Fractal And Topological Properties Of Sets In Beta Dynamical System

Let ?>1.This dissertation is devoted to investigating the size of some fractal sets in the ?-dynamical system in the sense of Lesbegue measure,Hausdorff dimension and topology.It is divided into the following four parts.1)We study the Hausdorff dimension and topological properties of the sets of ex-tremely divergence points on ?-expansions.The run-length function rn(x,?)is the max-imal length of consecutive zeros amongst the first n digits in the ?-expansion of x.Let?:N?R be an increasing function with limn???(n)=+?.We will consider the set of extremely divergence points related to the function ? which is defined as We show that Emax? is of full Hausdorff dimension and is residual.This result generalize the result of Li and Wu for the case of ?=2.2)We investigate the Lesbesgue measure,Hausdorff dimension and topological prop-erties of the sets of points related to the run-length function which corresponds to the Diophantine approximation.We create the relationship between the limit superior(re-spectively limit inferior)of rn(x,?)/n and the classical Diophantine approximation exponent(respectively uniform Diophantine approximation exponent).Then we successfully used this relationship to transfer the problem of irregular sets related to run-length function to the problem related to the Diophantine approximation exponents.We obtain the Les-besgue measure,Hausdorff dimension of the irregular set Ea,b(that is,the limit superior and limit inferior are a and b respectively where 0 ?a?b?1).Furthermore,we also show that the extremely divergence set E0,1 is residual.The same problems about rn(1,?)when regarding ? as a variable are also examined.3)We research on the set of irregular points on the length of basic intervals.The basic interval In(x)on ?-expansions is defined to be the set containing points with the same n prefix of the ?-expansion of x.It is known that the extremely exceptional set D:={x ?[0,1):limsupn??-log?|In(x)|/n=1+?(?)}is of Hausdorff dimension 0.We prove that the set D is residual.We give an example that a set have null Hausdorff dimension and Lebesgue measure but it is residual.4)We give the size of the set of points with prescribe uniform recurrence rate.The size of the set of points with asymptotic recurrence properties is shown in Shen and Wang's result.In the final part of our thesis,we study the uniform recurrence properties of the orbit of a point x ?[0,1)under the ?-transformation returning to the point itself in a uniform way.