Mathematical Sciences: Combinatorial and Measure-Theoretic Structure of Dynamical Systems

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Abstract Blokh We study attractors in the sense of Milnor in complex dynamics and for piecewise-continuous interval maps. The latter requires the proof of the absence of wandering intervals for such maps as well as developing for them new analytical and dynamical methods. In the case of polynomial/rational maps we hope to combine ideas which led to the solution of the problem in question for smooth interval maps with tools from complex dynamics. We plan to work on the specification property and the space of limit sets with Hausdorff metric. Also we pose the problem of introducing new kinds of rotation numbers for orbit portraits of quadratic maps. A series of problems deals with the growth of pointwise itineraries, multipliers at periodic points and Lyapunov exponents in smooth families of interval maps. Another topic is studying rotation numbers for one-dimensional maps. We define rotation numbers as ergodic averages of a function. A specific choice of this function allows one to compute the union of all rotation numbers in interval case. This generalizes the Sharkovskii theorem and gives rise to a number of problems (connections to surface and graph/tree dynamics, rotation sets with respect to a periodic orbit, forcing relation, analogs of circle rotations and their connection with Fibonacci maps, growth and typical behavior of the rotation set in spaces of smooth interval maps in connection with the problem of hyperbolicity of a typical map, the monotonicity of the entropy in one- and multi-parameter families including unimodal families and cubic family). Dynamical systems theory describes processes that develop over time. In particular, dynamical systems arise in physics (Lorenz map), biology and environmental studies (population dynamics) and chemistry (e.g. modeling the Belousov-Zhabotinskii reaction). The development of the system depends on its initial state and parameters coming from the "environment." It is of great importance that we understand the future behavior which may be exhibited by the system for the "majority" of its initial states; this corresponds to the description of so-called attractors to which the first part of the project is devoted. Also in the first part of the proposal we study how properties of some important dynamical systems depend on "environmental" parameters. Sometimes phenomena exhibited by a system are related to one another and information about some of them allows one to make a judgment about the existence of others; in other words phenomena coexist. A good example here is periodicity, i.e. cyclic occurrence of the same states in the system. It turns out that the existence of a cyclic process with a given period for some initial state guarantees that another cyclic process with a different period can be realized for a different initial state and the same environment. In the second part of the project we plan to thoroughly study this relationship between different periods of cyclic processes in the same system.