Schmidt’s Game And The Combinatorial Properties Of Some Dynamical Systems

In this dissertation, we studied the winning properties for the sets of points with non-dense orbits under a class of expanding maps in the sense of Schmidt’s game and the combinatorial properties for two classes of systems. The combinatorial properties contain forbidden patterns and the perturbation of invariant sets. The thesis contains three parts.In the first part, we study the forbidden patterns for a class of piecewise monotone decreasing and onto maps. The forbidden patterns of piecewise onto and increasing expanding maps on an interval, such as f(x)=Nx(mod1) on [0,1), which is corresponding to one-sided shift system with the usual lexicographic order, were first considered in[4]. And in[16], a complete characterization of forbidden patterns of shift systems and an enumeration of them were obtained. By now, the only non-trivial family of maps whose forbidden patterns can be revealed are shift maps and/3-shifts, see[16,17]. In[3], some necessary conditions of the forbidden patterns for signed shift systems are obtained. We point out that the necessary conditions for a pattern to be forbidden are not sufficient. And we shall give a complete characterization for the forbidden patterns of piecewise decreasing maps such as f(x)=Nx(mod1).In the second part, we shall study the winning properties of the set of points with non-dense orbits under a class of expanding maps. Let BAD(ξ) be the set of points whose forward orbits are bounded away from ξ. Schmidt[601proved that when T is the Gauss transformation, BAD(0) is1/2-winning. Farm[22] proved that for a class of expanding maps including the Gauss transformation, for any ξ∈[0,1] with a well-defined expansion, the set BAD(ξ) is1/4-winning. And Farm proved that if β>1is a simple beta number and T is the β-transformation, then for any ξ∈[0,1], the set BAD(ξ) is1/4-winning. Farm, Schmeling and Persson[23] proved that for any β>1if T is the β-transformation, then for any ξ∈[0,1] the set BAD(ξ) is strong α-winning for some α<1/64. Mance and Tseng[49] proved that the set of points with bounded expansions, or equivalently BAD(0), is1/8-winning and strong winning if T is the Liiroth transformation. In this thesis, we shall prove that for a class of piecewise C1+s expanding maps, for any ξ∈[0.1], the set BAD(ξ) is1/2-winning in the sense of Schmidt’s game. This class of maps includes the Gauss transformation, the Liiroth transformation and all beta transformations.In the third part, we study the perturbation of invariant sets for a class of piecewise linear sys- tems. For a piecewise monotone increasing map T on [0,1], we consider the set of points x such that Tn(x) is always in some interval [a, b], i.e., Ta,b:=∩T-n[a, b]. Let0(a, b)=dim H Ta,b. We consider the combinatorial construction of the set Ta,b. As an application, for the function Φ(a, b) with two variables, we get the condition under which Φ(a,b) is invariant with the perturbation of a, b.