Laminations and Low-Dimensional Dynamical Systems

摘要本提案旨在发展低维动力学的新工具。研究层积/多项式需要技术的结合，这些技术已经允许我们展示层积的徘徊间隙的存在，并且应该导致对单个多项式的动力学描述，以及对连通性轨迹边界的组合模拟的描述（即，高次多项式的Mandelbrot集）。对于多项式的循环，我们计划发展类似于区间映射的旋转理论，以及Sharkovskii定理的一个版本。我们继续发展c-tent平面映射理论，它将帐篷映射的扩展性质与复二次映射的性质结合起来，在二次复映射中发挥与帐篷映射在区间上同样的作用。我们还讨论了复映射的保形测度的Milnor吸引子，并研究了区间动力学中的一些问题。一些自然过程/映射用低维动力系统来描述。该项目旨在解决一些问题，帮助描述这些地图，以及它们的家庭。在维数大于1的不可逆映射中，就其性质而言，最好的是复多项式，我们计划研究维数大于2的组合类似物的族，以及就其循环（周期过程）的共存而言的单个多项式。我们还研究了c-tent映射、Milnor吸引子以及区间映射的一些问题。

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.