Hausdorff Measure Of Escaping Rays In The Cosine Family

In this thesis,we consider the dynamics rays of cosine map.Based on the result the Hausdorff dimension of the rays is one,we give the more finer result on it respect on gauge functions.First of all,according to the properties of fixed point,we construct the equation,and by using Veda theorem and monotonicity of function,we prove that the coefficient of cosine function S=aez+be-z satisfies the condition of 0<a<1/e and-1/e<b<0,so that the cosine function S has three real fixed points ?1,? and ?2 satisfying ?1<?<?2,with a attracting and ?1,?2 repelling.Secondly,according to the standard results of the function equation of Schr(?)der and the extended properties of the function,the fractional iteration of cosine function S=aez+be-z and inverse function Ir=S-r are defined,and re-parameterization of dynamic ray gs is obtained according to the iterative theory of the function,monotonicity and bijection:#12Finally,according to the iterative properties and the concavity and convexity of the function,the Hausdorff measures for gauge functions h(t)=t/(log(1/t))s and h(t)=t/Is(1/t)are obtained as follows:(1)Let s>1,then H~h(J(S)\C)=0 for h(t)=t/(log(1/t))s,(2)Let s>1,then H~h(J(S)\C)=? for h(t)=t/(Is(1/t).