We construct a self-affine sponge in R 3 whose dynamical dimension, i.e. the supremum of the Hausdorff dimensions of its invariant measures, is strictly less than its Hausdorff dimension. This resolves a long-standing open problem in the dimension theory of dynamical systems, namely whether every expanding repeller has an ergodic invariant measure of full Hausdorff dimension. More generally we compute the Hausdorff and dynamical dimensions of a large class of self-affine sponges, a problem that previous techniques could only solve in two dimensions. The Hausdorff and dynamical dimensions depend continuously on the iterated function system defining the sponge, implying that sponges with a dimension gap represent a nonempty open subset of the parameter space.
Abstract. We establish a new connection between metric Diophantine approximation and the parametric geometry of numbers by proving a variational principle facilitating the computation of the Hausdorff and packing dimensions of many sets of interest in Diophantine approximation. In particular, we show that the Hausdorff and packing dimensions of the set of singular m × n matrices are both equal to mn 1 − 1 m+n , thus proving a conjecture of Kadyrov, Kleinbock, Lindenstrauss, and Margulis as well as answering a question of Bugeaud, Cheung, and Chevallier. Other applications include computing the dimensions of the sets of points witnessing conjectures of Starkov and Schmidt.Résumé. Nousétablissons une nouvelle connexion entre l'approximation métrique diophantine et la géométrie paramétrique des nombres en prouvant un principe variationnel facilitant le calcul des dimensions d'Hausdorff et de packing de nombreux ensembles d'intérêt dans l'approximation diophantienne. Nous montrons que les dimensions précitées de l'ensemble des matrices m × n singulières sont toutes deuxégales a mn 1 − 1 m+n , démontrant ainsi une conjecture de Kadyrov, Kleinbock, Lindenstrauss, et Margulis et répondantà une question de Bugeaud, Cheung, et Chevallier. D'autres applications comprennent le calcul des dimensions des ensembles des points témoignant des conjectures de Starkov et de Schmidt. Main resultsThe notion of singularity (in the sense of Diophantine approximation) was introduced by Khintchine, first in 1937 in the setting of simultaneous approximation [11], and later in 1948 in the more general setting of matrix approximation [12]. 1 Since then this notion has been studied within Diophantine approximation and allied fields, see Moshchevitin's 2010 survey [13]. An m × n matrix A is called singular if for all ε > 0, there exists Q ε such that for all Q ≥ Q ε , there exist integer vectors p ∈ Z m and q ∈ Z n such that Aq + p ≤ εQ −n/m and 0 < q ≤ Q.Here · denotes an arbitrary norm on R m or R n . We denote the set of singular m × n matrices by Sing(m, n). For 1 × 1 matrices (i.e. numbers), being singular is equivalent to being rational, and in general any matrix A which satisfies an equation of the form Aq = p, with p, q integral and q nonzero, is singular. However, Khintchine proved that there exist singular 2 × 1 matrices whose entries are linearly independent over Q [10, Satz II], and his argument generalizes to the setting of m×n matrices for all (m, n) = (1, 1). The name singular derives from the fact that Sing(m, n) is a Lebesgue nullset for all m, n, see e.g. [11, p.431] or [2, Chapter 5,§7]. Note that singularity is a strengthening of the property of Dirichlet improvability introduced by Davenport and Schmidt [6].In contrast to the measure zero result mentioned above, the computation of the Hausdorff dimension of Sing(m, n) has been a challenge that so far only met with partial progress. The first breakthrough was made in 2011 by Cheung [3], who proved that the Hausdorff dimension of Sing(2, 1) is 4/3; this was extended in 2...
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.
We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis ('98) resolving Sprindžuk's conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss ('04), hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, of the Patterson-Sullivan measures of all nonplanar geometrically finite groups, and of the Gibbs measures (including conformal measures) of infinite iterated function systems. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW's sufficient conditions for extremality. In Part I, we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, thus proving a conjecture of KLW. We also prove the "inherited exponent of irrationality" version of this theorem, describing the relationship between the Diophantine properties of certain subspaces of the space of matrices and measures supported on these subspaces. In subsequent papers, we exhibit numerous examples of quasi-decaying measures, in support of the thesis that "almost any measure from dynamics and/or fractal geometry is quasi-decaying". We also discuss examples of non-extremal measures coming from dynamics, illustrating where the theory must halt.
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