**MSC:**- 11J70 Continued fractions and generalizations, See also {11A55, 11K50}
- 11J83 Metric theory

conformal interated function systems. We show that for each of

these

systems there exists a variety of finite hyperbolic subsystems

such that the spectrum made of the Hausdorff dimensions of the limit sets

of these subsystems is dense in the interval between $0$ and the

Hausdorff dimension of the given conformal iterated function system.

This result has interesting applications in conformal dynamics and

elementary fractal number theory.