This project is based on results and concepts from probability theory and ergodic theory.
Ergodic theory studies the average properties of general time evolutions on measure spaces.
Induced automorphisms have been studied on subsets of phase space which have measure zero. The induced transformations are found to be stable convolution semigroups rather than translation groups. The possible limits are classified by the index of the stable measure. The infinitesimal generators of these fractional semigroups are fractional derivatives whose order is just the stable index of the convolution semigroup. Further implications and ramifications of these results are explored within the project.