Michael Ka Shing Ng, Pure Mathematics, University of Waterloo
"Some aspects of Cantor sets"
For
every
positive,
decreasing,
summable
sequence
$a
=
(a_i)$,
we
can
construct
a
Cantor
set
$C_a$
associated
with
$a$.
These
Cantor
sets
are
not
necessarily
self-similar.
Their
dimensional
properties
and
measures
have
been
studied
in
terms
of
the
sequence
$a$.
In
this
thesis,
we
extend
these
results
to
a
more
general
collection
of
Cantor
sets.
We
study
their
Hausdorff
and
packing
measures,
and
compare
the
size
of
Cantor
sets
with
the
more
refined
notion
of
dimension
partitions.
The
properties
of
these
Cantor
sets
in
relation
to
the
collection
of
cut-out
sets
are
then
considered.
The
multifractal
spectrum
of
$\mathbf{p}$-Cantor
measures
on
these
Cantor
sets
are
also
computed.
We
then
focus
on
the
special
case
of
homogeneous
Cantor
sets
and
obtain
a
more
accurate
estimate
of
their
exact
measures.
At
the
end,
we
prove
the
$L^p$-improving
property
of
the
$\mathbf{p}$-Cantor
measure
on
a
homogeneous
Cantor
set
as
a
convolution
operator.