题目： Fractal analysis of a class of singular functions arising from the alpha-Farey systems.

时间：2012年8月21日上午10：30－12:00

地点：闵行校区数学系102报告厅

报告人：Dr. Sara Munday

摘要：We will show, for any arbitrary alpha-Farey map, that this map is topologically conjugate to the tent map. Denoting the conjugating homeomorphism by theta_alpha, we will first show that each map theta_alpha is singular with respect to the Lebesgue measure and that the derivative can only take the values 0 or infinity. So, we obtain a partition of the unit interval into three sets: Theta_0:={x in [0,1]: the derivative of theta_alpha at x is equal to 0}, Theta_inf:={x in [0,1]: the derivative of theta_alpha at x is equal to infinity} and Theta_~:= [0,1]\ (Theta_0 union Theta_inf). The main result is that:

dim_H(Theta_inf) = dim_H(Theta_~) = dim_H(L(log2)) < dim_H(Theta_0) = 1,

where dim_H denotes the Hausdorff dimension and the set L(log2) is the level-set at log 2 of the Lyapunov spectrum for the alpha-Farey map described in talk 1.