Strong Regularity Of A Kind Of Weak Self-similar Sets

A set K ? Rd is called strong regular if its Hausdor? dimension coincides with its box-counting dimension. Since the sets with regularity have good properties. For example, theHausdor? dimension of Cartesian product, between a strong regular set and any set, equals tothe sum of their dimensions. Therefore, the study of the strong regularity becomes an importantsubject in fractal geometry.It is known that Self-similar sets are all strong regular; Cookie-Cutter sets are strong reg-ular in R1; and self-conformal sets are strong regular, too. At first, we notes that Cookie-Cuttersets in R1 can be viewed equivalently as the invariant sets of a family of contractions each ofwhich is of class C1+η, and the invariant sets satisfy the Strong Separation Condition. we posenaturedly the question: without the Strong Separation Condition or any separation condition, isan invariant set of a family of contractions each of which is of class C1+ηstrong regular?At first, we propose the weak self-similar set, which is the invariant set of a family ofcontractions each of which is of class C1+η. Then, by the implicit theorem, we derive that,under some certain conditions, weak self-similar sets are strong regular, which extends theresults of self-similar sets, self-conformal sets and Cookie-Cutter sets in R1. Using the result,we can derive the strong regularity of Cookie-Cutter sets in Rd. In addition, we find a little leakabout Cookie-Cutter sets in [6]. We set a counterexample, and give a weaker property fillingthe leap.