Dimension sets for infinite IFSs: the Texan Conjecture

Abstract

We consider the set of Hausdorff dimensions of limit sets of finite subsystems of an infinite conformal iterated function system and refer to it as the restricted dimension set. The corresponding set for all subsystems will be referred to as the complete dimension set. We give sufficient conditions for a point to belong to the complete dimension set and consequently to be an accumulation point of the restricted dimension set. We also give sufficient conditions on the system for both sets to be nowhere dense in some interval. Both general results are illustrated by examples. Applying the first result to the case of continued fraction we are able to prove the Texan Conjecture, that is we show that the set of Hausdorff dimensions of bounded type continued fraction sets is dense in the unit interval.

The dimension spectrum of conformal graph directed Markov systems

“…Indeed, apart from being compact, perfect, and containing the interval [0, θ) (keep in mind that we are now in the realm of IFSs), it may happen to be an interval (then necessarily DS(S) = [0, dim H (J)]), or it may have many non-degenerate connected components (so intervals) and connected components being singletons. Such examples, even for similarities, can be found in [21]. All of this leads us to the following conjecture.…”

“…Of special significance is the question of when an IFS S has full dimension spectrum, that is DS(S) = [0, dim H (J)]. Kesseböhmer and Zhu proved in [21] that the spectrum is full for the IFS resulting from the real continued fractions algorithm, resolving the so-called Texan conjecture. Recall that any irrational number in [0, 1] can be represented as a continued fraction 1 e 1 + 1…”

On the dimension spectrum of infinite subsystems of continued fractions

“…Arguing as before we deduce that Proof. If m = 1 the result is due to Kesseböhmer and Zhu [23]. We can thus assume that m ≥ 2.…”

“…Our proofs employ the machinery which we developed recently in [5] in order to show that complex continued fractions have full spectrum, as well as some key ideas of Kesseböhmer and Zhu from [23]. All the techniques used, depend on a well known remarkable feature of the sets J E ; they can be realized as attractors of iterated function systems consisting of conformal maps; see Section 2 for more details.…”

A generalization of the Jarník–Besicovitch theorem by continued fractions

“…The Hausdorff dimension of this set was well studied by Bugeaud [5,7] and Bugeaud and Moreira [8] by also using the tools of continued fractions. For more dimensional results relating the partial quotients, see [9,10,13,16,17,19,22,23,25,26,34,35,37] and references therein.…”