  We set up a multiresolution analysis on fractal sets derived from limit sets
of Markov Interval Maps. For this we consider the $\mathbb{Z}$-convolution of a
non-atomic measure supported on the limit set of such systems and give a
thorough investigation of the space of square integrable functions with respect
to this measure. We define an abstract multiresolution analysis, prove the
existence of mother wavelets, and then apply these abstract results to Markov
Interval Maps. Even though, in our setting the corresponding scaling operators
are in general not unitary we are able to give a complete description of the
multiresolution analysis in terms of multiwavelets.
