Celebratio Mathematica — Williams — Complete Bibliography

[1] R. F. Williams: “Local contractions of compact metric spaces which are not local isometries,” Proc. Am. Math. Soc. 5 : 4 (1954), pp. 652–654. MR 63028 Zbl 0057.14905 article

Following Albert Edrei [1952], if \( X \) is a compact metric space with metric \( \rho \), \( f \) is a mapping of \( X \) onto \( X \), and \( x \in X \), then \( x \) is said to be a point of contraction under \( f \) relative to \( X \) provided that there is a positive number \( \mu(x) \) such that if \( y\in X \) and \( \rho(x,y) < \mu(x) \), then \[ \rho[f(x),f(y)] \leq \rho(x,y) .\] Further, if each point of \( X \) is a point of contraction under \( f \) relative to \( X \), \( f \) will be said to be a local contraction of \( X \). Edrei posed the following question: if \( X \) is a compact metric space and \( f \) is a contraction of \( X \) onto \( X \), is \( f \) a local isometry? The purpose of this paper is to answer this question in the negative.

[2] R. F. Williams: Reduction of open mappings. Ph.D. thesis, 1954. Advised by G. T. Whyburn. A short article based on this thesis was published in Proc. Am. Math. Soc. 7:2 (1956). phdthesis

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

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

[5] 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

[6] 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.

[7] 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

[8] 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

[9] 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.

[10] R. Williams: “Open mappings and solenoids,” Notices Am. Math. Soc. 6 (1959), pp. 867. article

[11] 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.

[12] 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

[13] R. F. Williams: “A useful functor and three famous examples in topology,” Trans. Am. Math. Soc. 106 : 2 (1963), pp. 319–329. MR 146832 Zbl 0113.37803 article

The purpose of this note is to describe a functor which provides a framework for certain constructions in topology. It is related to the sets \( (E,\pi,B,X,q) \) described in [1960] and is particularly adapted to discussing the limit of repeated modifications of triangulable spaces. Roughly speaking, one forms a space \( X\Delta K \) by replacing each top dimensional simplex of a complex \( K \) with a copy of a space \( X \). If in addition there are mappings on the spaces \( X \), \( K \), these induce a mapping on the new space \( X\Delta K \).

It has been called to my attention that several authors have considered analogous functors (though not as far as I know, in written form). This is not surprising inasmuch as \( X \Delta K \) is defined just as the Whitney sum of two bundles.

Though the principal applications of this functor are to be found elsewhere, in a paper by Frank Raymond and the author [1960, 1963] and a forthcoming paper by the author, three famous examples [Pontrjagin 1930; Boltyanskii 1951; Kolmogoroff 1937] are given as applications in the last section. Two of these are in dimension theory proper, but the third is essentially about transformation groups.

It is hoped that the reader will find our description of Boltyanskii’s example easier than the original, as a simpler, more homogeneous version is given. In addition, in our version of Kolmogoroff’s example, the group acts without fixed points. This answers a question raised by Anderson [1957].

[14] F. Raymond and R. F. Williams: “Examples of \( p \)-adic transformation groups,” Ann. Math. (2) 78 : 1 (July 1963), pp. 92–106. These examples originated in an article published in Bull. Am. Math. Soc. 66:5 (1960). MR 150769 Zbl 0178.26003 article

[15] R. F. Williams: “The construction of certain 0-dimensional transformation groups,” Trans. Am. Math. Soc. 129 : 1 (1967), pp. 140–156. MR 212127 Zbl 0169.25903 article

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

[17] M. E. Mahowald and R. F. Williams: “The stable homotopy of \( K(Z,n) \),” Bol. Soc. Mat. Mex., II. Ser. 11 (1967), pp. 22–28. MR 235563 Zbl 0178.57203 article

[18] 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

[19] R. F. Williams: “Compact non-Lie groups,” pp. 366–369 in Proceedings of the conference on transformation groups (New Orleans, 8 May–2 June 1967). Edited by P. S. Mostert. Springer (Berlin), 1968. MR 245724 Zbl 0193.52502 incollection

[20] R. F. Williams: “Non-compact Lie group actions,” pp. 441–445 in Proceedings of the conference on transformation groups (New Orleans, 8 May–2 June 1967). Edited by P. S. Mostert. Springer (Berlin), 1968. Zbl 0193.52501 incollection

[21] R. F. Williams: “Expanding attractors,” pp. 79–89 in Colloque de topologie différentielle [Differential topology colloquium] (Mont-Aigoual, France, 1–6 June 1969). Edited by C. Godbillon and H. Rosenberg. Université de Montpellier, 1969. MR 287581 Zbl 0208.25801 incollection

Claude Godbillon	Related
Harold William Rosenberg	Related

[22] 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

[23] 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

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.

Stephen Smale	Related
S. S. Chern	Related	Volume

[24] 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

[25] 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

Stephen Smale	Related
S. S. Chern	Related	Volume

[26] 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

[27] R. F. Williams: “Classification of symbol spaces of finite type,” Bull. Am. Math. Soc. 77 : 3 (May 1971), pp. 439–443. MR 300263 Zbl 0213.50403 article

[28] 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].

[29] R. Williams: “Expanding attractors,” pp. 125–127 in Proceedings of the symposium on differential equations and dynamical systems (Coventry, UK, September 1968–August 1969). Edited by D. Chillingworth. Lecture Notes in Mathematics 206. Springer (Berlin), 1971. incollection

[30] 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

[31] R. F. Williams: “Classification of subshifts of finite type,” pp. 281–285 in Recent advances in topological dynamics: Proceedings of a conference in topological dynamics (New Haven, CT, 19–23 June 1972). Edited by A. Beck. Lecture Notes in Mathematics 318. Springer (Berlin), 1973. Conference in honor of Gustav Arnold Hedlund. MR 391060 Zbl 0267.54038 incollection

Gustav Arnold Hedlund	Related
Anatole Beck	Related

[32] 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

[33] R. F. Williams: “Paul Schweitzer’s solution of the Seifert problem,” pp. 301–304 in Global analysis and its applications (Trieste, Italy, 4 July–25 August 1972), vol. 3. IAEA (Vienna), 1974. MR 436164 Zbl 0306.57010 incollection

[34] R. F. Williams: “Strange attractors,” pp. 293–300 in Global analysis and its applications (Trieste, Italy, 4 July–25 August 1972), vol. 3. IAEA (Vienna), 1974. MR 440623 Zbl 0303.58013 incollection

[35] 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.

[36] R. F. Williams: “Errata to ‘Classification of subshifts of finite type’,” Ann. Math. (2) 99 : 2 (1974), pp. 380–381. Errata for an article published in Ann. Math. 98:1 (1973). article

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

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

[39] 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

[40] 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.

[41] 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

[42] 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].

[43] 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

[44] 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

[45] R. F. Williams: “The bifurcation space of the Lorenz attractor,” pp. 393–399 in Bifurcation theory and applications in scientific disciplines (New York, 31 October–4 November 1977). Edited by O. Gurel and O. E. Roessler. Annals of the New York Academy of Sciences 316. New York Academy of Sciencies, 1979. MR 556844 Zbl 0472.58016 incollection

In a joint paper [1979] John Guckenheimer and the present author showed that the space of the title is two-dimensional, in that the pair \( (k_l,k_r) \) of kneading sequences is a complete conjugacy class invariant for the flows near the three-dimensional Lorenz system. But there are reasons for obtaining more complete knowledge of the space \( \mathcal{K} \) of all such pairs, \( k \). Thus, one can use the “Parry coordinates” \( (\lambda,c) \) but a priori, one doesn’t know these all correspond to Lorenz models — i.e., to differential equations in \( \mathbb{R}^3 \). Secondly, a recurring problem in bifurcation theory is just how do the periodic orbits change. as one changes a bifurcation parameter? Thus we shall present a formula for computing the periodic orbits in terms of the parameter \( k \), i.e., the kneading sequences.

Okan Gürel	Related
Otto E. Rössler	Related

[46] 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

[47] 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

[48] 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

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

[50] R. F. Williams: “Attractors, strange and perverse,” pp. 473–495 in Proceedings of the 1981 Shanghai symposium on differential geometry and differential equations (Shanghai and Hefei, China, 20 August–13 September 1981). Edited by C. H. Gu. Scientific Press (Beijing), 1984. MR 825292 Zbl 0679.58029 incollection

[51] 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

[52] 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

[53] 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

[54] M. Shub, D. Tischler, and R. F. Williams: “The Newtonian graph of a complex polynomial,” SIAM J. Math. Anal. 19 : 1 (1988), pp. 246–256. MR 924558 Zbl 0653.58013 article

In a recent paper [4] Smale posed as an important problem in complexity theory, characterization of the graph \( G_f \) of the Newtonian vector field \( N_f \) for a complex polynomial \( f \). Such graphs are known to be connected and acyclic and Smale conjectured that these two properties completely characterize them. The purpose of this paper is to prove this conjecture, after adding an additional hypothesis. In addition we give an example and prove a proposition to show this is necessary.

We present the proof as Theorem C in §5 using the topological characterization of analytic maps given by Stöilow [5] in 1929. In addition we present direct proofs of three special cases as Theorem A, Theorem B and Example 7. While this was being written an independent proof was given in the generic case (Theorem A) in [2].

Sections 1 and 2 are devoted to basic properties and to a list of examples designed to acquaint the reader (and the writers) with various aspects of Newtonian graphs.

David C. Tischler	Related
Michael Ira Shub	Related

[55] R. F. Williams: “The braid index of an algebraic link,” pp. 697–703 in Braids (Santa Cruz, CA, 13–26 July 1986). Edited by J. S. Birman and A. Libgober. Contemporary Mathematics 78. American Mathematical Society (Providence, RI), 1988. MR 975103 Zbl 0673.57003 incollection

Until recently the braid index of an oriented link (\( = \) the minimal number of strands the link can be presented as a closed braid on) was essentially non-computable and hence a useless concept. But with the advent of the Jones polynomial [1985] and its relatives [Freyd et al. 1985; Lickorish and Millett 1985] computations are now possible. Jones himself has computed [1987] this invariant for all knots with ten or fewer crossings. In particular, a long outstanding conjecture on positive braids with a full twist (“very positive braids”) bas now been proved independently by Morton [1985] and Franks-Williams [Franks and Williams 1985]. Here we show that this theorem suffices to compute the braid index of an important family of links — the algebraic links. Briefly, at a singular point \( (w_0,z_0) \) of the variety \( P(w,z) = 0 \), \( w, z \in \mathbf{C} \) the complex numbers, the boundary \( \mathbb{S}^3 \) of a small neighborhood of \( (w_0,z_0) \) intersects the variety in an oriented one-manifold, \( L \). Then \( L \subset \mathbb{S}^3 \) is an oriented link. Any such link \( L \) is called an algebraic link. They have been studied for some time and are known to be positive braids. (i.e., can be presented as a closed braid with all crossings of the same sign — say positive or right-handed).

Anatoly S. Libgober	Related
Joan S. Birman	Related	Volume

[56] R. F. Williams: “Geometric theory of dynamical systems,” pp. 67–75 in Workshop on dynamical systems (Trieste, Italy, September 1988). Edited by Z. Coelho and E. Shiels. Pitman Research Notes in Mathematics 221. Longman Scientific & Technical (Harlow, UK), 1990. MR 1096341 Zbl 0688.58029 incollection

Zaqueu Coelho	Related
E. Shiels	Related

[57] R. F. Williams: “How big is the intersection of two thick Cantor sets?,” pp. 163–175 in Continuum theory and dynamical systems (Arcata, CA, 17–23 June 1989). Edited by M. Brown. Contemporary Mathematics 117. American Mathematical Society (Providence, RI), 1991. MR 1112813 Zbl 0734.54022 incollection

[58] R. F. Williams: “The braid index of generalized cables,” Pac. J. Math. 155 : 2 (1992), pp. 369–375. MR 1178031 Zbl 0811.57013 article

If one knot is fashioned into another, by replacing each strand with \( q \) strands, then something gets multiplied by \( q \). What? The answer should not be overly dependent on how these strands are intertwined. We show that an invariant called the braid index is an answer. This proposition is apparently new. Another answer covered by our proof is the bridge number, though this was proved by Shubert in 1954. It was only with the advent of the Jones polynomial and its relatives in the mid 1980s, that much attention has been given to the braid index. For example, the knots obtained by repeated period doubling were shown to obey the multiplication rule, though no one seems to have thought of it this way. Their braid indices are powers of 2. We first considered the current proposition in trying to show that a certain knot, known to have braid index 5, could not be a two-cabling of anything.

[59] R. F. Williams: “Strong shift equivalence of matrices in \( \mathrm{GL}(2,\mathbb{Z}) \),” pp. 445–451 in Symbolic dynamics and its applications (New Haven, CT, 28 July–2 August 1991). Edited by P. Walters. Contemporary Mathematics 135. American Mathematical Society (Providence, RI), 1992. MR 1185108 Zbl 0768.58039 incollection

[60] C. Tresser and R. F. Williams: “Splitting words and Lorenz braids,” pp. 15–21 in Homoclinic chaos (Brussels, 6–9 May 1991), published as Phys. D 62 : 1–4. Issue edited by P. Gaspard, A. Arneodo, R. Kapral, and C. Sparrow. Elsevier (Amsterdam), 1993. MR 1207414 Zbl 0783.58041 incollection

Colin Trevor Sparrow	Related
Charles Tresser	Related
Alain Arneodo	Related
Pierre Gaspard	Related
Raymond Kapral	Related

[61] 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

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.

Jerrold E. Marsden	Related
Michael Ira Shub	Related
Stephen Smale	Related
Morris W. Hirsch	Related

[62] 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

René Thom	Related
E. Christopher Zeeman	Related
Mauricio Matos Peixoto	Related
Jerrold E. Marsden	Related
Donald Jerome Brown	Related
Michael Ira Shub	Related
Stephen Smale	Related
Philip John Holmes	Related
Richard Manning Karp	Related
Oscar Eramus Lanford, III	Related
Morris W. Hirsch	Related

[63] S. Kennedy, M. Stafford, and R. F. Williams: “A new Cayley–Hamilton theorem,” pp. 247–251 in Global analysis in modern mathematics (Houston, TX). Edited by K. Uhlenbeck. Publish or Perish, 1993. Symposium in honor of Richard Palais’ sixtieth birthday. MR 1278758 Zbl 0972.15004 incollection

Stephen Francis Kennedy	Related
Matthew Wayne Stafford	Related
Karen Uhlenbeck	Related	Volume
Richard Sheldon Palais	Related	Volume

[64] R. F. Williams: “Pisot–Vijayarghavan numbers and positive matrices,” pp. 268–277 in Dynamical systems and chaos (Hachioji, Japan, 23–27 May 1994), vol. 1. Edited by N. Aoki, K. Shiraiwa, and Y. Takahashi. World Scientific (River Edge, NJ), 1995. MR 1479944 Zbl 0989.57500 incollection

Branched manifolds, were introduced in [1969, 1974]. (The 1-dimensional case was introduced in [1967] and was used to study Anosov diffeomorphisms in [1970]; this last was popularized under the name “train track” [Fathi et al. 1979] somewhat later.) Branched manifolds have tangent bundles, so that such concepts as orientability and immersion make sense. However, the top dimensional homology of our branched manifolds are typically of dimension \( k > 1 \), so that there are many choices of generators, consistent with the chosen orientation. Thus we say that an oriented branched \( n \)-manifold \( K \) is sensed provided the \( n \) dimensional homology is free abelian, and that generators are so chosen that the homomorphisms induced by immersions, relative to the chosen bases, are matrices whose entries are all \( \geq 0 \).(Or \( \leq 0 \). Here the inequalities depend upon whether the immersion preserves or reverses the chosen orientations.) Sensed, oriented, branched manifolds are called “SOB’s” for short. We use this concept to prove a partial converse to the Perron–Frobenius theorem, and expect to be able to prove that “weak shift equivalence” implies “strong shift equivalence” for certain matrices. Though our theory works, at least in principal, in all dimensions, we have carried out the proofs only in the 3 dimensional case.

Yōichirō Takahashi	Related
Nobuo Aoki	Related
Kenichi Shiraiwa	Related

[65] R. F. Williams: “Spaces that won’t say no,” pp. 236–246 in International conference on dynamical systems (Montivideo, Uruguay, 27 March–1 April 1995). Edited by F. Ledrappier, J. Lewowicz, and S. Newhouse. Pitman Research Notes in Mathematics 362. Longman Scientific & Technical (Harlow, UK), 1996. Proceedings dedicated to Ricardo Mañé. MR 1460809 Zbl 0876.57044 incollection

A class of spaces \( K \) and their natural maps \[ f:K\to K^{\prime} \] were introduced in [Williams 1974] which satisfies the defining property that \[ f_*:H_n(K) \to H_n(K^{\prime}) \] is always a “positive matrix”. Here \( n \) is the dimension of both \( K \) and \( K^{\prime} \); we follow the usage in symbolic dynamics and say that a matrix is positive provided all its entries are \( \geq 0 \). The purpose of this paper is to show how a natural construction related to the “DA” maps of Smale [Williams 1970] often leads to SOB’s and in particular shows that they exist in all dimensions. Here we present the easier conceptual parts of this theory and elicit a geometric property which when it holds leads to quite satisfactory proofs.

Ricardo Mañé	Related
Sheldon Earle Newhouse	Related
François Ledrappier	Related
Jorge Lewowicz	Related

[66] R. F. Williams: “The universal templates of Ghrist,” Bull. Am. Math. Soc. (N.S.) 35 : 2 (1998), pp. 145–156. MR 1602073 Zbl 0902.57001 article

[67] Z. Coelho, W. Parry, and R. Williams: “A note on Livšic’s periodic point theorem,” pp. 223–230 in Topological dynamics and applications (Minneapolis, MN, 5–6 April 1995). Edited by M. G. Nerurkar, D. P. Dokken, and D. B. Ellis. Contemporary Mathematics 215. American Mathematical Society (Providence, RI), 1998. Volume in honor of Robert Ellis. MR 1603197 Zbl 0897.58037 incollection

David Bennett Ellis	Related
Mahesh G. Nerurkar	Related
William Parry	Related
Zaqueu Coelho	Related
Douglas Paul Dokken	Related
Robert Mortimor Ellis	Related

[68] M. Barge and R. F. Williams: “Classification of Denjoy continua,” Topology Appl. 106 : 1 (September 2000), pp. 77–89. MR 1769334 Zbl 0983.37013 article

[69] 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.

[70] 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.

[71] R. F. Williams: 2D continued fractions and positive matrices. Preprint 10-28, University of Texas, 2010. techreport

The driving force of this paper is a local symmetry in lattices. The goal is two theorems: a partial converse to the Perron–Frobenius theorem in dimension 3 and a characterization of conjugacy in \( Sl^3(\mathbb{Z}) \). In the process we develop a geometric approach to higher dimension continued fractions, HDCF. HDCF is an active area with a long history: see for example [Lagarias 1993; Brentjes 1981].

The algorithm: Let \( Z_r \) be the set of all lattice points within \( r > 0 \) of a ray \[ L = \{mP \in \mathbb{R}^n : m > 0\} .\] Let \( z_1 \) denote the point in \( Z_r \) closest to the origin. Having defined \( z_1,\dots \), \( z_i \), \( 1 \leq i < n \), let \( z_{i+1} \) be the point of \( Z_r \) closest point into the origin, which is independent of \( z_1,\dots \), \( z_i \).

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

William Williams	Related
Marcy Mason Barge	Related

[73] M. Barge, S. Štimac, and R. F. Williams: “Pure discrete spectrum in substitution tiling spaces,” Discrete Contin. Dyn. Syst. 33 : 2 (2013), pp. 579–597. MR 2975125 Zbl 1291.37024 article

Sonja Štimac	Related
Marcy Mason Barge	Related

[74] R. F. Williams: “Anosov and expanding attractors,” Sci. China, Math. 63 : 9 (2020), pp. 1929–1934. MR 4145925 Zbl 1451.37032 article