ZERO SETS OF EIGENFUNCTIONS OF TRANSITION OPERATORS

ZERO SETS OF EIGENFUNCTIONS OF

TRANSITION OPERATORS

Wayne M. Lawton

Department of Mathematics

National University of Singapore

matwml@nus.edu.sg

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.