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

摘要Blokh研究了复动力学和分段连续区间映射的Milnor意义下的吸引子。后者需要证明这类映射不存在游荡区间，并为此发展新的分析和动力学方法。在多项式/有理映射的情况下，我们希望将导致光滑区间映射问题的解决方案的想法与复杂动力学的工具相结合。我们计划研究Hausdorff度量的规范性质和极限集空间。我们还提出了为二次映射的轨道画像引入新的旋转数的问题。一系列问题涉及区间映射族中逐点路线的增长性、周期点处的乘子和Lyapunov指数。另一个主题是研究一维映射的旋转数。我们将旋转数定义为函数的遍历平均值。该函数的特定选择允许计算区间情况下所有旋转数的并集。这推广了Sharkovskii定理，并产生了许多问题(与曲面和图/树动力学的联系，关于周期轨道的旋转集，强迫关系，圆旋转的类比及其与Fibonacci映射的联系，光滑区间映射空间中旋转集的增长和典型行为，与典型映射的双曲性问题有关，单参数和多参数族中的熵的单调性，包括单峰族和三次族)。动力系统理论描述了随着时间的推移而发展的过程。特别是，动力系统出现在物理学(洛伦茨图)、生物学和环境研究(种群动力学)和化学(例如，对Belousov-Zhabotinskii反应建模)中。系统的发展取决于它的初始状态和来自“环境”的参数。非常重要的是，我们要了解系统在其初始状态的“大多数”时可能表现出的未来行为；这与本项目第一部分致力于的所谓吸引子的描述相对应。另外，在建议的第一部分，我们研究了一些重要的动力系统的性质如何依赖于“环境”参数。有时，一个系统表现出的现象是相互关联的，关于其中一些现象的信息允许人们对其他现象的存在做出判断；换句话说，现象是共存的。这里的一个很好的例子是周期性，即系统中相同状态的循环发生。结果表明，对于某一初始状态，给定周期的循环过程的存在保证了对于不同的初始状态和相同的环境，可以实现另一个不同周期的循环过程。在项目的第二部分，我们计划深入研究同一系统中不同周期过程之间的这种关系。

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.