Some Properties Of Topological Dynamical Systems In The Mean Sense And Nilsystem

The thesis consists of two topics on topological dynamics and ergodic theory. One is about the generalisation of some notions in topological dynamics in the mean sense, and the other is about nilsystem and its relationship with number theory.The thesis is organised as follows.In chapter1, we briefly recall the origin, developments and main objectives in topological dynamics and ergodic theory, and then we shall introduce the research backgrounds and main results of the thesis.In Chapter2, we shall briefly introduce some notions and results in topological dynamics and ergodic theory. Some tools which are useful for the following chapters will also be discussed.In Chapter3, we shall consider some the generalisation of some notions in topo-logical dynamics in the mean sense, including Banach proximal, mean equicontinu-ous, mean Banach equicontinuous and so on. First we get several equivalent conditions of a topological dynamical system being strongly proximal. By some intensive study of the structure of the equicontinuous system, we solved an open problem of Scarpellini: for a mean equicontinuous system, any ergodic measure on it has the discrete spectrum, we also introduce the notions of mean Banach equicontinuous, almost mean equicon-tinuous and almost Banach equicontinuous, and we show that while the entropy of an almost mean equicontinuous system need not to be zero, the entropy of an almost mean Banach equicontinuous system must be zero. Besides, we also prove the follow-ing theorem:a transitive is almost mean eqicontinuous or mean sensitive and a minimal system is mean equicontinuous or mean sensitive. For mean Banach equicontinuous system and mean sensitive system, we have the similar results.In Chapter4, the notion of dynamical parallelepiped in a minimal system will be discussed. For a topological dynamical system (X, T) and d∈N, Host-Kra-Maass gave the definition of the associated dynamical parallelepiped Q[d]. For a minimal distal system it was shown by them that the relation～d-1defined on Q[d-1] by x～d-1x’ if and only if (x,x’)∈Q[d] is an equivalence relation; the closing parallelepiped property holds, and for each x∈X the collection of points in Q[d] with first coordinate x is a minimal subset under the face transformations. By some detailed study of the structure of Q[d], we will give examples showing that the results can not extend to general minimal systems. This indicates that to to deal with the dynamical parallelepipeds in for a general minimal system some other method such as Ellis semi-group theory should be involved.In Chapter5, we shall take a look at the application of the nilsystem in number theory, we shall consider the relationship between the recurrence time of a dynamical system with the Nil Bohr-set. This is closely related to a classical problem in combi-natorial number theory about Bohr set:Let S be a syndetic set of Z, whether the set S-S is a Bohro set. Veech showed this problem has a positive answer in the sense of a difference of a set of zero density, and a recent work of Huang-Shao-Ye gave a higher order form of Veech’s result. We shall extend the research of Huang-Shao-Ye to the the polynomial form in this chapter.We also study the connection between the recurrence time set of a dynamical system and Bohr0set. We get the following two results:(1) We give a polynomial form of Huang-Shao-Ye’s result.(2) For the recurrence time set of the classical skew product systems, the Nild-Bohro sets generated by several polynomials, and the Nil2-Bohro sets, the difference set of any type of the above set is a Bohr0set.We shall use some deep tools such as generalised polynomials and the equidistri-bution of polynomial sequences in a nilmanifold.