Abstract
The
4
N
{4N}
-carpets are a class of infinitely ramified self-similar fractals with a large group of symmetries. For a
4
N
{4N}
-carpet F, let
{
F
n
}
n
≥
0
{\{F_{n}\}_{n\geq 0}}
be the natural decreasing sequence of compact pre-fractal approximations with
⋂
n
F
n
=
F
{\bigcap_{n}F_{n}=F}
. On each
F
n
{F_{n}}
, let
ℰ
(
u
,
v
)
=
∫
F
N
∇
u
⋅
∇
v
d
x
{\mathcal{E}(u,v)=\int_{F_{N}}\nabla u\cdot\nabla v\,dx}
be the classical Dirichlet form and
u
n
{u_{n}}
be the unique harmonic function on
F
n
{F_{n}}
satisfying a mixed boundary value problem corresponding to assigning a constant potential between two specific subsets of the boundary. Using a method introduced by [M. T. Barlow and R. F. Bass,
On the resistance of the Sierpiński carpet,
Proc. Roy. Soc. Lond. Ser. A
431 (1990), no. 1882, 345–360], we prove a resistance estimate of the following form: there is
ρ
=
ρ
(
N
)
>
1
{\rho=\rho(N)>1}
such that
ℰ
(
u
n
,
u
n
)
ρ
n
{\mathcal{E}(u_{n},u_{n})\rho^{n}}
is bounded above and below by constants independent of n. Such estimates have implications for the existence and scaling properties of Brownian motion on F.
Resistance scaling on 4N-carpets
