Multifractal analysis of Birkhoff averages for some self-affine IFS

Jordan, Thomas; Simon, Károly

doi:10.1080/14689360701524305

Cited by 13 publications

(8 citation statements)

References 18 publications

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“…Barral and Mensi [2], Barral and Feng [1] and Reeve [32,33]. Jordan and Simon [24] have given a conditional variational principle for typical members of parameterizable families of self-affine iterated function systems with a simultaneously diagonalizable linear part.…”

mentioning

confidence: 99%

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

Käenmäki¹,

Reeve²

2014

J. Fractal Geom.

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We develop a thermodynamic formalism for quasi-multiplicative potentials on a countable symbolic space and apply these results to the dimension theory of infinitely generated self-affine sets. The first application is a generalisation of Falconer's dimension formula to include typical infinitely generated self-affine sets and show the existence of an ergodic invariant measure of full dimension whenever the pressure function has a root. Considering the multifractal analysis of Birkhoff averages of general potentials Φ taking values in R N , we give a formula for the Hausdorff dimension of JΦ(α), the α-level set of the Birkhoff average, on a typical infinitely generated selfaffine set. We also show that for bounded potentials Φ, the Hausdorff dimension of JΦ(α) is given by the maximum of the critical value for the pressure and the supremum of Lyapunov dimensions of invariant measures µ for which Φ dµ = α. Our multifractal results are new in both the finitely generated and the infinitely generated setting.

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confidence: 99%

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

Käenmäki¹,

Reeve²

2014

J. Fractal Geom.

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“…This result was generalised for Gatzouras-Lalley carpets by Reeve [28]. Moreover, Jordan and Simon [22] studied the case of planar affine iterated function systems with diagonal matrices for generic translation vectors v i . As far as we are aware of, the only known result about non-diagonal matrices comes from Käenmäki and Reeve [24], who investigated irreducible matrices with generic translation vectors (see Section 2 for the precise definition of irreducibility).…”

Section: Introductionmentioning

confidence: 85%

“…In the situation, where the system is not strongly irreducible, the results are no longer always true; for example, see Reeve [28] where the author considers self-affine carpets. In the diagonal case, it would be possible to combine Hochman [20] and Jordan and Simon [22] to get results for a large class of systems based on the dimension of projections of measures onto the x-axis and y-axis.…”

Section: Main Theoremsmentioning

confidence: 99%

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Bárány

Jordan

Käenmäki

et al. 2020

International Mathematics Research Notices

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“…(6) It is worth to point out that Falconer gave a variational formula for the Hausdorff dimension for "almost all" self-affine sets under some assumptions [15], and for this case Käenmäki showed the existence of ergodic measures of full Hausdorff dimension on the typical self-affine sets [27]. See [26] for a related result on the multifractal analysis.…”

Section: Introductionmentioning

confidence: 99%

Weighted thermodynamic formalism on subshifts and applications

Barral¹,

Ding²

2012

Asian Journal of Mathematics

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Let (X, T ) and (Y, S) be two subshifts so that Y is a factor of X. For any asymptotically sub-additive potential Φ on X and a = (a, b) ∈ R 2 with a > 0, b ≥ 0, we introduce the notions of a-weighted topological pressure and a-weighted equilibrium state of Φ. We setup the weighted variational principle. In the case that X, Y are full shifts with one-block factor map, we prove the uniqueness and Gibbs property of aweighted equilibrium states for almost additive potentials having the bounded distortion properties. Extensions are given to the higher dimensional weighted thermodynamic formalism. As an application, we conduct the multifractal analysis for a new type of level sets associated with Birkhoff averages, as well as for weak Gibbs measures associated with asymptotically additive potentials on self-affine symbolic spaces.1991 Mathematics Subject Classification. Primary 37D35, Secondary 37B10, 37A35, 28A78.

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Multifractal analysis of Birkhoff averages for some self-affine IFS

Abstract: Abstract. In this paper we consider self-affine IFS {S i } m0 i=1 on the plane of the formWe describe the multifractal analysis of Birkhoff averages of the continuous functions. In Section 6 we compute it numerically in a special case (see Figure 1).

Cited by 13 publications

References 18 publications

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

Multifractal analysis of Birkhoff averages for typical infinitely generated self-affine sets

Birkhoff and Lyapunov Spectra on Planar Self-affine Sets

Weighted thermodynamic formalism on subshifts and applications

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