-
Previous Article
Stable ergodicity for partially hyperbolic attractors with negative central exponents
- JMD Home
- This Issue
-
Next Article
On the spectrum of a large subgroup of a semisimple group
Dirichlet's theorem on diophantine approximation and homogeneous flows
1. | Goldsmith 207, Brandeis University, Waltham, MA 02454-9110 |
2. | Ben Gurion University, Be'er Sheva, 84105, Israel |
$|Y_i\q - p_i| < \varepsilon e^{-t_i}\,$ $i = 1,\ldots, m$
$|q_j| < \varepsilon e^{t_{m+j}}\,$ $j = 1,\ldots, n$
(here $Y_1,\ldots,Y_m$ are rows of $Y$). We show that for any $\varepsilon<1$ and any $\mathcal{T}$ 'drifting away from walls', see (1.8), Dirichlet's Theorem cannot be $\epsilon$-improved along $\mathcal{T}$ for Lebesgue almost every $Y$. In the case $m = 1$ we also show that for a large class of measures $\mu$ (introduced in [14]) there is $\varepsilon_0>0$ such that for any drifting away from walls unbounded $\mathcal{T}$, any $\varepsilon<\varepsilon_0$, and for $\mu$-almost every $Y$, Dirichlet's Theorem cannot be $\varepsilon$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals.
Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\SL_{m+n}(\mathbb R)$/$SL_{m+n}(\mathbb Z)$.
[1] |
Michael Schmidt, Emmanuel Trélat. Controllability of couette flows. Communications on Pure & Applied Analysis, 2006, 5 (1) : 201-211. doi: 10.3934/cpaa.2006.5.201 |
[2] |
Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems - A, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597 |
[3] |
Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597 |
[4] |
Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 |
[5] |
Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399 |
[6] |
Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192 |
[7] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[8] |
Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 |
[9] |
Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119 |
[10] |
Xue-Ping Luo, Yi-Bin Xiao, Wei Li. Strict feasibility of variational inclusion problems in reflexive Banach spaces. Journal of Industrial & Management Optimization, 2020, 16 (5) : 2495-2502. doi: 10.3934/jimo.2019065 |
[11] |
Charles Amorim, Miguel Loayza, Marko A. Rojas-Medar. The nonstationary flows of micropolar fluids with thermal convection: An iterative approach. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2509-2535. doi: 10.3934/dcdsb.2020193 |
[12] |
Zhigang Pan, Chanh Kieu, Quan Wang. Hopf bifurcations and transitions of two-dimensional Quasi-Geostrophic flows. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021025 |
[13] |
Misha Bialy, Andrey E. Mironov. Rich quasi-linear system for integrable geodesic flows on 2-torus. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 81-90. doi: 10.3934/dcds.2011.29.81 |
[14] |
A. Kochergin. Well-approximable angles and mixing for flows on T^2 with nonsingular fixed points. Electronic Research Announcements, 2004, 10: 113-121. |
[15] |
Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 |
[16] |
Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115 |
2019 Impact Factor: 0.465
Tools
Metrics
Other articles
by authors
[Back to Top]