Celebratio Mathematica

R. F. Williams: “A note on unstable homeomorphisms,” Proc. Am. Math. Soc. 6 : 2 (1955), pp. 308–309. MR 68211 Zbl 0067.15402 article

R. F. Williams: “Local properties of open mappings,” Duke Math. J. 22 : 3 (September 1955), pp. 339–346. MR 75580 Zbl 0065.38201 article

R. F. Williams: “Reduction of open mappings,” Proc. Am. Math. Soc. 7 : 2 (1956), pp. 312–318. This is based on the author’s 1954 PhD thesis. MR 77112 Zbl 0073.17901 article

R. F. Williams: “The effect of maps upon the dimension of subsets of the domain space,” Proc. Am. Math. Soc. 8 : 3 (1957), pp. 580–583. MR 87920 Zbl 0079.38801 article

“Space filling curves” constitute simple examples of maps which raise the dimension of each closed subset of the domain space of positive dimension. That is, in its usual construction, a map \( f \) of a 1-cell \( X \) onto an \( m \)-cell \( Y \) is such that each closed subset of positive dimension, or what amounts to the same, each nondegenerate subcontinuum of \( X \), has an image of dimension \( m \).

In case the dimension of \( X \) is greater than 1, maps with such strong dimension raising properties do not always exist. For example, there does not exist a map of a 2-cell into a 3-cell such that the image of each nondegenerate sub-continuum is 3, or even \( \geq 2 \). This fact is contained in the results below, all of which give some bound to the dimension raising possibilities of maps.

R. F. Williams: “Local contractions and the size of a compact metric space,” Duke Math. J. 26 : 2 (June 1959), pp. 277–289. MR 105074 Zbl 0085.17002 article

R. F. Williams: “Lebesgue area of maps from Hausdorff spaces,” Acta Math. 102 : 1–2 (1959), pp. 33–46. MR 110785 Zbl 0144.30001 article

R. F. Williams: “Lebesgue area zero, dimension and fine cyclic elements,” Riv. Mat. Univ. Parma 10 (1959), pp. 131–143. MR 140663 Zbl 0107.27501 article

“If a surface has area zero, it is really a curve”, is the proposition to be discussed in this paper. With a suitably loose interpretation of “really a curve”, the proposition is valid in a very general setting. A more satisfying interpretation leads to an affirmative solution in a special case. This is a generalisation of a theorem of Radó [1943]. Lebesgue \( m \)-dimensional “area” and extensions of it are used.

An \( m \)-dimensional area is a function whose domain is a certain class of maps \[ f:X\to E_n ,\] \( m\leq n \). The largest such class considered here consists of arbitrary maps (continuous transformations) from compact, \( m \)-dimensional Hausdorff spaces, \( X \). This generality seems justified by our corollary that Lebesgue area zero is a topological property.

F. Raymond and R. F. Williams: “Examples of \( p \)-adic transformation groups,” Bull. Am. Math. Soc. 66 : 5 (1960), pp. 392–394. Full descriptions of these examples are given in an article published in Ann. Math. 78:1 (1963). MR 123634 Zbl 0096.17202 article

Our purpose here is to outline the construction of an \( n \)-dimensional space \( X^n \), \( n\geq 2 \), upon which the \( p \)-adic group \( A_p \) acts so that the orbit space \( X^n/A_p \) is of dimension \( n+2 \). Though such examples are new, it had been known [Williams et al. 1961; Yang 1960], that either they do exist or a certain long standing conjecture on transformation groups must be true. The conjecture states that every compact effective group acting on a (generalized) manifold must be a Lie group; it may well be false.

Another question concerns the amount, \( k \), by which the (cohomology) dimension of a compact space can be raised under the decomposition map \[ X \to X/A_p .\] By [Williams et al. 1961; Yang 1960], \( k\leq 3 \). (An example in which \( k = 1 \) is essentially due to Kolmogoroff [1937].) No example is known for which \( k = 3 \). The authors expect to have more to say on this subject.

G. E. Bredon, F. Raymond, and R. F. Williams: “\( p \)-adic groups of transformations,” Trans. Am. Math. Soc. 99 : 3 (1961), pp. 488–498. MR 142682 Zbl 0109.15901 article

There is the important conjecture that every compact group \( G \) of transformations on a manifold \( M \) is a Lie group. As is well known, if \( G \) is not a Lie group then \( G \) must contain a \( p \)-adic group which in turn acts effectively on \( M \).

Recently, Yang has shown [1960] that if a \( p \)-adic group does act on \( M \), then the orbit map raises the integral cohomology dimension by 2. To obtain this result Yang extends the Smith special homology theory to maps of prime power period, using the reals mod 1 as coefficients.

In this paper the authors compute the cohomology of the universal classifying space \( B_G \), for \( G = A_P \), the \( p \)-adic group, and \( G = \Sigma_p \), the \( p \)-adic solenoid. These results are then used with methods fitting the general scheme of [Borel 1960] to prove the dimension theorems of Yang.

We conclude the paper with fixed point theorems for \( p \)-adic solenoids

Frank Albert Raymond	Related
Glen Eugene Bredon	Related

R. F. Williams: “One-dimensional non-wandering sets,” Topology 6 : 4 (November 1967), pp. 473–487. MR 217808 Zbl 0159.53702 article

R. F. Williams: “The zeta function of an attractor,” pp. 155–161 in Conference on the topology of manifolds (East Lansing, MI, 15–17 March 1967). Edited by J. G. Hocking. Complementary Series in Mathematics 13. Prindle, Weber & Schmidt (Boston), 1968. MR 235573 Zbl 0179.51902 incollection

M. Shub and R. F. Williams: “Future stability is not generic,” Proc. Am. Math. Soc. 22 : 2 (1969), pp. 483–484. MR 242193 Zbl 0181.51402 article

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

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.

Stephen Smale	Related
S. S. Chern	Related	Volume

R. F. Williams: “The structure of attractors,” pp. 947–951 in Actes du Congrès International des Mathématiciens [Proceedings of the International Congress of Mathematicians] (Nice, France, 1–10 September 1970), vol. 2. Gauthier-Villars (Paris), 1971. MR 650645 Zbl 0228.58008 incollection

R. F. Williams: “Composition of contractions,” Bol. Soc. Brasil. Mat. 2 : 2 (1971), pp. 55–59. MR 367962 Zbl 0335.54026 article

A map \( f: X \to X \) of a metric space is a contraction if for some \( \lambda \), \( 0 \leq \lambda \leq 1 \), \[ d(f(x),f(y)) \leq \lambda d(x,y) \quad\text{for all }x,y \in X .\] The least such \( \lambda \) is the Lipschitz constant, \( L(f) \). If \( X \) is complete, a contraction \( f \) has a unique fixed point which we call \( F(f) \).

Now suppose \( m \) contractions \[ f_1,\dots,f_m: X \to X \] are given, where \( X \) is complete. Then each composite “word” \[ w = f_{i_1} \circ \cdots \circ f_{i_r} \] has a unique fixed point \( F(w) \). Here we are concerned with the closure \( F \) of the set \[ F = F(f_1,\dots,f_m) \] of all such fixed points, \( F(w) \). This paper can be regarded as a step toward studying generic properties of the action of free (non-abelian) groups on manifolds. See [Smale 1967].

R. C. Robinson and R. F. Williams: “Finite stability is not generic,” pp. 451–462 in Dynamical systems (Salvador, Brazil, 26 July–14 August 1971). Edited by M. M. Peixoto. Academic Press (New York), 1973. MR 331430 Zbl 0305.58009 incollection

Rex Clark Robinson	Related
Mauricio Matos Peixoto	Related

R. F. Williams: “Classification of subshifts of finite type,” Ann. Math. (2) 98 : 1 (July 1973), pp. 120–153. Errata were published in Ann. Math. 99:2 (1974). MR 331436 Zbl 0282.58008 article

R. F. Williams: “Expanding attractors,” Inst. Hautes Études Sci. Publ. Math. 43 (1974), pp. 169–203. MR 348794 Zbl 0279.58013 article

The purpose of this paper is to present complete proofs of the results toward characterizing attractors with hyperbolic structure (see Smale [1967]), as announced in [Williams 1969, 1971]. The crucial additional assumption (beyond hyperbolic structure) is that the attractor is expanding (that is, its set-theoretic dimension is equal to the dimension of the fiber of the unstable bundle). One other (technical) assumption is made that the stable manifold foliation \[ \{W^s(x)\mid x \textrm{ is in the attractor }\} \] is of class \( C^r \), \( r \geq 1 \). We expect that this assumption can be dropped with additional work.

Thus this paper carries out a portion of the program we felt was possible in [1967]. In some sections (e.g. the one on periodic points) more detailed proofs are given than in [1967]. In turn, this is a part of a large program evolving from the work of many authors, with S. Smale being the prime mover. See the important survey paper of Smale [1967], and the volume [Smale and Chern 1970] on global analysis.

M. Shub and R. F. Williams: “Entropy and stability,” Topology 14 : 4 (November 1975), pp. 329–338. MR 415680 Zbl 0329.58010 article

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

D. Sullivan and R. F. Williams: “On the homology of attractors,” Topology 15 : 3 (1976), pp. 259–262. MR 413185 Zbl 0332.58011 article

C. Robinson and R. Williams: “Classification of expanding attractors: An example,” Topology 15 : 4 (1976), pp. 321–323. MR 415682 Zbl 0338.58013 article

There are two ways to classify attractors. The first way is to ask, when are the restrictions of two diffeomorphisms to their attractors conjugate? The second way is to ask, when are two diffeomorphisms conjugate in neighborhoods of their attractors? R. Williams has studied the first question for expanding attractors (hyperbolic structure with dimension of the attractor equal the dimension of the unstable splitting) under the assumption that the stable foliation is \( C^1 \) [1967, 1970, 1974]. In this note we give an example that shows these are different questions, i.e. we give two diffeomorphisms \( f \) and \( g \) with attractors \( \Lambda_f \) and \( \Lambda_g \) such that \[ f: \Lambda_f \to \Lambda_f \] is conjugate to \[ g: \Lambda_g \to \Lambda_g ,\] but there is not even a homeomorphism from a neighborhood of \( \Lambda_f \) to a neighborhood of \( \Lambda_g \), taking \( \Lambda_f \) to \( \Lambda_g \). (They are embedded differently.) We also exhibit a technique (for this one example) that may overcome the assumption that the stable foliation is \( C^1 \) in the work of R. Williams. We end with an appendix that proves directly that an expanding attractor is locally homeomorphic to a Cantor set cross a \( u \)-dimensional disk. Here \( u \) is the dimension of the unstable bundle.

R. F. Williams: “The structure of Lorenz attractors,” pp. 94–112 in Turbulence seminar (Berkeley, CA, 1976–1977). Edited by P. Bernard and T. Ratiu. Lecture Notes in Mathematics 615. Springer (Berlin), 1977. Lecture VII. With appendix “Computer pictures of the Lorenz attractor.”. This appears to have been adapted for an article published in Inst. Hautes Études Sci. Publ. Math. 50 (1979). MR 461581 Zbl 0363.58005 incollection

The system of equations \begin{align*} \dot{x} &= -10x + 10y \\ \dot{y} &= 28x -y - xz \\ \dot{z} &= -\frac{8}{3}z + xy \end{align*} of E. N. Lorenz has attracted much attention lately, in part because of its supposed relation to turbulence. As was mentioned in [Marsden 1977], it is obtained by the truncation of the Navier–Stokes Equation. How this system really relates to turbulene is not known. yet and — as we saw in earlier talks — there are many pros and cons for this. One thing is sure: this system certainly gives rise to a type of attractor which is nonclassical and not even strange in the sense of Smale [1977]. Notice that according to the bifurcation classification of [Marsden 1977], the system above gives rise to the “standard” Lorenz attractor and the goal of this lecture is to study it.

Peter Simon Bernard	Related
Tudor Ștefan Rațiu	Related

W. Parry and R. F. Williams: “Block coding and a zeta function for finite Markov chains,” Proc. London Math. Soc. (3) 35 : 3 (1977), pp. 483–495. MR 466490 Zbl 0383.94011 article

For an irreducible Markov chain, movement from one state to another is specified stochastically by transition probabilities depending only on the present. In this way possible (positive probability) finite sequential movements can be specified. The topological or qualitative version of this type of behaviour motivates the concept of a topological Markov chain (or shift of finite type or intrinsic Markov chain [3]).

The basic classification problem of ergodic theory is to decide when two ergodic measure-preserving transformations of probability spaces are isomorphic. This problem specialises to a coding problem for stationary Markov chains. In this paper we are concerned with block codes or block isomorphisms between two such chains. In other words, codes shall have bounded memory and bounded anticipation. Without going into the (simple) details. a block isomorphism is a homeomorphism of the essential spaces on which the Markov chains aro supported, which preserves measure. In fact we shall take this as a definition. We see then that part of the problem of block code classification is the topological classification problem which was investigated in [4].

J. Guckenheimer and R. F. Williams: “Structural stability of Lorenz attractors,” Inst. Hautes Études Sci. Publ. Math. 50 (1979), pp. 59–72. Dedicated to the memory of Rufus Bowen. MR 556582 Zbl 0436.58018 article

Rufus Bowen	Related
John Mark Guckenheimer	Related

R. F. Williams: “The structure of Lorenz attractors,” Inst. Hautes Études Sci. Publ. Math. 50 (1979), pp. 73–99. Dedicated to the memory of Rufus Bowen. Seemingly based on a lecture published in Turbulence seminar (1977). MR 556583 Zbl 0484.58021 article

J. Franks and B. Williams: “Anomalous Anosov flows,” pp. 158–174 in Global theory of dynamical systems (Evanston, IL, 18–22 June 1979). Edited by Z. Nitecki and C. Robinson. Lecture Notes in Mathematics 819. Springer, 1980. MR 591182 Zbl 0463.58021 incollection

Rex Clark Robinson	Related
Zbigniew Henry Nitecki	Related
John Moore Franks	Related

J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical systems, I: Lorenz’s equations,” Topology 22 : 1 (1983), pp. 47–82. Part II was published in Low-dimensional topology (1983). MR 682059 Zbl 0507.58038 article

J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical system, II: Knot holders for fibered knots,” pp. 1–60 in Low-dimensional topology. Edited by S. J. Lomonaco\( Jr. \). Contemporary Mathematics 20. American Mathematical Society (Providence, RI), 1983. Part I was published in Topology 22:1 (1983). MR 718132 Zbl 0526.58043 incollection

Joan S. Birman	Related	Volume
Samuel James Lomonaco, Jr.	Related

R. F. Williams: “Lorenz knots are prime,” Ergodic Theory Dynam. Systems 4 : 1 (March 1984), pp. 147–163. MR 758900 Zbl 0595.58037 article

J. Franks and R. F. Williams: “Entropy and knots,” Trans. Am. Math. Soc. 291 : 1 (1985), pp. 241–253. MR 797057 Zbl 0587.58038 article

P. Holmes and R. F. Williams: “Knotted periodic orbits in suspensions of Smale’s horseshoe: Torus knots and bifurcation sequences,” Arch. Rational Mech. Anal. 90 : 2 (1985), pp. 115–194. MR 798342 Zbl 0593.58027 article

We construct a suspension of Smale’s horseshoe diffeomorphism of the two-dimensional disc as a flow in an orientable three manifold. Such a suspension is natural in the sense that it occurs frequently in periodically forced nonlinear oscillators such as the Duffing equation. From this suspension we construct a knot-holder or template — a branched two-manifold with a semiflow — in such a way that the periodic orbits are isotopic to those in the full three-dimensional flow. We discuss some of the families of knotted periodic orbits carried by this template. In particular we obtain theorems of existence, uniqueness and non-existence for families of torus knots. We relate these families to resonant Hamiltonian bifurcations which occur as horseshoes are created in a one-parameter family of area preserving maps, and we also relate them to bifurcations of families of one-dimensional ‘quadratic like’ maps which can be studied by kneading theory. Thus, using knot theory, kneading theory and Hamiltonian bifurcation theory, we are able to connect a countable subsequence of “one-dimensional” bifurcations with a subsequence of “area-preserving” bifurcations in a two parameter family of suspensions in which horseshoes are created as the parameters vary. One implication is that infinitely many bifurcation sequences are reversed as one passes from the one dimensional to the area-preserving family: there are no universal routes to chaos

J. Franks and R. F. Williams: “Braids and the Jones polynomial,” Trans. Amer. Math. Soc. 303 : 1 (1987), pp. 97–108. MR 896009 Zbl 0647.57002 article

R. F. Williams: “The Penrose, Ammann and DA tiling spaces are Cantor set fiber bundles,” Ergodic Theory Dynam. Systems 21 : 6 (2001), pp. 1883–1901. MR 1869076 Zbl 1080.37012 article

Anderson and Putnam have recently shown that the space of all tilings of a substitution tiling scheme is a special case of the present author’s ‘expanding attractor’. This resurrects an earlier conjecture of the present author in this special case: The tiling space of a substitution tiling space is a fiber bundle over the torus, with Cantor set fiber. If true, this could be important in the classification of these spaces, just as in the one-dimensional case. These spaces were known (in these two disciplines) to be locally the product of a disk and a Cantor set. Farrell and Jones gave an example showing our more inclusive conjecture is false. Here we show that the Penrose, Ammann and ‘stepped plane’ tiling spaces are in fact such bundles. Among dynamicists, the stepped plane tiling space (in special cases) is known as the ‘DA’ attractor of Smale. The formalism of substitution systems provides a Markov partition, and this is certainly part of their intuitive appeal. However, in dynamics such partitions are rare in dimensions greater than 2 and do not occur for the DA attractor. We consider a weaker concept, that omits this requirement. Finally, we construct an example like the Farrell–Jones example, in sufficient detail that it can be visualized as a smooth tiling space, as opposed to the more rigid geometric tiling space.

L. Sadun and R. F. Williams: “Tiling spaces are Cantor set fiber bundles,” Ergodic Theory Dynam. Systems 23 : 1 (February 2003), pp. 307–316. MR 1971208 Zbl 1038.37014 article

We prove that fairly general spaces of tilings of \( \mathbb{R}^d \) are fiber bundles over the torus \( T^d \), with totally disconnected fiber. This was conjectured (in a weaker form) in the second author’s recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to the \( d \)-fold suspension of a \( \mathbb{Z}^d \) subshift (or equivalently, a tiling space whose tiles are marked unit \( d \)-cubes). The only restrictions on our tiling spaces are that

the tiles are assumed to be polygons (polyhedra if \( d > 2 \)) that meet full-edge to full-edge (or full-face to full-face),
only a finite number of tile types are allowed, and
each tile type appears in only a finite number of orientations.

The proof is constructive and we illustrate it by constructing a ‘square’ version of the Penrose tiling system.

M. Barge and R. Williams: Asymptotic structures in Penrose, Tübingen and octagon tilings, 2012. Conference paper. misc

William Williams	Related
Marcy Mason Barge	Related

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Robert Fones Williams

An Autobiography

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