Sierpinski carpet
From Freepedia
The Sierpinski carpet is a plane fractal first described by Wacław Sierpiński. The carpet is one generalization of the Cantor set to two dimensions (the other is Cantor dust). Higher-dimensional generalizations such as the 3-dimensional Menger sponge are also possible.
Construction
The construction of the Sierpinski carpet begins with a square. The square is cut into 9 congruent subsquares in a 3-by-3 grid, and the central subsquare is removed. The same procedure is then applied recursively to the remaining 8 subsquares, ad infinitum. The illustration below shows the first few iterations in the construction process.
| Order 0 | Order 1 | Order 2 | Order 3 | Order 4 |
The Hausdorff dimension of the carpet is log 8/log 3 ≈ 1.8928.
Brownian motion on the Sierpinski carpet
The topic of Brownian motion on the Sierpinski carpet has attracted scientific interest in recent years. Martin Barlow and Richard Bass have shown that a random walk on the Sierpinski carpet diffuses at a slower rate than an unrestricted random walk in the plane. The latter reaches a mean distance proportional to n1/2 after n steps, but the random walk on the discrete Sierpinski carpet reaches only a mean distance proportional to n1/β for some β > 2. They also showed that this random walk satisfies stronger large deviation inequalities (so called "sub-gaussian inequalities") and that it satisfies the elliptic Harnack inequality without satisfying the parabolic one. The existence of such an example was an open problem for many years.
