  For any conformal iterated function system (CIFS) consisting of finitely or
countably many maps, and any closed shift-invariant set of right-infinite
sequences of such maps, one can associate a limit set, which we call a
shift-generated conformal iterated construction. We define the extended
Hausdorff dimension spectrum of a CIFS to be the set of Hausdorff dimensions of
all such limit sets. We prove that for any CIFS with finitely or countably many
maps, the extended Hausdorff dimension spectrum is maximal, i.e. all
nonnegative dimensions less than or equal to the dimension of the limit set of
the CIFS are realized. We also prove a version of this result even for
so-called conformal graph directed Markov systems, obtained via
nearest-neighbor restrictions on the CIFS. %when there are nearest-neighbor
restrictions on the CIFS (similar to those in the so-called conformal graph
directed Markov systems).
  The main step of the proof is to show that for the family $(X_\beta)$ of
so-called $\beta$-shifts, the Hausdorff dimension of the limit set associated
to $X_\beta$ varies continuously as a function of $\beta$.
