The Principal Investigator will perform research in the theory of low-dimensional dynamical systems, both smooth and continuous. He will concentrate mainly on smooth interval maps. Old problems of density of Axiom A maps among r times differentiable ones (for both unimodal and polymodal cases) and monotone dependence on a parameter for families of unimodal maps are still open. Problems encountered in the theory of real maps in dimension one have often their counterparts in the theory of complex maps in dimension one and real maps in dimension two. These problems will be also studied. The last part of the project is to study general rotation theories, which deal with properties of the set of all limits of ergodic averages of a given observable (a function from the phase space to some linear space), and combinatorial dynamics for tree maps. For almost every phenomenon from Physics, Chemistry, Biology, Medicine, Economy and other sciences, one can make a mathematical model that can be regarded as a dynamical system. Sometimes the study of this system is extremely difficult, but quite often it can be reduced to the study of a much simpler, low dimensional one. This project is devoted to the investigation of the properties of this type of systems from the mathematical point of view. One of its main goals is to show that typical one-dimensional systems are not chaotic.
