If G is any Kleinian group, we show that the dimension of the limit set Λ is always equal to either the dimension of the bounded geodesics or the dimension of the geodesics that escape to infinity at linear speed. Suppose G is a discrete group of isometries on hyperbolic space B, n ≥ 2. The limit set Λ ⊂ Sn−1 is defined to be the accumulation set of the G-orbit of 0 ∈ B. A point x ∈ Λ can be as...

We consider infinite conformal function systems on Rd. We study the geometric structure of the limit set of such systems. Suppose this limit set intersects some l-dimensional C1-submanifold with positive Hausdorff t-dimensional measure, where 0 < l < d and t is the Hausdorff dimension of the limit set. We then show that the closure of the limit set belongs to some l-dimensional affine subspace ...

Positive set theory is at present a subject on its own, from the initial work of R.J. Malitz [16], to the further developments including topological models [6], [7], and very strong variants, in particular Esser’s theory GPK ∞ where the class of all well-founded sets interprets ZF [3], [4]. In another direction, namely in languages allowing terms, does positive set theory appear as a weakening ...