Celebratio Mathematica

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

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

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

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

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

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Robert Fones Williams

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