The Symbolic Representation Of A Class Of Dynamical Systems

For the representation of smooth dynamical systems, there have been systematic studies and fruitful results, but for the symbolic representation of the dynamical systems induced by non-smooth or discontinuous maps, there are only a few results. In this thesis, we will discuss the symbolic representation of a class of two-dimensional and three-dimensional torus maps. In the first chapter, there are the backgrounds and the main results.In the second chapter, based on the studies of iterated maps, we discuss the topological conjugate representations and quasi-representations of the iterated maps, and obtain some useful results.In the third chapter, we introduce a dynamical system on the 2-torus and discuss in detail the periodic points of the system. By using the arrangements and the computation of symbolic sequences, we get that the system has one fixed point, some 3-periodic and 4-periodic points, and some special n-periodic points. But there is no any 2-periodic point for the system. We point out the reason for this is that the partition for the system has only two parts.In the forth chapter, we discuss a class of piecewise maps on the 3-torus. We construct the partition and get the corresponding symbolic sequences. By analyzing the symbolic sequences, we obtain the regulations of the maps on the 3-torus. Due to the difficulty for the study on the 3-torus, we only present some examples and obtain the admissibility conditions for the symbolic sequences.In the fifth chapter,we discuss the relation between the (∑(N),σ) and (∑(I₀),σ), and point out the limitation of using the (∑(N),σ) to study the symbolic representation of the dynamical systems.In the sixth chapter,we give a summary for the thesis and point out the foreground of the further study of the symbolic representation and symbolic analysis.