Stochastic fractal and Noether's theorem

We consider the binary fragmentation problem in which, at any breakup event, one of daughter segments either survives with probability $p$ or disappears $1\!-\!p$. It describes a stochastic dyadic Cantor set that evolves time, and eventually becomes fractal. investigate this phenomenon, through analytical methods Monte Carlo simulation, for generic class models, where segment points follow symmetric beta distribution shape parameter $\alpha$, which also determines rate. For fractal dimension $d_f$, we find $d_f$-th moment $M_{d_f}$ is conserved quantity, independent $\alpha$. use idea data collapse -- consequence dynamical scaling symmetry to demonstrate system exhibits self-similarity. In an attempt connect reinterpret equation as continuity Euclidean quantum-mechanical system. Surprisingly, Noether charge corresponding trivial, while relates purely mathematical symmetry: phase rotation time.