How inhomogeneous Cantor sets can pass a point

摘要

For x > 0, let @@@ Upsilon(x) = {(a,b) : x is an element of E-a,E-b,E- a > 0,b > 0, a + b <= 1}, @@@ where E-a,E-b is the unique nonempty compact invariant set generated by the inhomogeneous IFS @@@ Psi(a,b) = {f(0)(x) = ax, f(1)(x) = b(x + 1)}. @@@ We show that the set Upsilon(x) is a Lebesgue null set with full Hausdorff dimension and the intersection of the sets Upsilon(x(1)), ..., Upsilon(x(l)) still has full Hausdorff dimension for any finite number of positive numbers x(1), ..., x(l).

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