Disjointness of interval exchange transformations from systems of probabilistic origin

Mobius disjointness for interval exchange transformations on three intervals

We show that Sarnak's conjecture on Mobius disjointness holds for interval exchange transformations on three intervals (3-IETs) that satisfy a mild diophantine condition.

Möbius Disjointness for Nilsequences Along Short Intervals

For a nilmanifold $G/Γ$, a $1$-Lipschitz continuous function $F$ and the Möbius sequence $μ(n)$, we prove a bound on the decay of the averaged short interval correlation $$\frac1{HN}\sum_{n\leq N}\Big|\sum_{h\leq H} μ(n+h)F(g^{n+h}x)\Big|$$ as $H,N\to\infty$. The bound is uniform in $g\in G$, $x\in G/Γ$ and $F$.

关于3-IETs与混合系统及轻混合系统的不交性

On disjointness of 3-IETs from mixing and lightly mixing systems

本文证明所有3-IETs与混合系统谱不交、与轻混合系统不交，改进了此前仅几乎所有3-IETs满足该性质的结论，还据此得出部分3-IETs虽弱混合但与轻混合系统Furstenberg不交的结果。

Hausdorff dimension of a class of three-interval exchange maps

In \cite{B} Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of Three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show, that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

Weak mixing properties of interval exchange transformations and translation flows

Let $d >1$. In this paper we show that for an irreducible permutation $π$ which is not a rotation, the set of $[λ]\in \mathbb{P}_+^{d-1}$ such that the interval exchange transformation $f([λ],π)$ is not weakly mixing does not have full Hausdorff dimension. We also obtain an analogous statement for translation flows. In particular, it strengthens the result of almost sure weak mixing proved by G. Forni and the first author. We adapt here the probabilistic argument developed in their paper in order to get some large deviation results. We then show how the latter can be converted into estimates o…

Fast Universal Algorithms for Robustness Analysis

In this paper, we develop efficient randomized algorithms for estimating probabilistic robustness margin and constructing robustness degradation curve for uncertain dynamic systems. One remarkable feature of these algorithms is their universal applicability to robustness analysis problems with arbitrary robustness requirements and uncertainty bounding set. In contrast to existing probabilistic methods, our approach does not depend on the feasibility of computing deterministic robustness margin. We have developed efficient methods such as probabilistic comparison, probabilistic bisection and ba…