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 Integer n x n expanding matrices (moduli of all eigenvalues > 1) real analytic functions periodic functions zero sets

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

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

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

7. Reformulation & Extension Theorem (Main)

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

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

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 Lemma submanifold a continuous Proof 1 st deg Taylor approx. error

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

13. Asymptotic Tangent Vectors Theorem submanifold

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

15. Invariance Properties an expanding endomorphism induces 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 use induction on then

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 makes this diagram commute is injective, paths in lift to paths with bounded lengths

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

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

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

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

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