Geometry, dynamics, and arithmetic of $S$-adic shifts

Simultaneous Diophantine Approximation in Non-degenerate p-adic Manifolds

S-arithmetic Khintchine-type theorem for products of non-degenerate analytic p-adic manifolds is proved for the convergence case. In the p-adic case the divergence part is also obtained.

算术动力学中的非常稀疏返回集

Very sparse return sets in arithmetic dynamics

本文研究动力莫德尔-朗猜想中返回集的稀疏性，探讨p进版本、复解析版本及后向轨道版本的类似现象。

格点在$S$-算术群中的流形作用

Actions of lattices in $p$-adic and $S$-arithmetic groups on manifolds

本文证明，当维数小于秩时，简单$p$-adic群中格点在紧流形上的$C^1$微分同胚作用是有限的，并将此结果推广到全不连通的$S$-算术群中的格点。

Microlocal lifts and quantum unique ergodicity on $GL(2,\mathbb{Q}_p)$

We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a repre…

Dynamics of the $p$-adic Shift and Applications

We present a novel way of realizing the Bernoulli shift on $p$ symbols on the $p$-adic integers, where $p$ is a prime. By showing that suitably small perturbations of the shift are still Bernoulli we find many "nice" maps, such as polynomials on $\mathbb Z_p$, that are isomorphic to the (noninvertible) Bernoulli shift.

$\mathcal{S}$-adic characterization of minimal ternary dendric shifts

Dendric shifts are defined by combinatorial restrictions of the extensions of the words in their languages. This family generalizes well-known families of shifts such as Sturmian shifts, Arnoux-Rauzy shifts and codings of interval exchange transformations. It is known that any minimal dendric shift has a primitive $\mathcal{S}$-adic representation where the morphisms in $\mathcal{S}$ are positive tame automorphisms of the free group generated by the alphabet. In this paper we investigate those $\mathcal{S}$-adic representations, heading towards an $\mathcal{S}$-adic characterization of this fa…