ABSTRACT
The chapter presents various forms of fractals in classical mechanics. In particular, the smallest scales in phase space are smeared out and fractals survive as a transient phenomenon only. The smallest scales of fractals are eliminated upon quantization. The several fractals contribute to eigenfunctions, thus even further blurring the self-similarity. Classical fractals persist in quantum mechanics only as transient phenomena. The completely soluble model of linear maps on a torus shows another effect of quantization, visible in eigenstates: Not only are classical fractals spread out, they are also mixed to form quantal eigenstates. The quantum mechanics is not as singular as classical mechanics. The Heisenberg uncertainty relations introduce an inner, smallest scale in quantum mechanics, below which all densities are smooth. Similar reasoning applies to the fat fractal stochastic layer. And finally, an inner scale in irregular scattering is introduced by the spreading of the wave packet and/or tunneling.
