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Basic Properties of the β Dynamical System

Author: WangXiaoKun
Tutor: WangBaoWei
School: Huazhong University of Science and Technology
Course: Basic mathematics
Keywords: β-expansion Pressure function Approximation from below
CLC: O19
Type: Master's thesis
Year: 2013
Downloads: 5
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This note is concerned with some important basic properties in dynamical system ([0,1],Tβ)for general β>1.First,we study β-expansion and the characteristic equation. Second,this note gets the distribution of simple β numbers.In the end,we study the relation of different β-admissible sequences.In some sense,it implies that almost all β-admissible sequence is βN-admissible when we approximates β from below by Parry number βN.Let the β expansion of1as follow:(ε1*(β),…,εN-1*(β),εN*(β)-1)∞Give β admissible sequence with the length of n: ε=(ε1,…,εn) We have the following way:we notice the word ε=(ε1,…,εn)from left to right,if there is the finite part of β expansion of1,then we change to(ε1*(β),…,εN-1*(β),εN*(β)-1).Otherwise,we needn’t do nothing.The way is also as follow:(ε1*(β),…,εN-1*(β),εN*(β))→(ε1*(β),…,βN-1*(β),εN*(β)-1). At last,we can get the sequence ε,and let π(ε)=ε.for all ε∈∑βNn.#π-1(ε)≤2n/N, We know: βn≤#∑βn≤βn+1/(β-1). This states that ε and ε are one to one.So,when we study problem in the sub system,we don’t lost admissible sequences.Consequently,we prove that the pressure function P(f,Tβ)with a continuous potential f is proven to be continuous with respect to β,where,f is a positive continuous function on [0,1].We can get some important results of the dimension by the continuity of pressure function.

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