Grant: Discovery Grants Program - Individual

The theory of one and higher dimensional transformations and random maps is a beautiful, important and challenging subject for research. Methods ranging from theory of numbers and elementary geometry to measure theory and functional analysis are used. The idea of using successive maxima of one variable in a higher dimensional dynamical system was introduced by Lorenz in his numerical studies of a 3-dimensional system of differential equations that model atmospheric effects. One of the arguments leading to the belief in the existence of a famous chaotic Lorenz attractor is the fact that the sequence of consecutive maxima of the z variable represented as a map z_n -> z_{n+1} form a chaotic map (similar to tent map). Can this purely speculative reasoning be turned into a mathematical proof? To study this problem we consider a map T and a sequence (x_n) of consecutive maxima of one of the variables and ask what chaotic properties of (x_n) imply analogous chaotic properties of T. In more general form we ask whether a chaotic behavior on the lower dimensional subset of a space implies chaos of the whole higher dimensional system. We also study other problems related to the invariant measures of dynamical systems: existence of such measures for a random map modeling market dynamics, various methods of approximation of invariant densities and modeling of some quantum mechanics phenomena: double slit experiment and the holographic principle for the information horizon of black holes.