Dimensions of compact invariant sets of some expanding maps

Yayama, Yuki

doi:10.1017/s014338570800014x

Cited by 16 publications

(30 citation statements)

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“…Using a coding map constructed by a Markov partition for T, one obtains a symbolic representation of the carpet which is a full shift on finitely many symbols. A more general set whose symbolic representation is a shift of finite type was considered in [19]. Such a set is called an SFT-NC carpet.…”

Section: Introductionmentioning

confidence: 99%

Existence of a measurable saturated compensation function between subshifts and its applications

Yayama

2010

Ergod. Th. Dynam. Sys.

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Link to this article: http://journals.cambridge.org/abstract_S0143385710000404How to cite this article: YUKI YAYAMA (2011). Existence of a measurable saturated compensation function between subshifts and its applications.Abstract. We show the existence of a bounded Borel measurable saturated compensation function for any factor map between subshifts. As an application, we find the Hausdorff dimension and measures of full Hausdorff dimension for a compact invariant set of an expanding non-conformal map on the torus given by an integer-valued diagonal matrix. These problems were studied in [23] for a compact invariant set whose symbolic representation is a shift of finite type under the condition of the existence of a saturated compensation function. By using the ergodic equilibrium states of a constant multiple of a Borel measurable compensation function, we extend the results to the general case where this condition might not hold, presenting a formula for the Hausdorff dimension for a compact invariant set whose symbolic representation is a subshift and studying invariant ergodic measures of full dimension. We study uniqueness and properties of such measures for a compact invariant set whose symbolic representation is a topologically mixing shift of finite type. Downloaded: 16 Mar 2015 IP address: 132.174.255.116 1564 Y. Yayama given by Shin [19] and necessary and sufficient conditions for the existence of a saturated compensation function were also studied by Shin [18]. In this paper, we first consider a Borel measurable function F : X → R that satisfies equation (1.1). If there is such an F, we say that F is a Borel measurable compensation function. F is a Borel measurable saturated compensation functionsubshifts there always exists a bounded Borel measurable saturated compensation function and characterize the equilibrium states. Replacing F in equation (1.1) by this Borel measurable compensation function, in Theorem 3.9 we also show that equation (1.1) holds for a larger class of potentials on Y called subadditive potentials (see §1 for the definition of subadditive potentials). Using Borel measurable saturated compensation functions, we then study for a fixed α > 0 the measures that maximize the weighted entropy functional φ αFinding such measures is useful in problems on Hausdorff dimension (see [8]). When there is a saturated compensation function between subshifts, the measures that maximize φ α are, according to Shin [20], the equilibrium states of a constant multiple of a saturated compensation function. We extend the results for any subshifts without assuming the existence of a saturated compensation function (Proposition 3.14 and Theorem 3.15).As an application, we study the problem on dimensions of compact invariant sets of non-conformal expanding maps; in particular, we consider the endomorphism of the torus given by T (x, y) = (lx mod 1, my mod 1), l > m ≥ 2, l, m ∈ N. Throughout this paper, by a measure µ of full dimension for a compact T -invariant set K , we mean that µ is a Borel probability m...

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Section: Introductionmentioning

confidence: 99%

Existence of a measurable saturated compensation function between subshifts and its applications

Yayama

2010

Ergod. Th. Dynam. Sys.

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“…† An analogue approach of this question has been developed in the recent paper by Yayama[51]; however, the present work was issued independently.http://journals.cambridge.org Downloaded: 13 Dec 2014 IP address: 169.230.243.252…”

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Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus

Olivier¹

2009

Ergod. Th. Dynam. Sys.

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We consider the variational principle for dimension on compact subsets of the 2-torus which are invariant under a non-conformal expanding diagonal endomorphism. Condition (H) ensures that the invariant measures with full dimension are the equilibrium states of some potential function. This result applies to the problem of uniqueness of the measure with full dimension on the sofic affine-invariant sets.

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“…For extensions of these formulae to sets modelled by subshifts we refer the reader to [10,11,14,18]. When considered as a subset of the torus T 2 , the set X is invariant under the expanding toral endomorphism T (x, y) = (nx, my) mod 1.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Dimension of self-affine sets with holes

Jordan¹,

Ferguson²,

Rams³

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper we compute the dimension of a class of dynamically defined nonconformal sets. Let X ⊆ T 2 denote a Bedford-McMullen set and T : X → X the natural expanding toral endomorphism which leaves X invariant. For an open set U ⊂ X we letWe investigate the box and Hausdorff dimensions of X U for both a fixed Markov hole and also when U is a shrinking metric ball. We show that the box dimension is controlled by the escape rate of the measure of maximal entropy through U , while the Hausdorff dimension depends on the escape rate of the measure of maximal dimension.

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Dimensions of compact invariant sets of some expanding maps

Cited by 16 publications

References 25 publications

Existence of a measurable saturated compensation function between subshifts and its applications

Existence of a measurable saturated compensation function between subshifts and its applications

Uniqueness of the measure with full dimension on sofic affine-invariant subsets of the 2-torus

Dimension of self-affine sets with holes

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