Open Set Conditions On Graph-directed Controlled-Moran Constructions Sets

The equivalence of "Separation properties" that includes the open set condition, the strong set condition and the positivity of the Hausdorff measure for self-similar sets and graph-directed self-similar sets have been worked out. In this article, we mainly consider "Separation properties" for a class of non-self-similar sets that we call it Graph-directed Controlled-Moran Constructions Sets.In the second chapter, we prove the equivalence of the ball condition, the finite clustering property, the uniform ball condition, the uniform finite clustering property and the positivity of t-dimension Hausdorff measure of Graph-directed Controlled-Moran Constructions Sets.We derive an important conclusion (Theorem2.14): For a tractable Graph-directed Controlled-Moran Constructions and its limit set, the following conditions are equivalent: (1)the ball condition; (2)the uniform ball condition; (3)the finite clustering property; (4)the uniform finite clustering property; (5) (?)~t (K) > 0, where t is the zero of the topological pressure.In the third chapter, we prove the equivalence of the open set condition, the strong open set condition, the ball condition and the positivity of t-dimension Hausdorff measure of Graph-directed Controlled-Moran Constructions Sets that generated by Graph-directed iterated function system.We derive: For a congruent graph-directed iterated function system which satisfies bounded condition, the ball condition, the open set condition, the strong open set condition, and (?)~t(K) > 0 are equivalent, where t is the zero of the topological pressure.