Dyadic Cantor set and its kinetic and stochastic counterpart

•A dyadic Cantor set and its two variants are studied both analytically and numerically.•Probability, time and randomness are added in a logical progression to understand their role in the emergent behavior.•Exact expression for fractal dimension and its connection to conservation law is pointed out.•The kinetic and the stochastic dyadic cantor sets exhibit dynamic scaling which shows self-similarity in time.•The idea of data-collapse for the interval size distribution function is used to verify the dynamic scaling.

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)(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.