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

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.

Self-translation surfaces

1993 ◽  
Vol 5 (5) ◽  
Author(s):  
María del Carmen Fuster ◽  
Clin McGrory ◽  
Juan Ballesteros
Keyword(s):  
Translation Surfaces

Weingarten Affine Translation Surfaces in Euclidean 3-Space

2017 ◽  
Vol 72 (4) ◽  
pp. 1839-1848 ◽  
Author(s):  
Seoung Dal Jung ◽  
Huili Liu ◽  
Yixuan Liu
Keyword(s):  
Translation Surfaces ◽  
Affine Translation

scholarly journals The Kontsevich–Zorich cocycle over Veech–McMullen family of symmetric translation surfaces

2019 ◽  
Vol 14 (1) ◽  
pp. 21-54
Author(s):  
Artur Avila ◽  
◽  
Carlos Matheus ◽  
Jean-Christophe Yoccoz ◽  
◽  
...  
Keyword(s):  
Translation Surfaces

L-cut splitting of translation surfaces and non-embedding of pseudo-Anosovs (in genus two)

2018 ◽  
Vol 153 (1) ◽  
pp. 51-79
Author(s):  
Andrew Bouwman ◽  
Jarosław Kwapisz
Keyword(s):  
Translation Surfaces ◽  
Genus Two

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.

scholarly journals An origami of genus 3 with arithmetic Kontsevich–Zorich monodromy

2019 ◽  
Vol 169 (1) ◽  
pp. 19-30
Author(s):  
PASCAL HUBERT ◽  
CARLOS MATHEUS SANTOS
Keyword(s):  
Moduli Space ◽  
Translation Surfaces

AbstractIn this we exploit the arithmeticity criterion of Oh and Benoist–Miquel to exhibit an origami in the principal stratum of the moduli space of translation surfaces of genus three whose Kontsevich–Zorich monodromy is not thin in the sense of Sarnak.

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