Positively Expansive Maps and Resolution of Singularities
Wayne Lawton
Department of Mathematics
National University of Singapore
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.
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