Fractals, coherent states and self-similarity induced noncommutative geometry

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.

► Fractals are studied in the framework of q-deformed algebra of coherent states. ► The dissipative structure of fractals is related to noncommutative geometry. ► The formation of fractals is related to coherent boson condensation. ► Fractal geometrical properties of coherent states become manifest. ► The fractal global nature emerges from microscopic coherent deformation processes.