Related papers: Rigidity of non-maximal torus actions, unipotent q…

We prove positive characteristic analogues of certain measure rigidity theorems in characteristic zero. More specifically we give a classification result for positive entropy measures on quotients of $\operatorname{SL}_d$ and a…

Let $\Gamma$ be a lattice in a simply connected nilpotent Lie group $G$. Given an infinite measure preserving action $T$ of $\Gamma$ and a "direction" in $G$ (i.e. an element $\theta$ of the projective space $P(\goth g)$ of the Lie algebra…

For an invariant probability measure for the Gauss map, almost all numbers are Diophantine if the log of the partial quotient function is integrable. We show that with respect to a ``continued fraction mixing'' measure for the Gauss map…

We study smooth actions by lattices in higher-rank simple Lie groups. Assuming one element of the action acts with positive topological entropy, we prove a number of new rigidity results. For lattices in $\mathrm{SL}(n,\mathbb{R})$ acting…

For a class of irrational numbers, depending on their Diophantine properties, we construct explicit rank-one transformations that are totally ergodic and not weakly mixing. We classify when the measure is finite or infinite. In the finite…

The main result of this paper is a construction of finitely additive measures for higher rank abelian actions on Heisenberg nilmanifolds. Under a full measure set of Diophantine conditions for the generators of the action, we construct…

We consider two numerical entropy--type invariants for actions of $\Zk$, invariant under a choice of generators and well-adapted for smooth actions whose individual elements have positive entropy. We concentrate on the maximal rank case,…

We study the rigidity properties of a class of algebraic Z^3-actions with entropy rank two. For this class, conditions are found which force an invariant measure to be the Haar measure on an affine subset. This is applied to show…

We prove absolute continuity of "high entropy" hyperbolic invariant measures for smooth actions of higher rank abelian groups assuming that there are no proportional Lyapunov exponents. For actions on tori and infranilmanifolds existence of…

We study the one-dimensional expanding Lorenz maps and show the existence of dense subset D of Lorens maps such that each f in D has an uncountable set of ergodic invariant probabilities with infinite Lyapunov exponent and positive entropy.…

We show that for a $\mathbb{Z}^{l}$-action (or $(\N\cup\{0\})^l$-action) on a non-empty compact metrizable space $\Omega$, the existence of a affine space dense in the set of continuous functions on $\Omega$ constituted by elements…

We study slow entropy invariants for abelian unipotent actions $U$ on any finite volume homogeneous space $G/\Gamma$. For every such action we show that the topological slow entropy can be computed directly from the dimension of a special…

In this article we establish two new results on quantitative Diophantine approximation for one-parameter families of diagonal ternary indefinite forms. In the first result, we consider quadratic forms taking values at prime points. In the…

Assuming positive entropy we prove a measure rigidity theorem for higher rank actions on tori and solenoids by commuting automorphisms. We also apply this result to obtain a complete classification of disjointness and measurable factors for…

We prove sharp estimates in a shrinking target problem for the action of an arbitrary subgroup $\Gamma$ of $SL_2(\mathbb{Z})$ on the 2-torus. This can also be viewed as a non-commutative Diophantine approximation problem. The methods…

We extend Lyons's tree entropy theorem to general determinantal measures. As a byproduct we show that the sofic entropy of an invariant determinantal measure does not depend on the chosen sofic approximation.

The rigidity propeties of higher rank diagonalizable actions is a major theme in homogenous dynamics, with origins in work of Cassels and Swinnerton-Dyer in the 1950s and Furstenberg. We survey both results and conjectures regarding such…

For any diagonal element $a$ with two eigenvalues, we construct a sequence of $a$-invariant probability measures on the space of unimodular lattices with high entropy but converging to the zero measure. This extends the result of Kadyrov…

We prove a local-global principle for representations of binary by quaternary quadratic forms. One of the main ingredients is a recent measure rigidity result of Einsiedler and Lindenstrauss for diagonalizable actions on quotients of…

Let G be a sofic group and X a compact group that G acts on by automorphisms. Using (and reformulating) the notion of doubly-quenched convergence developed by Austin, we show that in many cases the topological and the measure-theoretic…