In this paper, we use numerical and analytical methods to discuss the following problems. (1) The first is the relation between Hausdorff dimension d and Lyapunov exponent λ. We point out that in the relation N(ε) there is a multi-scaling-region phenomena which shows the structure of the point set at different scale. To analyse this phenomena seriously is significant for calculating d. (2) Acorrding to the relation between the one-dimensional map and the Poincare map of dynamic system, and the discontinuity and multivalue property of d( λ ) of one-dimensional map at λ=0, we conjecture that the relation d(λ) at λ= (0,0,-) of the dynamical system is also discontinuous and multivalued. (3) On the base of relation d(λ), we use λ>0 as a criterion to estimate the measure of chaotic solutions, of logistic model in chaotic region, on the axis of parameter. The result is mc= 0.893±0.022 (The total measure of chaotic region is normalized to unity).