arXiv subjects

Michael Chow

Publications and source records attributed to Michael Chow.

8 recordsLinked to original sources

arXiv2024

Exponential prime orbit theorems for Anosov subgroups

Let $\Gamma$ be a Zariski dense Anosov subgroup of a connected semisimple real algebraic group -- these are higher rank analogues of convex cocompact subgroups. Let us measure the Jordan projections with any linear form which is positive on the limit cone of $\Gamma$. We prove a corresponding counting theorem with a power saving error term for the conjugacy classes of loxodromic elements in $\Gamma$. The proof is based on interpreting the Jordan projections as periods of a natural flow associated to $\Gamma$ and proving exponential mixing. We also prove the existence of a spectral gap for the Selberg zeta function.

math.DS↗

arXiv2024

Multiple correlations of spectra for higher rank Anosov representations

We describe multiple correlations of Jordan and Cartan spectra for any finite number of Anosov representations of a finitely generated group. This extends our previous work on correlations of length and displacement spectra for rank one convex cocompact representations. Examples include correlations of the Hilbert length spectra for convex projective structures on a closed surface as well as correlations of eigenvalue gaps and singular value gaps for Hitchin representations. We relate the correlation problem to the counting problem for Jordan and Cartan projections of an Anosov subgroup with respect to a family of carefully chosen truncated {\it hypertubes}, rather than in tubes as in our previous work. Hypertubes go to infinity in a linear subspace of directions, while tubes go to infinity in a single direction and this feature presents a novel difficulty in this higher rank correlation problem.

math.DS↗

arXiv2023

Jordan and Cartan spectra in higher rank with applications to correlations

For a given $d$-tuple $\rho=(\rho_1,\dots,\rho_d):\Gamma \to G$ of faithful Zariski dense convex cocompact representations of a finitely generated group $\Gamma$, we study the correlations of length spectra $\{\ell_{\rho_i(\gamma)}\}_{[\gamma]\in[\Gamma]}$ and correlations of displacement spectra $\{\mathsf{d}(\rho_i(\gamma)o,o)\}_{\gamma\in\Gamma}$. We prove that for any interior vector $\mathsf v=(v_1,\dots,v_d)$ in the {\it{spectrum cone}}, there exists $\delta_\rho(\mathsf v) > 0$ such that for any $\varepsilon_1, \dots, \varepsilon_d>0$, there exist $c_1,c_2> 0$ such that \begin{align*} &\#\{[\gamma]\in [\Gamma]: v_iT \le \ell_{\rho_i(\gamma)} \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_1 \frac{e^{\delta_\rho (\mathsf{v})T}}{ T^{{(d+1)}/{2}}};\\ &\#\{\gamma\in \Gamma: v_iT \le \mathsf{d}(\rho_i(\gamma)o,o) \le v_i T+\varepsilon_i, \;1 \le i \le d \} \sim c_2 \frac{e^{\delta_\rho (\mathsf v)T}}{ T^{{(d-1)}/{2}}}. \end{align*} We deduce this result as a special case of our main theorem on the distribution of Jordan projections with holonomies and Cartan projections {\it{in tubes}} of an Anosov subgroup $\Gamma$ of a semisimple real algebraic group $G$. We also show that the growth indicator of $\Gamma$ remains the same when we use Jordan projections instead of Cartan projections and tubes instead of cones, except possibly on the boundary of the limit cone.

math.GT↗

arXiv2023

Joint equidistribution of maximal flat cylinders and holonomies for Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $P<G$ be a minimal parabolic subgroup with Langlands decomposition $P=MAN$. Let $\Gamma < G$ be a Zariski dense Anosov subgroup with respect to $P$. Since $\Gamma$ is Anosov, the set of conjugacy classes of primitive elements of $\Gamma$ is in one-to-one correspondence with the set of (positively oriented) maximal flat cylinders in $\Gamma\backslash G/M$. We describe the joint equidistribution of maximal flat cylinders and their holonomies as their circumferences tend to infinity. This result can be viewed as the Anosov analogue of the joint equidistribution result of closed geodesics and holonomies in rank one by Margulis--Mohammadi--Oh.

math.DS↗

arXiv2022

Exponential mixing of frame flows for convex cocompact locally symmetric spaces

Let $G$ be a connected center-free simple real algebraic group of rank one and $\Gamma < G$ be a Zariski dense torsion-free convex cocompact subgroup. We prove that the frame flow on $\Gamma \backslash G$, i.e., the right translation action of a one-parameter subgroup $\{a_t\}_{t \in \mathbb R} < G$ of semisimple elements, is exponentially mixing with respect to the Bowen-Margulis-Sullivan measure. The key step is proving suitable generalizations of the local non-integrability condition and the non-concentration property which are essential for Dolgopyat's method. This generalizes the work of Sarkar-Winter for $G = \operatorname{SO}(n, 1)^\circ$ and also strengthens the mixing result of Winter in the convex cocompact case.

math.DS↗

arXiv2021

Local mixing of one-parameter diagonal flows on Anosov homogeneous spaces

Let $G$ be a connected semisimple real algebraic group and $\Gamma < G$ be a Zariski dense Anosov subgroup with respect to a minimal parabolic subgroup. We prove local mixing of the one-parameter diagonal flow $\{\exp(t\mathsf{v}) : t \in \mathbb R\}$ on $\Gamma \backslash G$ for any interior direction $\mathsf{v}$ of the limit cone of $\Gamma$ with respect to the Bowen--Margulis--Sullivan measure associated to $\mathsf{v}$. More generally, we allow a class of deviations to this flow along a direction $\mathsf{u}$ in some fixed subspace transverse to $\mathsf{v}$. We also obtain a uniform bound for the correlation function which decays exponentially in $\|\mathsf{u}\|^2$. The precise form of the result is required for several applications such as the asymptotic formula for the decay of matrix coefficients in $L^2(\Gamma \backslash G)$ proved by Edwards--Lee--Oh.

math.DS↗

arXiv2021

Parallel Scale-wise Attention Network for Effective Scene Text Recognition

The paper proposes a new text recognition network for scene-text images. Many state-of-the-art methods employ the attention mechanism either in the text encoder or decoder for the text alignment. Although the encoder-based attention yields promising results, these schemes inherit noticeable limitations. They perform the feature extraction (FE) and visual attention (VA) sequentially, which bounds the attention mechanism to rely only on the FE final single-scale output. Moreover, the utilization of the attention process is limited by only applying it directly to the single scale feature-maps. To address these issues, we propose a new multi-scale and encoder-based attention network for text recognition that performs the multi-scale FE and VA in parallel. The multi-scale channels also undergo regular fusion with each other to develop the coordinated knowledge together. Quantitative evaluation and robustness analysis on the standard benchmarks demonstrate that the proposed network outperforms the state-of-the-art in most cases.

cs.CV↗

arXiv2017

Composite quasianalytic functions

We prove two main results on Denjoy-Carleman classes: (1) a composite function theorem which asserts that a function f(x) in a quasianalytic Denjoy-Carleman class Q, which is formally composite with a generically submersive mapping y=h(x) of class Q, at a single given point in the source (or in the target) of h, can be written locally as f(x) = g(h(x)), where g(y) belongs to a shifted Denjoy-Carleman class Q' ; (2) a statement on a similar loss of regularity for functions definable in the o-minimal structure given by expansion of the real field by restricted functions of quasianalytic class Q. Both results depend on an estimate for the regularity of an infinitely differentiable solution g of the equation f(x) = g(h(x)), with f and h as above. The composite function result depends also on a quasianalytic continuation theorem, which shows that the formal assumption at a given point in (1) propagates to a formal composition condition at every point in a neighbourhood.

math.CV↗