Dimensions of equilibrium measures on a class of planar self-affine sets

Fraser, Jonathan M.; Jordan, Thomas; Jurga, Natalia

doi:10.4171/jfg/85

Cited by 7 publications

(7 citation statements)

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“…Feng [7] has very recently shown that all ergodic invariant measures supported on the attractors of IFS composed of affine maps are exact dimensional and satisfy a formula in terms of the Lyapunov exponents and conditional entropies. This answered a folklore open question in the fractal community and unified previous partial results obtained in [1,10]. In the non-conformal setting, this formula for the exact dimension of μ is often called a "Ledrappier-Young formula", following the work of Ledrappier and Young on the dimension of invariant measures for C 2 diffeomorphisms on compact manifolds [13,14].…”

mentioning

confidence: 52%

“…Let r > 0 be sufficiently small so that k ∈ N that satisfies (27) is sufficiently large that (17) holds for ε. Then, using (10), ( 26) and (17) we get…”

Section: Lemma 44mentioning

confidence: 99%

“…Let r > 0 be sufficiently small so that k ∈ N that satisfies (31) is sufficiently large that (18) holds for ε. Therefore by using (10) as r → 0 (so k → ∞). Since ε > 0 was arbitrary, the upper bound follows.…”

Section: Lemma 44mentioning

confidence: 99%

“…3 we recall some technical results which were proved in [6] concerning the contractive properties of the maps in our IFS. Section 4 contains the proof of Theorem 2.5, which adapts an approach used in [10].…”

mentioning

confidence: 99%

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Ledrappier–Young formulae for a family of nonlinear attractors

Jurga

Lee

2021

Math. Z.

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We study a natural class of invariant measures supported on the attractors of a family of nonlinear, non-conformal iterated function systems introduced by Falconer, Fraser and Lee. These are pushforward quasi-Bernoulli measures, a class which includes the well-known class of Gibbs measures for Hölder continuous potentials. We show that these measures are exact dimensional and that their exact dimensions satisfy a Ledrappier–Young formula.

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mentioning

confidence: 52%

“…Let r > 0 be sufficiently small so that k ∈ N that satisfies (27) is sufficiently large that (17) holds for ε. Then, using (10), ( 26) and (17) we get…”

Section: Lemma 44mentioning

confidence: 99%

Section: Lemma 44mentioning

confidence: 99%

mentioning

confidence: 99%

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Ledrappier–Young formulae for a family of nonlinear attractors

Jurga

Lee

2021

Math. Z.

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“…It was shown recently by D.-J. Feng in [22] that every self-affine measure is exact-dimensional; previous partial results in this direction include [3,5,26]. Let us now describe the most natural generalisation of Hutchinson's dimension formula N i=1 r s i = 1 to the affine context.…”

Section: Introductionmentioning

confidence: 99%

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

Morris¹,

Sert²

2019

Preprint

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A well-known theorem of J.E. Hutchinson states that if an iterated function system consists of similarity transformations and satisfies the open set condition then its attractor supports a self-similar measure with Hausdorff dimension equal to the similarity dimension. In this article we prove the following result which may be regarded as a form of partial converse: if an iterated function system consists of invertible affine transformations whose linear parts do not preserve a common invariant subspace, and its attractor supports a self-affine measure with Hausdorff dimension equal to the affinity dimension, then the system necessarily consists of similarity transformations. We obtain this result by showing that the equilibrium measures of an affine iterated function system are never Bernoulli measures unless the system either is reducible or consists of similarity transformations. The proof builds on earlier work in the thermodynamic formalism of affine iterated function systems due to Feng, Käenmäki, Bochi and the first named author and also relies on the work of Benoist and Quint on the spectral properties of Zariski-dense subsemigroups of reductive linear groups.

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Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

2020

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In topics such as the thermodynamic formalism of linear cocycles, the dimension theory of self-affine sets, and the theory of random matrix products, it has often been found useful to assume positivity of the matrix entries in order to simplify or make feasible certain types of calculation. It is natural to ask how positivity may be relaxed or generalised in a way which enables similar calculations to be made in more general contexts. On the one hand one may generalise by considering almost additive or asymptotically additive potentials which mimic the properties enjoyed by the logarithm of the norm of a positive matrix cocycle; on the other hand one may consider matrix cocycles which are dominated, a condition which includes positive matrix cocycles but is more general. In this article we explore the relationship between almost additivity and domination for planar cocycles. We show in particular that a locally constant linear cocycle in the plane is almost additive if and only if it is either conjugate to a cocycle of isometries, or satisfies a property slightly weaker than domination which is introduced in this paper. Applications to matrix thermodynamic formalism are presented.

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Dimensions of equilibrium measures on a class of planar self-affine sets

Cited by 7 publications

References 10 publications

Ledrappier–Young formulae for a family of nonlinear attractors

Ledrappier–Young formulae for a family of nonlinear attractors

A converse statement to Hutchinson's theorem and a dimension gap for self-affine measures

Domination, almost additivity, and thermodynamic formalism for planar matrix cocycles

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