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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.

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ZERO SETS OF EIGENFUNCTIONS OF TRANSITION OPERATORS

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  1. ZERO SETS OF EIGENFUNCTIONS OF TRANSITION OPERATORS Wayne M. Lawton Department of Mathematics National University of Singapore matwml@nus.edu.sg

  2. 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

  3. REFINABLE FUNCTIONS If is continuous then

  4. EIGENFUNCTIONS and where If then and

  5. ZERO SETS If and then satisfies

  6. LAGARIUS-WANG HYPERPLANE-ZEROS CONJECTURE If f is real-analytic and satisfies Eq. 1 then rational subspaces of, points in

  7. A RESULT OF FRISCH Theorem. Ring of real analytic functions on T^d is Noetherian (Grothendieck conjecture) Corollary. If satisfies Eq. 1 then

  8. 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.

  9. 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.

  10. Jeffrey C. Lagarius and Yang Wang, “Integral self-affine tiles in R^n II. Lattice tilings”, J. Fourier Anal.& Appl., 3 (1997), 83-101. REFERENCES 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.

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