It is shown by Denker and Yuri [Contemp. Math. 63, 93–108 (2015)] that “general” iterated function systems admit conformal families of measures. Here, we relate this property to estimating Hausdorff dimension in the expanding case, including overlaps. A correct formulation of Theorem 3.11 in the above paper is provided as well.
Iterated function systems are a dynamical concept to study time series governed by several sources of behavior. Its set of initial values for which indefinite iteration is possible is one of the most important objectives. Here, the dimension of this collection of initial values is estimated using a novel approach based on thermodynamic formalism.
I. A REVIEW OF FACTS FOR “GENERAL” ITERATED FUNCTION SYSTEMS
Denker and Yuri introduced an approach to treat general iterated function systems in 2015. In order to review the associated basic definitions, let us begin with an example.
The existence of invariant measures for transfer operators (or Perron Frobenius operator) in our situation has been established in Denker and Yuri.1 We recall the result in Sec. II. In Sec. III, we relate the eigenvalues of the transfer operator and some growth condition of overlaps of the domains of the “general” iterated function system to the Hausdorff dimension of the set of infinite orbits. More evolved results are certainly possible.
II. EIGENMEASURES FOR TRANSFER OPERATORS
Recall1 that a conformal family of measures for a “general” iterated function system and the potential is a collection of measures where denotes a probability measure on and a measure on for satisfying
- For some and every ,
on the interior of for all .
The term on the left hand side in (2.2) is called the deficiency of the conformal family. So, is a conformal measure if the deficiency vanishes.
Theorem 3.11 in Denker and Yuri1 is not correctly formulated as one crucial assumption is missing. Here is the correct formulation. (This is a corrigendum of Conformal Families Of Measures For General Iterated Function Systems authored and approved by Michiko Yuri, Department of Mathematics, Hokkaido University. We thank Rodrigo Bissacot, Thiago Costa Raszeja, and, in particular, Rodrigo S. Frausino for pointing out this omission.) For the definitions of , and of a proper potential, we refer to Denker and Yuri.1 The notation in the correction below follows that of the present paper and differs slightly from the notation used in the former. It gives a condition to obtain conformal measures. Their existence is a basic assumption in Sec. III.
III. HAUSDORFF DIMENSION
The transfer operator in (2.1) can be connected to the estimation of the Hausdorff dimension of . Here, we successively develop that relation.
Our next assumption is uniform distortion:
- If for a measurable set , andthen
- If for a measurable set andthen
- Let be a measurable set, and . Assume thatandThen,
We first observe
can be chosen uniformly for if .
Let be a “general” iterated function system and be a strictly negative potential satisfying uniform distortion and admitting the existence of conformal measures for the potential with eigenvalue , .
Use Besicovic’s covering theorem2,3 and Lemmas III.3 and III.4.
Note that the corollary reduces to the classical Bowen–Manning–McCluskey formula for dimension in the case of a Ruelle expanding map. It also applies to the example presented in the introduction, especially in the case where the potential is given by the derivative of the maps.
I would like to thank two referees for valuable comments and suggestions improving the clarity of notions, proofs, and formulas.
Conflict of Interest
The authors have no conflicts to disclose.
Manfred Denker: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (equal).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.