Title : Dynamics for/with fractals and applications
The comprehension of the world through fractals promises to be one of the effective ways for creative scientific, biological, social, medical, and industrial activities. We are confident that self-similarity extended to arbitrary small scales of real world subjects is in the core of effectiveness of researches of the universe and helps to optimize many processes as well as structures of objects. Thus, it is of significant interest to consider fractals in dynamics. Surprisingly, even the task of deformation of the sets, which keeps the fractal structure invariant, have not been investigated in the literature despite the fact that the problem is of interest for geometry, measure theory, and topology. The first steps in that direction are performed in this book. This makes it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. These achievements allow us to analyze several practically useful problems.
To provide strong arguments for the genericity of chaos in the real and abstract universe, we suggest the concept of abstract similarity. The self-similar space is equipped with chaos if a special property is assumed. This provides the way to prove chaos for fractals. We believe that the concept and its developments can become a universal instrument for chaos and fractal investigation. It may unite different definitions of chaos as well as ways of chaos detection. This is true also for new methods to determine and analyze fractal structures.