Spectrality Of The Moran Measures With Two-element Digit Sets On The Plane

Let μ be a Borel probability measure in R2 with compact support.If there is a countable set A such that E(Λ):={e-2πi<λ,x>:λ∈Λ} forms an orthonormal basis for L2(μ),the measure μ is called a spectral measure,and accordingly Λ is called a spectrum of μ.This paper mainly studies the spectral properties of Moran measureμ{Mk},D on R2,where(?)∈M2(Z)is the entire matrix expansion,and digital set(?).The main work of this graduation thesis is divided into two chapters:In the second chapter,we introduce the basic knowledge and known conclusion properties needed to study the spectral measures.It mainly introduces the related concepts and properties of Hilbert space,spectral measure and fractal measure.In the third chapter,we consider the Moran measure μ{Mk},D where the notation of Mk and D is expressed as above.We get the following conclusions,μ{Mk},D is a spectral measure if and only if for all k≥2,ak is even.