  Michael Barnsley introduced a family of fractals sets which are repellers of
piecewise affine systems. The study of these fractals was motivated by certain
problems that arose in fractal image compression but the results we obtained
can be applied for the computation of the Hausdorff dimension of the graph of
some functions, like generalized Takagi functions and fractal interpolation
functions.
  In this paper we introduce this class of fractals and present the tools in
the one-dimensional dynamics and nonconformal fractal theory that are needed to
investigate them. This is the first part in a series of two papers. In the
continuation there will be more proofs and we apply the tools introduced here
to study some fractal function graphs.
