued to paint throughout our twenty years of marriage which cemented my interest in art and the chemistry of oil painting. My thesis was “Reduction of open mappings” [2], [3] and my Ph.D. was granted in 1954.

Bob Plunkett and I accepted assistant professor positions at the newly created Florida State University. I left this position for a one-year appointment to the Mathematics Department of The University of Wisconsin. This continued my contact with Professor RH Bing, whom I knew at Texas. I concentrated on his ongoing work on surfaces: that a 2-sphere topologically embedded in a 3-sphere can be approximated by one which has a three-dimensional disk on one side; Bing solved it himself. I was attracted to and wrote a paper on dimension theory [4].

In 1956 I became Assistant Professor at Purdue University. I took the summer of 1957 off and toured Europe with my wife Catherine. My daughter Ellen Lindsay was born in Lafayette, Indiana, April of 1958.

There was little topology at Purdue and I began working with the group around Lamberto Cesari and Wendell Fleming [6]. I also made contact with Mo Hirsch and Steve Smale, who were nearby at the University of Chicago. I got a one-year NSF fellowship to attend the Institute for Advanced Study (extended to a second year.)

This was a real learning period — much of it with Smale, Hirsch and Frank Raymond. I began a long period working with Raymond on the Hilbert–Smith conjecture [8], [9]. The IAS professors helped me get a two-year post-doc at the University of Chicago. I finally realized that I needed to learn linear algebra and got to teach it. Chicago was a friendly place and I learned a lot. I had some success here [4] but a long project concerning the agreement of cohomology dimension with topological dimension failed because of an elementary and serious mistake.

While I was at the University of Chicago, two undergraduates,
Rich Goldstone
and Steve Derenzo,
introduced me to
rock climbing. They had discovered a rock wall near the university
that made for
challenging traverses. And on weekends, there was the fine wall at
Devil’s Lake, in Wisconsin. Like many climbers in these early days,
Goldstone is a mathematician, as are
Mike Freedman,
John Gill,
Rob Kirby,
John Milnor
and
George de Rham,
all of whom
I have had great pleasure climbing with.
De Rham wrote the guide book to
* l’Argentine*
(a mountain in the Vaud Alps, Switzerland). For the next
four decades I was an
avid climber. Later, trips in the US and to roughly ten countries for
mathematics
meetings, I easily found climbers, mountains, rock and
snow for my main sport.

I moved to nearby Northwestern University as an associate professor
without tenure, where I finished my work on the Hilbert–Smith conjecture.
I rode the Chicago “L” down to the
colloquia at the University of
Chicago: Smale gave one about hyperbolic subsets of a manifold under a
diffeo (Axiom “A”). Example:
the doubling diffeo on the dyadic solenoid (DS), just as in
[1].
Suddenly I was at home in dynamical systems.
The additional structure
suited me.
Struggling with the zeta function of the dyadic
solenoid, I was delighted to understand the following:
a zeta function is rational
if and only if there are two matrices __\( A \)__ and __\( B \)__ such the number
of fixed points of __\( f^n = |f^n|= \operatorname{tr}A^n- \operatorname{tr}B^n \)__.

I used a preliminary version of Smale’s “Differential dynamical systems” paper [e2] to teach a seminar. Paul Schweitzer was visiting Northwestern at this time and was enthusiastic; we also thought about Smale’s newly formulated “stable manifold theorem”, later proved by Hirsch, Pugh, and Shub (eventually there were several other proofs). The climax of this seminar was my introduction of one-dimensional “branched manifolds” (later given the great name “train tracks” by Thurston) which I used to characterize all one-dimensional hyperbolic attractors [10].

Smale was enthusiastic and invited me to visit Berkeley. He had
constructed the “__\( DA \)__” diffeomorphism of the torus; I saw its beauty
and showed him how it fit into my characterization
[10].
The __\( DA \)__ was
clearly structurally
stable and a great addition to his “horse shoe” from
seven years earlier.
But I had no idea
how much of my work in the next years would involve this example.
(My visit happened
to coincide with thousands marching through San Francisco to protest
the war in Vietnam.)

This was the beginning of an exciting era for dynamics and most of my life’s work. At this time I began to work on the Weil-type zeta function introduced by Artin and (Barry) Mazur and a higher-dimensional version of [10], which eventually appeared in 1971 [14].

I computed the periodic points in several examples
[10]
and saw the
formula
__\[
|f^n| = [\operatorname{tr} (f_*^n) + \operatorname{tr} (\tilde f_*)^n]/2,
\]__
where __\( \tilde f \)__ is
the lift to the double cover orienting the expanding bundle of
__\( df \)__,
__\( |f| \)__ is the number of fixed points of __\( f \)__, and __\( * \)__ indicates the map
induced on homology. This led to the paper
[11]
showing that all hyperbolic
attractors had rational
zeta functions. I had a sabbatical year at the
University of Geneva where I completed that work.

#### The global analysis meeting in Berkeley, summer of 1968

*shift equivalence*and

*strong shift equivalence*at this time and used these to classify one-dimensional hyperbolic attractors [10]. These were shown to be equivalent in geometric situations, such as expanding attractors. Subshifts of finite type came into dynamics when Bowen showed that “basic sets” are semi-conjugate to such. Subshifts of finite type

*are*classified by strong shift equivalence. But by itself, strong shift equivalence is not computable and my proof that the two equivalences are the same failed.

In fact this led to eighteen years of work (see [e6]) on the “Williams problem.” Meanwhile, at Warwick, Bill Parry found the error in [13] while we were working on finite Markov chains [24]. In a series of fine papers, John Wagoner, Kim and Roush showed that the strong version is in fact stronger.

The Turbulence Seminar
was organized in Berkeley
following the suggestion by
Ruelle
and
Takens
[e3]
of the “strange attractor”
as
a model for turbulence: a contemporary definition of strange attractor
was an attractor
that is is neither finite not a submanifold. This
included those I had worked on.
It also includes the Lorenz attractor, but this last
was essentially unnoticed until
Guckenheimer’s
“A strange strange attractor” appeared as a
preprint
[e4].
My
paper
[26]
was next; it made use of branched manifolds
[18]
and gave
additional insight into the Lorenz attractor. Our model was referred to
as the * geometric Lorenz attractor*. I introduced a zeta
function to characterize them, at the Turbulence Seminar
[23].
I
later wrote about these knots with
Joan Birman
[28].

At this point we had no complete proof connecting the geometric LA of Guckenheimer and myself; I had done lots of verification. For example, I would “locate” a periodic of say five or six trips around the periodic points, then James Curry would find it with an ODE solver. I wrote a program to “find” the strong stable manifolds and the pictures were striking.

Curry is a longtime member of the Mathematics Department of the University of Colorado, which I had the honor of visiting as holder of the Stanisław Ulam position. It was here that I first met Marcy Barge, whose thesis was under Wes Wilson. Later Marcy and I worked together on tiling theory.

At the Turbulence Seminar, I showed that the periodic orbits
corresponded
one-to-one with the aperiodic words in two letters, say
__\( \{x,y\} \)__, and
pointed out that
they are mostly knotted.
All my life I have liked knots, how to tie them and where
to use them, but never as a mathematician. Now I had a chance and
spent many hours tying these “Lorenz” knots, using
the symbolic dynamics I had found.
I built a “knot loom” using this symbolic dynamics
for tying them — and the very first time I used it, the knot fell
into a shape that I saw was important: each Lorenz knot is a positive
braid with a full twist.
Shortly after this, I met Joan Birman,
whose knot work was largely on braids. Notably every knot can be
written as
a braid, and the Lorenz knots are given as braids.
Together we wrote
[28]
on Lorenz knots and
[29].

Earlier, three new dynamicists from Berkeley, Clark Robinson, Sheldon Newhouse and John Franks came to Northwestern. Together we began The Midwestern Dynamical Systems Seminar with Joel Robbin and Charles Conley (at Wisconsin) which met weekly for a while. This was the beginning of MWDS which remains important 50 years later; I am proud of this — especially, my listening closely to the much younger Newhouse, Robinson and Franks. True to his many important contributions to dynamics, Robinson kept the records and mailing list. Also important were the series of other young dynamicists that had a year or two at NU: Bob Devaney, Paul Blanchard, and Lai-Sang Young. I was invited to present a paper at the International Congress in Nice, 1970.

#### Warwick University, IMPA (Brazil), IHES, Berkeley

Over the next three decades, I took part in many meetings for dynamical
systems. I had always wanted to visit Brazil and my four visits there
were fulfilling.
In 1971 at the Instituto de Matemática Pura e Aplicada (IMPA) I gave my slightly simplified version of the
__\( DA \)__, and
a paper on the composition of contractions that I had worked out on
a train ride from Geneva to Warwick University.
Sheldon Newhouse and I drove our “fuscas” through the jungle, to the
month-long
international meeting in
Salvador, Bahia (Brazil) in 1971.

Knot theory had been an active section of mathematics for over fifty
years when
Vaughan Jones
found a new polynomial
invariant. Defined via __\( C^* \)__ algebras, many knot
theorists figured out, at the about the same time, how to compute it
via standard “knot moves”.
My colleague John Franks persuaded the
undergrad
Henry Cejtin,
to write a program to compute this. After many
computations we noticed
a certain
exponent was a lower bound for the braid index (BI) of a knot
[33].
As every knot can be written as a braid, the smallest number of strands
for such a braid is a knot invariant. This makes BI a
useful invariant for the first time.

That
one-dimensional attractors have nontrivial homology was proved in my 1967 paper in *Topology*
[10].
But for expanding
attractors, a new idea is needed:
Dennis Sullivan
and I showed this
using cochains with bounded coefficients
[21].
Though I had known
Dennis for many years, I had the most contact with him while a member
of the Institut des Hautes Études Scientifiques. This also gave me a
lot of contact with
René Thom.
His friendship has been important to me.

#### Tilings

__\( T \)__of

__\( \mathbb{R}^n \)__, there are only finitely many local patterns, and the collection

__\( X \)__of all tilings of

__\( \mathbb{R}^n \)__that have only these patterns has a natural metric making it into a compact space, called the

*hull*or

*tiling space*

__\( T \)__.

__\( X \)__includes all translations of

__\( T \)__and for aperiodic tilings this space is easily shown to be locally the product of an

__\( n \)__-dimensional disk and a Cantor set. Thus

__\( \mathbb{R}^n \)__acts on

__\( X \)__, giving more structure. The subdivision of

__\( T \)__induces a subdivision

__\( t \circlearrowleft X \)__. In 1996, Anderson and Putnam [e5] revolutionized tiling theory by showing that this dynamical system is an expanding attractor as in [18]. They constructed a branched manifold

__\( K \)__from finitely many tiles; the subdivision acts on

__\( K \)__, say

__\( s \circlearrowleft K \)__. Taking inverse limits gives

__\( \hat s \circlearrowleft \hat K \)__, topologically conjugate to

__\( t \circlearrowleft X \)__.

Inspired by this theorem which unites two areas of mathematics, I began working on tilings which led to [34], [35] and [36].

The mathematician,
Karen Uhlenbeck
and I got together in 1975; the
outdoors with
back-packing, and
mountain-climbing has been continuous and great together; we are married. I
left Northwestern
University to
accept
a position at the University of Texas in 1986.
During this period we spent summers in the West. We now have a house
in Bozeman, Montana and been given much hospitality by Montana State
University.
We lived
in a house in the Texas Hill Country, surrounded by trees. After a
visit to Hawaii, I built an outdoor shower
with circulating hot water.
I retired in 1999 and continued as an Editor for dynamics for
*Transactions of the AMS* for several years.
We moved to Princeton when Karen retired in 2014. Until this move we
have always had at least two cats. We are as busy as
ever with our new connections to the Institute for Advanced Study. The
IAS
has been very accommodating; Karen is a Visiting Distinguished
Professor.

As I write this we are hopeful, along with millions of others, of living through the COVID-19 coronavirus pandemic.