Tushar Das

The Hausdorff and dynamical dimensions of self-affine sponges: a dimension gap result

We construct a self-affine sponge in

A variational principle in the parametric geometry of numbers, with applications to metric Diophantine approximation

\mathbb {R}^3

The Bishop–Jones relation and Hausdorff geometry of convex-cobounded limit sets in infinite-dimensional hyperbolic space

R3 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.

The geometry of Baire spaces

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.

Geometry and dynamics in gromov hyperbolic metric spaces : with an emphasis on non-proper settings

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 m n ( 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.

Extremality and dynamically defined measures, part II : Measures from conformal dynamical systems

We construct a non-elementary Fuchsian group that admits two non-elementary subgroups with trivial intersection and whose radial limit sets intersect non-trivially. This disproves a conjecture attributed to Perry Susskind that was published by James W. Anderson (2014).

Badly approximable points on self-affine sponges and the lower Assouad dimension

We generalize the mass redistribution principle and apply it to prove the Bishop–Jones relation for limit sets of metrically proper isometric actions on real infinite-dimensional hyperbolic space. We also show that the Hausdorff and packing measures on the limit sets of convex-cobounded groups are finite and positive and coincide with the conformal Patterson measure, up to a multiplicative constant.

A proof of the matrix version of Baker's conjecture in Diophantine approximation

We introduce the concept of Baire embeddings and we classify them up to C 1+ϵ conjugacies. We show that two such embeddings are C 1+ϵ-equivalent if and only if they have exponentially equivalent geometries. Next, we introduce the class of iterated function system (IFS)-like Baire embeddings and we also show that two Hölder equivalent IFS-like Baire embeddings are C 1+ϵ conjugate if and only if their scaling functions are the same. In the remaining sections, we introduce metric scaling functions and we show that the logarithm of such a metric scaling function and the logarithm of Sullivans scaling function multiplied by the Hausdorff dimension of the Baire embedding are cohomologous up to a constant. This permits us to conclude that if the Bowen measures coincide for two IFS-like Baire embeddings, then the embeddings are bi-Lipschitz conjugate.