  We introduce a broad class of fractal jet observables that recursively probe
the collective properties of hadrons produced in jet fragmentation. To describe
these collinear-unsafe observables, we generalize the formalism of
fragmentation functions, which are important objects in QCD for calculating
cross sections involving identified final-state hadrons. Fragmentation
functions are fundamentally nonperturbative, but have a calculable
renormalization group evolution. Unlike ordinary fragmentation functions,
generalized fragmentation functions exhibit nonlinear evolution, since fractal
observables involve correlated subsets of hadrons within a jet. Some special
cases of generalized fragmentation functions are reviewed, including jet charge
and track functions. We then consider fractal jet observables that are based on
hierarchical clustering trees, where the nonlinear evolution equations also
exhibit tree-like structure at leading order. We develop a numeric code for
performing this evolution and study its phenomenological implications. As an
application, we present examples of fractal jet observables that are useful in
discriminating quark jets from gluon jets.
