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# Comparison of Fractal Measures, Periodic β Expansions and Higher Dimensional Periodic Words

Author: ZuoYanHui
Tutor: WenZhiYing
School: Tsinghua University
Course: Mathematics
Keywords: gauge function recurrent sequence periodicβexpansion maximal pattern complexity analytic type
CLC: O174.12
Type: PhD thesis
Year: 2006
Downloads: 120
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### Abstract

 The thesis consists of three independent problems, which are respectively the com-parison of usual fractal measures and the measures induced by gauge functions; de-termining the periods of theβ-expansions of real numbers, and characterizing higherdimensional periodic words by a kind of new complexity. These problems come fromfractal geometry, number theory and combinatorics respectively.Chapter one is devoted to an introduction of the backgrounds of the problemsmentioned above, and gives a short survey about the related fields. By the way, wepresent also how these problems are introduced in our researches.In chapter two, we establish the following relation between usual fractal measuresand the measures induced by gauge functions: suppose the gauge function is equivalentto power function td, then all the measures are the same up to a constant, where d is thedimension of the space. These constants are determined by the upper and lower limitsof the quotient of gauge function and td when t tends to 0+. But if gauge function isequivalent to power function ts with 0 < s < d, then the results above fail by showingseveral counter examples. The main technique used is the density theory of measures,of course, it should be generalized to the case of measures induced by gauge functions.In chapter three, we characterize the periods of theβ-expansions of real numbers.We obtain the following results: If x∈Q(β) has strictly periodicβ-expansion, thenthe periods of the expansion are determined by a linear recurrent sequence related toβand x, whereβis a quadratic Pisot unit. More precisely, the periods of the expan-sion coincide with that of the linear recurrent sequence modulo certain integer numberwhich is completely determined byβand x. Particularly, ifβ= (√5 + 1)/2 is thegolden number, then the periods of theβ-expansions are determined by the Fibonaccisequence. To deal with this purely number-theoretic problem, we introduced someideas and techniques of dynamical system and Tiling theory.In chapter four, we characterize the higher dimensional periodic words by theoryof maximal pattern complexity, developed by [59, 60]. This result is a generalizationof  about two dimensional words. The most difficult part is to treat the monotoneword to which the known ideas are not valid no longer. We overcome this difficulty byintroducing the concept of analytic type and using certain topological methods.

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