The Finite Strong Open Set Condition For A Class Of Generalized Self-Similar Sets

A.Schief first researched the equivalences of the OSC,SOSC and the Hausdorff measure of self-similar set of invariant set that generated by finite similar contractive mapping greater than zero.Later Jing Ling Wang generalized it to graph-directed self-similar set also abtain the equivalence of OSC,SOSC and Hausdorff measure of invariant set greater than zero.But those research all based on the case of finete number of similar contractive mapping.Resently Ye Yuan Ling and Chen Xiao Peng give a sufficient condition for Hausdorff measure that invariant set generated by infinite number of similar contractive mapping greater than zero imply finite strong open set condition.A proof when the number of mapping is infinite the OSC doesn't imply SOSC was given by T.Szarek and S.Wedrychowicz.This article generalize the research object to a class of generalized fractal s,studied when the number of mapping of graphdirected fractal set's Hausdorff measure bigger than zreo,the finite strong open set condition is satisfied.