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

AbstractIn 1997 Lagarias and Wang asserted a conjecture thatcharacterized 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.

nn ZR /

NotationFields: Complex, real,

rational numbers

Ring: of integers,

Set: natural numbers

...}3,2,1{,,,, NZQRC

nEInteger n x n expanding matrices

(moduli of all eigenvalues > 1)

)( nRAreal analytic functions

}0)(:{ xhRxZ nh

nn ZpRxxhpxh ,),()(periodic functions

zero sets

periodic )(, nn RAhEE

rational

subspaces

nhh ZZEZ )(

ni

p

i ih ZxVZ

)(1

Hyperplane Zeros Conjecture

}{span nii ZVV

ni Rx Lagarias & Wang, JFAA, 1997

More Notationtorus group and canonical homomorphism

opennTU

nnnn ZRTR n /

})(:{)( UAhZUV h ))(()( 1 OAhUAh nn

analytic

varieties

analytic

sets} var.anal.locally {)( US

nEn TT and1E preserve these sets

More Notation

gives a one to one correspondences between

)( nc TG

n

Closed

connected

subgroups

subspaces of

)({)(1p

i iin xGTF

nR and connected subgroups ofnT

rational subspaces correspond to elements in )( nc TG

)( nci TGG

ni Tx

)()( TGTG cE

c )()(1

TFTF E

)(),(, SESTVSEE nn

Reformulation & Extension

)( nTFS

)(),(, SESTSSEE nn

)( nTFS

Theorem (Main)

)dim(),( SdTSS n Regular Points

USSRx d )(

xU

d-dim manifold

for some openFacts

}),dim(:{)( dxSSxSRd )()(\ n

d TSSRS dSRS d ))(\dim(

NarasimhanBruhat-Whitney

Reduction

)(),( SEETSS n

)()()3( n

d TFSR

*Y

SSE )()1(

Theorem (Reduced) nEE

Meta Theorem : Reduced theorem equivalent to main theorem

)())(\()2( nd TSSRS

Intersection of all real analytic sets containing Y

Stationarity

Theorem (Narasimhan) The intersection of any collection of real analytic subsets of the torus group equals the intersection of a finite subcollectionCorollary Properties (1) and (2) of the reduced theorem are valid.

)(0

SESp

i

ip

Proof (1) Else

Asymptotic Tangent Vectors

metric space),( X XBA ,),(infsup),( baBA

BbAa

asymmetric

distance unit ball }),(:{),( ryxXyxrB

Lemma nRM submanifold2Ca),0(: M continuous

]1,0[,,)())(),(( 2 rMxrxMTxrB x

Proof 1st deg Taylor approx. error

Asymptotic Tangent Vectors

0,,}),({),( vRvxvspanxvx nA triplet

Theorem

),,( vxM is asymptotic if

VVREE nn ,

asymptotic

0))(),,(),1((lim

MEvExExEB jjjj

j

dominant, complement eigenspaces),,( vxMVv

Proof Derive/exploit inequality

22

||||

||||||||)())(),,(),1((

vE

ExxMEvExExEB

j

jjjjj

Asymptotic Tangent Vectors

Theorem )(, nn TSSEE

submanifold VMTMxSM xn )(,)(

nRMSSE ,)(

)(,or nc TGHSy

SHyH and1)dim(

Invariance Properties

Definition )(, nc

n TGGTS The G-invariant subset of S

GgG gSSGxSxS

)(}:{

SHyH and1)dim(

Lemma )()( nG TSSTSS

Invariance Properties

GTTTGG nnnc

n /)(

)dim()dim())(dim()1( GSS GGG )/()()()2( GTSSTSS n

GGn

G

)/()()()3( GTFSTFS nGG

nG

EGGEEE n )(,)4(an expanding endomorphism

induces

GTGT nEn // and ))(()()( GnGG SESSES

Invariance Properties

)( nc TG

GGGG jj

with Hausdorff topology is

compact, countable, and

,,, SSTSEE nn

for large j

NpTGGSS ncH ),(

Theorem

,1)dim(),(,)( HTGHSSE nc

Gp SSGGEG ,)(,1)dim(

Invariance Properties

Lemma ),(),(, SESTSSEE nn

Hn

c SSHTGH ,1)dim(),(

,1)dim(),(,)( HTGHSSE nc

)( nTFSProof Use previous theorem,replaceE by SEE n

p , by ),(SGuse induction on ))(dim()dim( SS G

then

Invariance Properties

Proposition ,,, nnn RVTxEE

and every pair of points in K can be

,)(, yyETy n

)( nrat ZVspanV

connected by a smooth path with a uniform bound on the lengths, then

,)(,)(,)( KKExVKVVE n

yVK ratn )(

Invariance Properties

Proof Find yVxVyyETy nnn )()(,)(,

construct unique homomorphism

that

JJEVJyKJ n )(),(,

WVV Wrat

GVV Gnratn )()( n n

makesthis

diagramcommute

is injective, paths in )(JGlift to paths with bounded lengths

)(}0{))((1ratnG VJJ

Resolution of Singularities

Theorem ),(,)(),(, SESSRSTSSEE dn

n

SS E

S~ real analytic manifold no bdy

WLOG assume S~

Finite # connected components

SS E ~~ ~

is connected

immersion

Brower Invariance Domain&Baire Category VS or )( n

c TGH inv )( nTFS

surjective

Resolution of Singularities

Construction Sx xMS

0

~

is an

by0

~S

mapping

is a real analytic submanifold of S,

germ of

Mx

analytic and Riemannian manifold

,x topologizexM

talee

M at

SS 0

~MMyM y },:{ an.sub. S

0

~S

Resolution of Singularities

0

~ completion

~SS

real analytic sets

wrt geodesic metric

unif. cont.

above is surjective

Lojasiewicz’s structure theorem for

SS ~

)~

()( 0SSRd

0

~\

~SSK

0

~S is Hausdorff,

S~

is connected, locally connected

compact

KSKSS E \~

\~~

0 open

SS~~

0 by Hiraide

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).

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.

nR

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.