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Andrei E. Ghenciu

Publications and source records attributed to Andrei E. Ghenciu.

5 recordsLinked to original sources
arXiv2023

The extended Hausdorff dimension spectrum of a conformal iterated function system is maximal

For any conformal iterated function system (CIFS) consisting of finitely or countably many maps, and any closed shift-invariant set of right-infinite sequences of such maps, one can associate a limit set, which we call a shift-generated conformal iterated construction. We define the extended Hausdorff dimension spectrum of a CIFS to be the set of Hausdorff dimensions of all such limit sets. We prove that for any CIFS with finitely or countably many maps, the extended Hausdorff dimension spectrum is maximal, i.e. all nonnegative dimensions less than or equal to the dimension of the limit set of the CIFS are realized. We also prove a version of this result even for so-called conformal graph directed Markov systems, obtained via nearest-neighbor restrictions on the CIFS. %when there are nearest-neighbor restrictions on the CIFS (similar to those in the so-called conformal graph directed Markov systems). The main step of the proof is to show that for the family $(X_β)$ of so-called $β$-shifts, the Hausdorff dimension of the limit set associated to $X_β$ varies continuously as a function of $β$.

math.DS↗
arXiv2014

Dynamical properties of S-gap shifts and other shift spaces

We study the dynamical properties of certain shift spaces. To help study these properties we introduce two new classes of shifts, namely boundedly supermultiplicative (BSM) shifts and balanced shifts. It turns out that any almost specified shift is both BSM and balanced, and any balanced shift is BSM. However, as we will demonstrate, there are examples of shifts which are BSM but not balanced. We also study the measure theoretic properties of balanced shifts. We show that a shift space admits a Gibbs state if and only if it is balanced. Restricting ourselves to $S$-gap shifts, we relate certain dynamical properties of an $S$-gap shift to combinatorial properties from expansions in non-integer bases. This identification allows us to use the machinery from expansions in non-integer bases to give straightforward constructions of $S$-gap shifts with certain desirable properties. We show that for any $q\in(0,1)$ there is an $S$-gap shift which has the specification property and entropy $q$. We also use this identification to address the question, for a given $q\in(0,1),$ how many $S$-gap shifts exist with entropy $q?$ For certain exceptional values of $q$ there is a unique $S$-gap shift with this entropy.

math.DS↗
arXiv2007

Conformal Graph Directed Markov Systems

We present the main concepts and results for Graph Directed Markov Systems that have a finitely irreducible incidence matrix. We then see how these results change when the incidence matrix is not assumed to be finitely irreducible.

math.DS↗