Linear and affine mappings

2018 ◽  
pp. 72-94
Author(s):  
Karlheinz Spindler
Keyword(s):  
Affine Mappings

Falconer's formula for the Hausdorff dimension of a self-affine set in R2

1995 ◽  
Vol 15 (1) ◽  
pp. 77-97 ◽  
Author(s):  
Irene Hueter ◽  
Steven P. Lalley
Keyword(s):  
Hausdorff Dimension ◽  
Dimensional Hausdorff Measure ◽  
Similarity Transformations ◽  
Affine Mappings ◽  
Finite Set ◽  
Self Similar ◽  
Image Position ◽  
Affine Set ◽  
Small Overlap ◽  
Box Dimensions

Let A1, A2,…,Ak be a finite set of contractive, affine, invertible self-mappings of R2. A compact subset Λ of R2 is said to be self-affine with affinitiesA1, A2,…,Ak ifIt is known [8] that for any such set of contractive affine mappings there is a unique (compact) SA set with these affinities. When the affine mappings A1, A2,…,Ak are similarity transformations, the set Λ is said to be self-similar. Self-similar sets are well understood, at least when the images Ai(Λ) have ‘small’ overlap: there is a simple and explicit formula for the Hausdorff and box dimensions [12, 10]; these are always equal; and the δ-dimensional Hausdorff measure of such a set (where δ is the Hausdorff dimension) is always positive and finite.

An L 2 convergence theorem for random affine mappings

1995 ◽  
Vol 32 (01) ◽  
pp. 183-192 ◽  
Author(s):  
Robert M. Burton ◽  
Uwe Rösler
Keyword(s):  
Hilbert Space ◽  
Lyapunov Exponent ◽  
Bilinear Form ◽  
Convergence Theorem ◽  
Convergence In Distribution ◽  
Wasserstein Metric ◽  
Affine Mappings ◽  
Affine Maps ◽  
Second Moments ◽  
Contraction Condition

We consider the composition of random i.i.d. affine maps of a Hilbert space to itself. We show convergence of thenth composition of these maps in the Wasserstein metric via a contraction argument. The contraction condition involves the operator norm of the expectation of a bilinear form. This is contrasted with the usual contraction condition of a negative Lyapunov exponent. Our condition is stronger and easier to check. In addition, our condition allows us to conclude convergence of second moments as well as convergence in distribution.

Affine mappings in the geometries of algebras

1972 ◽  
Vol 2 (2) ◽  
pp. 115-143
Author(s):  
Peter Ashley Lawrence
Keyword(s):  
Affine Mappings

On stability of classes of affine mappings

1995 ◽  
Vol 36 (5) ◽  
pp. 930-942 ◽  
Author(s):  
A. A. Egorov
Keyword(s):  
Affine Mappings

Affine mappings of translation surfaces: geometry and arithmetic

2000 ◽  
Vol 103 (2) ◽  
pp. 191-213 ◽  
Author(s):  
Chris Judge ◽  
Eugene Gutkin
Keyword(s):  
Translation Surfaces ◽  
Affine Mappings

scholarly journals On Fixed Point Property under Lipschitz and Uniform Embeddings

2018 ◽  
Vol 2018 ◽  
pp. 1-6 ◽  
Author(s):  
Jichao Zhang ◽  
Lingxin Bao ◽  
Lili Su
Keyword(s):  
Banach Space ◽  
Fixed Point ◽  
Convex Subset ◽  
Convex Sets ◽  
Fixed Point Property ◽  
Point Property ◽  
Lipschitz Mappings ◽  
Affine Mappings ◽  
Reflexive Space ◽  
Gateaux Differentiability

We first present a generalization of ω⁎-Gâteaux differentiability theorems of Lipschitz mappings from open sets to those closed convex sets admitting nonsupport points and then show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for isometries if it Lipschitz embeds into a super reflexive space. With the application of Baudier-Lancien-Schlumprecht’s theorem, we finally show that every nonempty bounded closed convex subset of a Banach space has the fixed point property for continuous affine mappings if it uniformly embeds into the Tsirelson space T⁎.

Renormings and fixed point property in non-commutative -spaces II: Affine mappings

2012 ◽  
Vol 75 (13) ◽  
pp. 5357-5361 ◽  
Author(s):  
Carlos A. Hernández-Linares ◽  
Maria A. Japón
Keyword(s):  
Fixed Point ◽  
Fixed Point Property ◽  
Point Property ◽  
Affine Mappings
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