K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, New York (1990). Home/Search Document Not in Database Summary Related Articles Check |
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The Hausdorff Dimension Of Hilbert's Coordinate Functions - McClure (1998) (Correct)
....xy : x( 3 t) 4) 1 y(t) 2 d yx : y( 3 t) 4) 1=2 x(t) 22 2 Hausdorff Measure and Dimension In this section, we recall the definitions of Hausdorff measure and dimension, show that X and Y have dimension 3=2, and state some useful lemmas. Our notation has been influenced by [Edg2] and [Fal2], where one mayfind proofs of the basic facts. Let s 0. We will define the s dimensional Hausdorff measure, on Euclidean space, R n .LetF . The diameter of F will be denoted by diam(F ) Let 0. An cover, C,ofF is a countable collection of sets such that F ae[ U2C U and diam(U ....
....Lower bounds for Hausdorff measure are, typically, more difficult. Our strategy will be to show that (x ;1 (z) 0 for all z [0# 1] The lower bound for X will then follow from a result of Besicovitch. We will obtain the lower bound for level sets by using following measure comparison lemma ([Fal2], page 55) Lemma 2.1 Let beaBorel measure on the Borel set F and suppose that for some s 0,thereare numbers c# ffi 0 such that (U ) for all open sets U with diam(U ) ffi . Then, F ) c. We will, also, need the following scaling property of Hausdorff measure ( Fal2] page 27) ....
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K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley and Sons, West Sussex, England, 1990.
Digraph Self-Similar Sets and Aperiodic Tilings - McClure (2001) (Correct)
....at tilings of the plane. Furthermore, a generalization of self similarity, called digraph self similarity,provides a way to construct aperiodic tilings. 2 Self similarity and tiling A set which is composed of several scaled images of itself maybethoughtof as self similar as described in [5] and [6]. We will write a more mathematical definition shortly. A square is a simple example of a self similar set in the plane, being composed of four copies of itself, scaled by the factor 2 .Eachof these four copies are in turn composed of smaller copies, etc. By iterating the decomposition and ....
K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley and Sons, West Sussex, UK, 1990.
Entropy Dimensions Of The Hyperspace Of Compact Sets - McClure (Correct)
....is somewhat more concrete. Corollary 1.1 If E is totally bounded and Delta(E) i (finite dimension s 0 ) then Proof: If s s 0 ,then Delta(E) So Delta(K(E) 2 ,by lemma 1.1. If s 0 s 1 s,then Delta(E) OE ,by lemma 1.3.2 Equation 3.29 and theorem 9. 3 of [Fal] show that this corollary applies to a self similar set E ae R n of finite upper entropy index s 0 . Analogous statements for the Hausdorff dimension of the hyperspace of such sets will be studied in a subsequent paper ( Mcc] It is natural to ask if Delta(E) i implies Delta(K(E) i 2 . ....
....by considering finite dimensional sets. Corollary 2.1 If E is totally bounded and satisfies Delta(E) i i Delta(E) Proof: If s 0 s 1 s, then Delta(E) OE , by lemma 2.2. If s s 0 ,then Delta(E) by theorem 2.1.2 Equation 3.29 and theorem 9. 3 of [Fal] show that this corollary applies to a self similar set E ae R n of finite upper entropy dimension s 0 . ....
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K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, New York (1990)
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K. J. Falconer, `Fractal Geometry: Mathematical Foundations and Applications ' John Wiley and Sons, New York (1990)
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K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications. John Wiley & and Sons, 1990.
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