Visibility Of A Class Of Fractal Sets And Problems On Random ? Dynamical Systems

In this thesis,we investigate the visibility of a class of fractal sets and some dynamical properties of random ?-transformations.The main results are as follows:Using some techniques in real analysis,we establish a connection between the visible problem and arithmetic on the fractal sets.We give a complete description of the visible set for Cartesian products of generalized Cantor sets,containing its Hausdorff dimension and topological properties.Using a suitable partition of ‘fat' Sierpinski gaskets without holes,we define the greedy,lazy and random ?-transformations and study their properties.Using the multidimensional notion of variation,we show that absolutely continuous invariant measures exist for the three transformations.We also prove the uniqueness and existence of the invariant measure of maximal entropy for the random ?-transformation.Using a linear map,we obtain the range of the parameter ? for ‘fat' Sierpinski gaskets with radial holes,i.e.,the holes are all centered on three radial lines originating from the center of the attractor and extending to the three vertices.We modify the definition of the random ?-transformation and prove the uniqueness and existence of the invariant measure of maximal entropy.