Lagrange spectrum and dynamics on the invariant sets of linear-fractional IFS-s

A geometrical approach to investigation of Lagrange spectrum over finite fields, i.e. for the real numbers whose continues fraction expansions contain coefficients from a fixed finite set of natural numbers, is proposed. The real number under investigation defines dynamics on the invariant sets of iterated functional system (IFS) of linear-fractional transformations with the coefficients from this fixed set of natural numbers. The value of the point of Lagrange spectrum corresponding to the real number then can be expressed by means of some geometrical characteristic of this dynamics. Proposed approach is illustrated by demonstrations of known results on structure of Lagrange spectrum for the numbers whose continued fractions coefficients are 1 or 2 (i.e. left part of the spectrum).