Positively Expansive Maps and Resolution of Singularities

1. Positively Expansive Maps and Resolution of Singularities Wayne Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg http://www.math.nus.edu.sg/~matwml

2. Abstract In 1997 Lagarias and Wang asserted a conjecture that characterized the structure of certain real analytic subvarieties of the torus group . This talk describes how, during proving [a stronger version of] this conjecture, we were led to construct a resolution of singularities of a real analytic subset. In contrast to Hironaka's proof in his acclaimed 1964 paper (described by Grothendieck as the most difficult theorem in the 20th century), our proof uses a simple [modulo Lojaciewicz's theorem] e'tale covering of the set of regular points followed by an application of Hiraide's 1990 result showing that a compact connected manifold that admits a positively expansive map has empty boundary. The class of analytic sets that satisfy the hypothesis of the conjecture [theorem] include zero sets of eigenfunctions of Frobenius-Ruelle operators that play a crucial role in both refinable functions and statistical mechanics. The methods developed in the paper may also be useful for investigations related to Lehmer's conjecture about heights of polynomials and Mahler's measure.

3. Notation Fields: Complex, real, rational numbers Ring: of integers, Set: natural numbers real analytic functions periodic functions zero sets Integer n x n expanding matrices (moduli of all eigenvalues > 1)

4. Hyperplane Zeros Conjecture rational subspaces Lagarias & Wang, JFAA, 1997

5. More Notation torus group and canonical homomorphism analytic varieties analytic sets and preserve these sets

6. More Notation Closed connected subgroups gives a one to one correspondences between subspaces of and connected subgroups of rational subspaces correspond to elements in

7. Reformulation & Extension Theorem (Main)

8. Regular Points d-dim manifold Facts for some open Narasimhan Bruhat-Whitney

9. Reduction Theorem (Reduced) Intersection of all real analytic sets containing Y Meta Theorem : Reduced theorem equivalent to main theorem

10. Stationarity Theorem (Narasimhan) The intersection of any collection of real analytic subsets of the torus group equals the intersection of a finite subcollection Corollary Properties (1) and (2) of the reduced theorem are valid. Proof (1) Else

11. Asymptotic Tangent Vectors metric space asymmetric distance unit ball submanifold Lemma a continuous Proof 1st deg Taylor approx. error

12. Asymptotic Tangent Vectors is asymptotic if A triplet Theorem dominant, complement eigenspaces asymptotic Proof Derive/exploit inequality

13. Asymptotic Tangent Vectors Theorem submanifold

14. Invariance Properties Definition The G-invariant subset of S Lemma

15. Invariance Properties induces an expanding endomorphism and

16. Invariance Properties with Hausdorff topology is compact, countable, and for large j Theorem

17. Invariance Properties Lemma Proof Use previous theorem,replace by by then use induction on

18. Invariance Properties Proposition and every pair of points in K can be connected by a smooth path with a uniform bound on the lengths, then

19. Invariance Properties Proof Find construct unique homomorphism that diagram commute makes this is injective, paths in lift to paths with bounded lengths

20. Resolution of Singularities Theorem real analytic manifold no bdy Finite # connected components surjective immersion WLOG assume is connected Brower Invariance Domain&Baire Category or inv

21. Resolution of Singularities Construction is a real analytic submanifold of S, at germ of topologize an.sub. by is an mapping analytic and Riemannian manifold

22. Resolution of Singularities wrt geodesic metric unif. cont. above is surjective Lojasiewicz’s structure theorem for real analytic sets is Hausdorff, is connected, locally connected open compact by Hiraide

23. References Hauser, H., The Hironaka theorem on resolution of singularities, Bull. Amer. Math. Soc. 40, 323-403 (2003). Hiraide, K., Nonexistence of positively expansive on compact connected manifolds with boundary, Proceedings of the American Mathematical Society, 104#3(1988),934-941

24. References Hironaka, H., Resolution of singularities of an algebraic variety over a field of characteristc zero,I, II, Annals of Mathematics 79 (1964),109 203; 79 (1964), 205-326 Lagarias, J. C., and Wang, Y., Integral self-affine tiles in . Part II: Lattice tilings, The Journal of Fourier Analysis and Applications, 3#1(1997), 83-102.

25. References Lojasiewicz, S., Introduction to the Theory of Complex Analytic Geometry, Birkhauser,Boston,1991. Narasimhan, R., Introduction to the Theory of Analytic Spaces, Lecture Notes on Mathematics, Volume 25, Springer, New York, 1966.