Upload
marsden-lopez
View
25
Download
2
Embed Size (px)
DESCRIPTION
ZERO SETS OF EIGENFUNCTIONS OF TRANSITION OPERATORS. Wayne M. Lawton Department of Mathematics National University of Singapore [email protected]. TRANSITION OPERATORS. - PowerPoint PPT Presentation
Citation preview
ZERO SETS OF EIGENFUNCTIONS OF
TRANSITION OPERATORS
Wayne M. Lawton
Department of Mathematics
National University of Singapore
TRANSITION OPERATORS
where M is an expansive endomorphsim and u is a nonnegative analytic mask function on the d-dimensional torus group T^d = R^d / Z^d and
d
xMyu T,u(y)f(y)f)(x)(P
x
d
xMy
T,0u(y)
x
REFINABLE FUNCTIONS
d
Zp
Rx,p)-x(M)p(c(x)d
T
d
1k
Ry,)yM(u)0(ˆ)y(ˆ k
m/cu If is continuous then
EIGENFUNCTIONS
)R(L d2 df
dRx
dZp),px()x()p(d
If whereand
then ffPu and
d2
Zq
Ry,0|)qy(|f(y)d
ZERO SETS
0f
0)y(f|RyV df
and then
satisfies
ffPu If
dff ZMVV 1 Eq.
LAGARIUS-WANG HYPERPLANE-ZEROS
CONJECTURE
If f is real-analytic and satisfies Eq. 1 then
dm
1i
iif ZxSV
rational subspaces of, points inii x,S dR
A RESULT OF FRISCH
Theorem. Ring of real analytic functions on T^d is Noetherian (Grothendieck conjecture)
satisfies Eq. 1 thenfVCorollary. If
dff ZMVV
LOJACIEWICZ STRUCTURE THEOREM for REAL ANALYTIC VARIETIES
Theorem (see Krantz and Parks) Generalizes of the local Puiseux expansion for d = 2. Derived from Weierstrass Preparation.
Provides ‘easy’ proof of the corollary.
SUBSPACE ITERATION
The QR decomposition of M^k for large k describes the asymptotic geometry of the dynamical system
Combined with the LST and induction it yields a proof of the LWC.
ff VV:M
REFERENCESJeffrey C. Lagarius and Yang Wang, “Integral self-affine tiles in R^n II. Lattice tilings”, J. Fourier Anal.& Appl., 3 (1997), 83-101. Jacques Frisch,”Points de platitude d’un morphism d’espaces analytiques complexes”, Inventiones Math., 4 (1967), 118-138.
S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions, Birkhauser, Boston, 1992.
B. N. Parlett and W. G. Pool, Jr., “A geometric theory for the QR, LU and power iterations”, SIAM J. Numer. Analy. 10#2 (1973), 389-412.