For any finite horizon Sinai billiard map
 Award ID(s):
 1607260
 NSFPAR ID:
 10324455
 Date Published:
 Journal Name:
 Journal für die reine und angewandte Mathematik (Crelles Journal)
 Volume:
 2020
 Issue:
 767
 ISSN:
 00754102
 Page Range / eLocation ID:
 1 to 16
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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on the twotorus, we find\begin{document}$ T $\end{document} such that for each\begin{document}$ t_*>1 $\end{document} there exists a unique equilibrium state\begin{document}$ t\in (0,t_*) $\end{document} for\begin{document}$ \mu_t $\end{document} , and\begin{document}$  t\log J^uT $\end{document} is\begin{document}$ \mu_t $\end{document} adapted. (In particular, the SRB measure is the unique equilibrium state for\begin{document}$ T $\end{document} .) We show that\begin{document}$  \log J^uT $\end{document} is exponentially mixing for Hölder observables, and the pressure function\begin{document}$ \mu_t $\end{document} is analytic on\begin{document}$ P(t) = \sup_\mu \{h_\mu \int t\log J^uT d \mu\} $\end{document} . In addition,\begin{document}$ (0,t_*) $\end{document} is strictly convex if and only if\begin{document}$ P(t) $\end{document} is not\begin{document}$ \log J^uT $\end{document} a.e. cohomologous to a constant, while, if there exist\begin{document}$ \mu_t $\end{document} with\begin{document}$ t_a\ne t_b $\end{document} , then\begin{document}$ \mu_{t_a} = \mu_{t_b} $\end{document} is affine on\begin{document}$ P(t) $\end{document} . An additional sparse recurrence condition gives\begin{document}$ (0,t_*) $\end{document} .\begin{document}$ \lim_{t\downarrow 0} P(t) = P(0) $\end{document} 
Abstract Let
denote the matrix multiplication tensor (and write$M_{\langle \mathbf {u},\mathbf {v},\mathbf {w}\rangle }\in \mathbb C^{\mathbf {u}\mathbf {v}}{\mathord { \otimes } } \mathbb C^{\mathbf {v}\mathbf {w}}{\mathord { \otimes } } \mathbb C^{\mathbf {w}\mathbf {u}}$ ), and let$M_{\langle \mathbf {n} \rangle }=M_{\langle \mathbf {n},\mathbf {n},\mathbf {n}\rangle }$ denote the determinant polynomial considered as a tensor. For a tensor$\operatorname {det}_3\in (\mathbb C^9)^{{\mathord { \otimes } } 3}$ T , let denote its border rank. We (i) give the first handcheckable algebraic proof that$\underline {\mathbf {R}}(T)$ , (ii) prove$\underline {\mathbf {R}}(M_{\langle 2\rangle })=7$ and$\underline {\mathbf {R}}(M_{\langle 223\rangle })=10$ , where previously the only nontrivial matrix multiplication tensor whose border rank had been determined was$\underline {\mathbf {R}}(M_{\langle 233\rangle })=14$ , (iii) prove$M_{\langle 2\rangle }$ , (iv) prove$\underline {\mathbf {R}}( M_{\langle 3\rangle })\geq 17$ , improving the previous lower bound of$\underline {\mathbf {R}}(\operatorname {det}_3)=17$ , (v) prove$12$ for all$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.32\mathbf {n}$ , where previously only$\mathbf {n}\geq 25$ was known, as well as lower bounds for$\underline {\mathbf {R}}(M_{\langle 2\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1$ , and (vi) prove$4\leq \mathbf {n}\leq 25$ for all$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+1.6\mathbf {n}$ , where previously only$\mathbf {n} \ge 18$ was known. The last two results are significant for two reasons: (i) they are essentially the first nontrivial lower bounds for tensors in an “unbalanced” ambient space and (ii) they demonstrate that the methods we use (border apolarity) may be applied to sequences of tensors.$\underline {\mathbf {R}}(M_{\langle 3\mathbf {n}\mathbf {n}\rangle })\geq \mathbf {n}^2+2$ The methods used to obtain the results are new and “nonnatural” in the sense of Razborov and Rudich, in that the results are obtained via an algorithm that cannot be effectively applied to generic tensors. We utilize a new technique, called
border apolarity developed by Buczyńska and Buczyński in the general context of toric varieties. We apply this technique to develop an algorithm that, given a tensorT and an integerr , in a finite number of steps, either outputs that there is no border rankr decomposition forT or produces a list of all normalized ideals which could potentially result from a border rank decomposition. The algorithm is effectively implementable whenT has a large symmetry group, in which case it outputs potential decompositions in a natural normal form. The algorithm is based on algebraic geometry and representation theory. 
Abstract We obtain new quantitative estimates on Weyl Law remainders under dynamical assumptions on the geodesic flow. On a smooth compact Riemannian manifold ( M , g ) of dimension n , let $$\Pi _\lambda $$ Π λ denote the kernel of the spectral projector for the Laplacian, $$\mathbb {1}_{[0,\lambda ^2]}(\Delta _g)$$ 1 [ 0 , λ 2 ] (  Δ g ) . Assuming only that the set of near periodic geodesics over $${W}\subset M$$ W ⊂ M has small measure, we prove that as $$\lambda \rightarrow \infty $$ λ → ∞ $$\begin{aligned} \int _{{W}} \Pi _\lambda (x,x)dx=(2\pi )^{n}{{\,\textrm{vol}\,}}_{_{{\mathbb {R}}^n}}\!(B){{\,\textrm{vol}\,}}_g({W})\,\lambda ^n+O\Big (\frac{\lambda ^{n1}}{\log \lambda }\Big ), \end{aligned}$$ ∫ W Π λ ( x , x ) d x = ( 2 π )  n vol R n ( B ) vol g ( W ) λ n + O ( λ n  1 log λ ) , where B is the unit ball. One consequence of this result is that the improved remainder holds on all product manifolds, in particular giving improved estimates for the eigenvalue counting function in the product setup. Our results also include logarithmic gains on asymptotics for the offdiagonal spectral projector $$\Pi _\lambda (x,y)$$ Π λ ( x , y ) under the assumption that the set of geodesics that pass near both x and y has small measure, and quantitative improvements for Kuznecov sums under nonlooping type assumptions. The key technique used in our study of the spectral projector is that of geodesic beams.more » « less

Abstract A subset E of a metric space X is said to be starlikeequivalent if it has a neighbourhood which is mapped homeomorphically into $\mathbb{R}^n$ for some n , sending E to a starlike set. A subset $E\subset X$ is said to be recursively starlikeequivalent if it can be expressed as a finite nested union of closed subsets $\{E_i\}_{i=0}^{N+1}$ such that $E_{i}/E_{i+1}\subset X/E_{i+1}$ is starlikeequivalent for each i and $E_{N+1}$ is a point. A decomposition $\mathcal{D}$ of a metric space X is said to be recursively starlikeequivalent, if there exists $N\geq 0$ such that each element of $\mathcal{D}$ is recursively starlikeequivalent of filtration length N . We prove that any null, recursively starlikeequivalent decomposition $\mathcal{D}$ of a compact metric space X shrinks, that is, the quotient map $X\to X/\mathcal{D}$ is the limit of a sequence of homeomorphisms. This is a strong generalisation of results of Denman–Starbird and Freedman and is applicable to the proof of Freedman’s celebrated disc embedding theorem. The latter leads to a multitude of foundational results for topological 4manifolds, including the fourdimensional Poincaré conjecture.more » « less

null (Ed.)Abstract The classical Aronszajn–Donoghue theorem states that for a rankone perturbation of a selfadjoint operator (by a cyclic vector) the singular parts of the spectral measures of the original and perturbed operators are mutually singular. As simple direct sum type examples show, this result does not hold for finite rank perturbations. However, the set of exceptional perturbations is pretty small. Namely, for a family of rank $d$ perturbations $A_{\boldsymbol{\alpha }}:= A + {\textbf{B}} {\boldsymbol{\alpha }} {\textbf{B}}^*$, ${\textbf{B}}:{\mathbb C}^d\to{{\mathcal{H}}}$, with ${\operatorname{Ran}}{\textbf{B}}$ being cyclic for $A$, parametrized by $d\times d$ Hermitian matrices ${\boldsymbol{\alpha }}$, the singular parts of the spectral measures of $A$ and $A_{\boldsymbol{\alpha }}$ are mutually singular for all ${\boldsymbol{\alpha }}$ except for a small exceptional set $E$. It was shown earlier by the 1st two authors, see [4], that $E$ is a subset of measure zero of the space $\textbf{H}(d)$ of $d\times d$ Hermitian matrices. In this paper, we show that the set $E$ has small Hausdorff dimension, $\dim E \le \dim \textbf{H}(d)1 = d^21$.more » « less