Quasi Harmonic Functions On Finite-Type Fractals

Quasi-Harmonic Functions on Finite Type Fractals
Nguyen Viet Hung Mathias Mesing

Department of Mathematics and information processing system Science University of Greifswald, Germany

Workshop Fractal abstract 2005 - 09 - 15

Introduction

Fractals of ?nite vitrine

Constructing algorithmic functions

Example

Outline

1

Introduction Fractals of ?nite type Constructing recursive functions Example

2

3

4

Hung, Mesing (Greifswald)

analysis on Finite Type Fractals

Fractal outline 2005

2 / 23

Introduction

Fractals of ?nite type

Constructing recursive functions

Example

Notations
Let fi : Rd ? Rd contractions with same ratio r , i ? S = {1, . . . , m}, and E the belonging unceasing set. Set S n := {(ui )i=1,...,n | ui ? S ? i = 1, . . . , n}, S ? := n?N S n . For u := u1 . . . un ? S ? de?ne fu := fu1 ? . . . ? fun and Eu := fu (E ).





Example (Christmas channelize fractal)
fi : C ? C, i = 1, 2, 3 fi (z) = a(z + ci ) with ci = e a=
2?(i?1) 3

,
7 4

1/( 1 2

?

+

+i

3 2 )

Hung, Mesing (Greifswald)

Analysis on Finite Type Fractals

Fractal Analysis 2005

3 / 23

Introduction

Fractals of ?nite type

Constructing recursive functions

Example

Motivation
Harmonic functions on Sierpinski gasket
Regard functions on points of E . envision the values at new points by averaging rules:
g (x) = g (y ) = g (z) =
g (a)+2g (b)+2g (c) 5 2g (a)+g (b)+2g (c) 5 2g (a)+2g (b)+g (c) 5

New approach
Regard functions on subpieces of E . Use averaging rules to calculate the values on smaller pieces.
Hung, Mesing (Greifswald) Analysis on Finite Type Fractals Fractal Analysis 2005 4 / 23

Introduction

Fractals of ?nite type

Constructing recursive functions

Example

Motivation
Harmonic functions on Sierpinski gasket
Regard functions on points of E . Calculate the values at new points by averaging rules:
g (x) = g (y ) = g (z) =
g (a)+2g (b)+2g (c) 5 2g (a)+g (b)+2g (c) 5 2g (a)+2g (b)+g (c) 5...If you want to get a full essay, order it on our website: Orderessay



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