Buffon's needle probability of rational product Cantor sets - The Abel

82 Pages·2012·0.77 MB·English

by  Izabella Łaba

Buffon’s needle probability of rational product
Cantor sets
Izabella L(cid:32) aba
The Abel Symposium, Oslo, August 2012
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
Assume that E has Hausdorff dimension 1.
∞
We are interested in the average (wrt angle) length of linear
projections of E .
n
The problem is of interest in ergodic theory as well as theory of
analytic functions (analytic capacity).
The Favard length problem
Let E = (cid:84)∞ E be a self-similar Cantor set in the plane.
∞ n=1 n
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
We are interested in the average (wrt angle) length of linear
projections of E .
n
The problem is of interest in ergodic theory as well as theory of
analytic functions (analytic capacity).
The Favard length problem
Let E = (cid:84)∞ E be a self-similar Cantor set in the plane.
∞ n=1 n
Assume that E has Hausdorff dimension 1.
∞
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
The problem is of interest in ergodic theory as well as theory of
analytic functions (analytic capacity).
The Favard length problem
Let E = (cid:84)∞ E be a self-similar Cantor set in the plane.
∞ n=1 n
Assume that E has Hausdorff dimension 1.
∞
We are interested in the average (wrt angle) length of linear
projections of E .
n
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
The Favard length problem
Let E = (cid:84)∞ E be a self-similar Cantor set in the plane.
∞ n=1 n
Assume that E has Hausdorff dimension 1.
∞
We are interested in the average (wrt angle) length of linear
projections of E .
n
The problem is of interest in ergodic theory as well as theory of
analytic functions (analytic capacity).
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
The 4-corner set, 1st iteration
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
The 4-corner set, 2nd iteration
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
The 1-dimensional Sierpin´ski triangle, 1st iteration
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
(cid:73) Divide it into L2 congruent squares of sidelength 1.
(cid:73) Choose sets A,B ⊂ {0,1,...,L−1} so that |A|,|B| ≥ 2 and
|A||B| = L.
(cid:73) Keep those squares whose bottom left vertices have
coordinates in A×B.
(cid:73) Iterate the construction.
Product Cantor sets
A generalization of the 4-corner set construction:
(cid:73) Start with a L×L square, where L ≥ 4 is a positive integer.
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets
(cid:73) Choose sets A,B ⊂ {0,1,...,L−1} so that |A|,|B| ≥ 2 and
|A||B| = L.
(cid:73) Keep those squares whose bottom left vertices have
coordinates in A×B.
(cid:73) Iterate the construction.
Product Cantor sets
A generalization of the 4-corner set construction:
(cid:73) Start with a L×L square, where L ≥ 4 is a positive integer.
(cid:73) Divide it into L2 congruent squares of sidelength 1.
IzabellaL(cid:32)aba Buffon’sneedleprobabilityofrationalproductCantorsets