Laminations and Low-Dimensional Dynamical Systems

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ABSTRACTThe proposal aims at developing new tools in low-dimensional dynamics. Studying laminations/polynomials requires a combination of techniques which already allowed us to show the existence of wandering gaps of laminations and should lead to the description of dynamics of an individual polynomial, and to the description of the combinatorial analog of the boundary of the connectedness locus (i.e., the Mandelbrot set for high degree polynomials). For the cycles of polynomials we plan to developthe rotation theory analogous to that for interval maps, and a version of the Sharkovskii theorem. We continue developing theory of c-tent plane maps which combine expansive properties of tent maps with properties of complex quadratic maps and should play the same role for quadratic complex maps as tent maps play on the interval. We also deal with Milnor attractors for conformal measures of complex maps, and work on some problems in interval dynamics.Some natural processes/maps are described by low-dimensional dynamical systems.The project aims at solving a few problems helping describe these maps,as well as their families. Among non-invertible maps in dimensions higherthan 1 the best in terms of their properties are complex polynomials, andwe plan to study the families of their combinatorial analogs for dimensions higher than 2, as well as individual polynomials in terms of the coexistenceof their cycles (periodic processes). We also study so-called c-tent map,Milnor attractors, and some problems for map of the interval.