题目：Sierpinski垫上的正交系

本文研究了自仿测度的非谱问题.主要目标是在一类自仿测度下,估计Sierpinski垫上正交系的基数.本文的主要结果如下:
1.联系到扩张整矩阵和数字集:$$ M= left[                                egin{array}{cc}                                 a & b                                  c & d                                  end{array}                  ight], D=left{                                           egin{array}{c}                                             left(                                 egin{array}{c}                                   0                                    0                                    end{array}                               ight),                              left(                                 egin{array}{c}                                   1                                    0                                    end{array}                               ight),left(                                 egin{array}{c}                                   0                                    1                                  end{array}                               ight)                                 end{array}                                ight}$$的自仿测度$mu_{M,D}$的支撑在迭代函数系${phi_{d} (x)=M^{-1}(x+d)}_{din D}$的吸引子上,其中$(a+d)^2=4(ad-bc)$并且3不整除a+d. 首先得到了自仿测度的傅里叶变换$hat{mu}_{M,D}$的零集$Z(hat{mu}_{M,D})$的一些表示性质,其次用简单的方法得到了几个有用的结果.最后利用自仿测度傅里叶变换$hat{mu}_{M,D}$的零集$Z(hat{mu}_{M,D})$的这些性质证明了$L^2(mu_{M,D})$空间中有且只有3个相互正交的指数函数.
2.联系到扩张整矩阵和数字集:$$M=left[egin{array}{ccc}p & 0 & m & p & 0 & 0 & p end{array}ight], D=left{                egin{array}{c}                  left(egin{array}{c}0 end{array}ight),left(egin{array}{c}1 end{array}ight),left(egin{array}{c}0 1 end{array}ight),left(egin{array}{c}0 1  end{array} ight)  end{array} ight}$$                               的自仿测度$mu_{M,D}$的支撑在广义三维Sierpinski垫T(M,D)上,其中p是奇数.在第三章先证明了在$L^2(mu_{M,D})$空间中至多存在7个相互正交的指数函数,并利用这个方法估计了当数字集有四个元素时平面Sierpinski垫上正交指数函数的个数.最后当m=0时,证明了在三维Sierpinski垫上至多存在4个相互正交的指数函数.
上述研究结果分别推广了Dutkay, Jorgensen, Li以及Pedersen等人的相应结论,对进一步深刻了解自仿测度的谱性质有重要作用.