Exact Hausdorff and packing measures of linear Cantor sets with overlaps
Abstract
Let
K
be the attractor of a linear iterated function system (IFS)
S
j
(x)=
ρ
j
x+
b
j
,
j=1,⋯,m
, on the real line satisfying the generalized finite type condition (whose invariant open set
O
is an interval) with an irreducible weighted incidence matrix. This condition was introduced by Lau \& Ngai recently as a natural generalization of the open set condition, allowing us to include many important overlapping cases. They showed that the Hausdorff and packing dimensions of
K
coincide and can be calculated in terms of the spectral radius of the weighted incidence matrix. Let
α
be the dimension of
K
. In this paper, we state that
H
α
(K∩J)≤|J
|
α
for all intervals
J⊂
O
¯
¯
¯
¯
, and
P
α
(K∩J)≥|J
|
α
for all intervals
J⊂
O
¯
¯
¯
¯
centered in
K
, where
H
α
denotes the
α
-dimensional Hausdorff measure and
P
α
denotes the
α
-dimensional packing measure. This result extends a recent work of Olsen where the open set condition is required. We use these inequalities to obtain some precise density theorems for the Hausdorff and packing measures of
K
. Moreover, using these densities theorems, we describe a scheme for computing
H
α
(K)
exactly as the minimum of a finite set of elementary functions of the parameters of the IFS. We also obtain an exact algorithm for computing
P
α
(K)
as the maximum of another finite set of elementary functions of the parameters of the IFS. These results extend previous ones by Ayer \& Strichartz and Feng, respectively, and apply to some new classes allowing us to include linear Cantor sets with overlaps.