The self-similarity properties of fractals are studied in the framework of
the theory of entire analytical functions and the q-deformed algebra of
coherent states. Self-similar structures are related to dissipation and to
noncommutative geometry in the plane. The examples of the Koch curve and
logarithmic spiral are considered in detail. It is suggested that the dynamical
formation of fractals originates from the coherent boson condensation induced
by the generators of the squeezed coherent states, whose (fractal) geometrical
properties thus become manifest. The macroscopic nature of fractals appears to
emerge from microscopic coherent local deformation processes.Comment: 2 figure
