  Firstly, we propose and investigate a dyadic Cantor set (DCS) and its kinetic
counterpart where a generator divides an interval into two equal parts and
removes one with probability $(1-p)$. The generator is then applied at each
step to all the existing intervals in the case of DCS and to only one interval,
picked with probability according to interval size, in the case of kinetic DCS.
Secondly, we propose a stochastic DCS in which, unlike the kinetic DCS, the
generator divides an interval randomly instead of equally into two parts.
Finally, the models are solved analytically; an exact expression for fractal
dimension in each case is presented and the relationship between fractal
dimension and the corresponding conserved quantity is pointed out. Besides, we
show that the interval size distribution function in both variants of DCS
exhibits dynamic scaling and we verify it numerically using the idea of
data-collapse.
