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Some Problems on Fractal Geometry and Dynamical Systems

Author: ShuLin
Tutor: ChenErCai
School: Nanjing Normal University
Course: Basic mathematics
Keywords: Figure iterated function systems Self -shaped collection Figure since common collection Strong Open Set Condition Open set condition Topological Entropy Part One Bifurcation point set
CLC: O19
Type: Master's thesis
Year: 2003
Downloads: 127
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Abstract


This paper focuses on the two themes of the fractal geometry and dynamical systems . One , two chapters of iterated function systems , the third chapter is about the power system topology entropy . In the first chapter , we study the Hausdorff dimension and measure of self - conformal sets general metric space , mainly to consider the separation properties and Hausdorff measure of the relationship between since commonality collection is . This chapter as from the G. A. Edgar and J. Golds (1999) , proposed general complete metric space CPC conformal mapping defined general complete metric space since the first attempt of the Chinese Communists shaped iterated function systems research , while promoting the A. Schief (1996) years of self-similar set of conclusions complete metric space . In the second chapter , we iterated function systems R d space of two conclusions . The first is with respect to FIG iteration since the common set of open set condition , strong open set condition , and the equivalence between positive Hausdorff measure . Second is Figure iterated function systems to meet bi-Lipschitz conditions and strong open set condition , we obtain estimates of the upper and lower bounds of the the measure dimension on Figure attractor . In the third chapter , we mainly study the bifurcation point set topological entropy . Let f : XX is a compact metric space (X, d) to meet the specification of transformation . For any n ∈ N, the definition the L n x = 1 / N SUM from k = 0 to n -1 the δ f k x where δ x is the atomic measure at the point x . Let Y is a linear compatible metric vector space , M (X) is a space consisting of all probability measure on X ,: M (X) Y serials, affine . Defined set of bifurcation point D (f,) as follows : D ( f ) = {x ∈ X | limit the L n x does not exist } . If not all points x ∈ X , { the L n x } have the same limit points , D (f,) topological entropy and topological entropy of the entire space . In addition , we also consider the topological entropy of the Part . The topological entropy Let M (f, X) for the collection of all f - invariant probability measure on X , C Y a closed convex set , Part Δ (C) meet which the h μ < / sub > (f) representatives with respect to the measure μ entropy map f measure .

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CLC: > Mathematical sciences and chemical > Mathematics > Dynamical systems theory
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