Diagonal actions in positive characteristic

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 classification of joinings for higher rank actions on simply connected absolutely almost simple groups.Comment: 44 page

Escape of mass and entropy for diagonal flows in real rank one situations

Let G be a connected semisimple Lie group of real rank 1 with finite center, let be a non-uniform lattice in G and a any diagonalizable element in G. We investigate the relation between the metric entropy of a acting on the homogeneous space \G and escape of mass. Moreover, we provide bounds on the escaping mass and, as an application, we show that the Hausdorff dimension of the set of orbits (under iteration of a) which miss a fixed open set is not full

Simultaneous dense and nondense orbits for commuting maps

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.V. B. acknowledges support received from the National Science Foundation via Grant DMS-1162073 M. E. acknowledges support by the SNF (200021-152819). J. T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 291147

Simultaneous dense and nondense orbits for commuting maps

This is the author accepted manuscript. The final version is available from the publisher via the DOI in this record.We show that, for two commuting automorphisms of the torus and for two elements of the Cartan action on compact higher rank homogeneous spaces, many points have drastically different orbit structures for the two maps. Specifically, using measure rigidity, we show that the set of points that have dense orbit under one map and nondense orbit under the second has full Hausdorff dimension.V. B. acknowledges support received from the National Science Foundation via Grant DMS-1162073 M. E. acknowledges support by the SNF (200021-152819). J. T. acknowledges the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013)/ERC Grant Agreement n. 291147

Tropical polyhedra are equivalent to mean payoff games

We show that several decision problems originating from max-plus or tropical convexity are equivalent to zero-sum two player game problems. In particular, we set up an equivalence between the external representation of tropical convex sets and zero-sum stochastic games, in which tropical polyhedra correspond to deterministic games with finite action spaces. Then, we show that the winning initial positions can be determined from the associated tropical polyhedron. We obtain as a corollary a game theoretical proof of the fact that the tropical rank of a matrix, defined as the maximal size of a submatrix for which the optimal assignment problem has a unique solution, coincides with the maximal number of rows (or columns) of the matrix which are linearly independent in the tropical sense. Our proofs rely on techniques from non-linear Perron-Frobenius theory.Comment: 28 pages, 5 figures; v2: updated references, added background materials and illustrations; v3: minor improvements, references update

Asymptotic diophantine approximation:the multiplicative case

Let $\alpha$ and $\beta$ be irrational real numbers and 0<\F<1/30. We prove a precise estimate for the number of positive integers $q\leq Q$ that satisfy \|q\alpha\|\cdot\|q\beta\|<\F. If we choose \F as a function of $Q$ we get asymptotics as $Q$ gets large, provided \F Q grows quickly enough in terms of the (multiplicative) Diophantine type of $(\alpha,\beta)$, e.g., if $(\alpha,\beta)$ is a counterexample to Littlewood's conjecture then we only need that \F Q tends to infinity. Our result yields a new upper bound on sums of reciprocals of products of fractional parts, and sheds some light on a recent question of L\^{e} and Vaaler.Comment: To appear in Ramanujan Journa

Partner orbits and action differences on compact factors of the hyperbolic plane. Part I: Sieber-Richter pairs

Physicists have argued that periodic orbit bunching leads to universal spectral fluctuations for chaotic quantum systems. To establish a more detailed mathematical understanding of this fact, it is first necessary to look more closely at the classical side of the problem and determine orbit pairs consisting of orbits which have similar actions. In this paper we specialize to the geodesic flow on compact factors of the hyperbolic plane as a classical chaotic system. We prove the existence of a periodic partner orbit for a given periodic orbit which has a small-angle self-crossing in configuration space which is a `2-encounter'; such configurations are called `Sieber-Richter pairs' in the physics literature. Furthermore, we derive an estimate for the action difference of the partners. In the second part of this paper [13], an inductive argument is provided to deal with higher-order encounters.Comment: to appear on Nonlinearit