scholarly journals Badziahin-Pollington-Velani's theorem and Schmidt's game

2013 ◽  
Vol 45 (4) ◽  
pp. 721-733 ◽  
Author(s):  
Jinpeng An
Keyword(s):  
Schmidt’S Game

Schmidt’s game on fractals

2009 ◽  
Vol 171 (1) ◽  
pp. 77-92 ◽  
Author(s):  
Lior Fishman
Keyword(s):  
Schmidt’S Game

scholarly journals SCHMIDT'S GAME ON HAUSDORFF METRIC AND FUNCTION SPACES: GENERIC DIMENSION OF SETS AND IMAGES

Mathematika ◽  
2020 ◽  
Vol 67 (1) ◽  
pp. 196-213
Author(s):  
Ábel Farkas ◽  
Jonathan M. Fraser ◽  
Erez Nesharim ◽  
David Simmons
Keyword(s):  
Hausdorff Metric ◽  
Function Spaces ◽  
And Function ◽  
Schmidt’S Game

scholarly journals Schmidt’s game, fractals, and orbits of toral endomorphisms

2010 ◽  
Vol 31 (4) ◽  
pp. 1095-1107 ◽  
Author(s):  
RYAN BRODERICK ◽  
LIOR FISHMAN ◽  
DMITRY KLEINBOCK
Keyword(s):  
Hausdorff Dimension ◽  
Recent Result ◽  
Lacunary Sequence ◽  
Alternative Proof ◽  
Integer Matrix ◽  
Image Position ◽  
Schmidt’S Game ◽  
Badly Approximable

AbstractGiven an integer matrix M∈GLn(ℝ) and a point y∈ℝn/ℤn, consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y∈ℚn/ℤn, the set $ \tilde E(M,y)$ has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M∈GLn(ℝ)∩Mn×n(ℤ) and y∈ℝn/ℤn, and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m×n matrices. Furthermore, we show that sets of the form $ \tilde E(M,y)$ and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝn. As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint, arXiv:0912.2445] on badly approximable systems of affine forms.

scholarly journals Schmidt’s game and nonuniformly expanding interval maps

Nonlinearity ◽  
2020 ◽  
Vol 33 (11) ◽  
pp. 5611-5628
Author(s):  
Jason Duvall
Keyword(s):  
Interval Maps ◽  
Schmidt’S Game

scholarly journals Schmidt’s game, fractals, and numbers normal to no base

2010 ◽  
Vol 17 (2) ◽  
pp. 307-321 ◽  
Author(s):  
Ryan Broderick ◽  
Yann Bugeaud ◽  
Lior Fishman ◽  
Dmitry Kleinbock ◽  
Barak Weiss
Keyword(s):  
Schmidt’S Game

scholarly journals Schmidt games and non-dense forward orbits of certain partially hyperbolic systems

2015 ◽  
Vol 36 (5) ◽  
pp. 1656-1678 ◽  
Author(s):  
WEISHENG WU
Keyword(s):  
Hausdorff Dimension ◽  
Hyperbolic Systems ◽  
Countable Subset ◽  
Open Set ◽  
Forward Orbit ◽  
Winning Set ◽  
Partially Hyperbolic ◽  
Schmidt’S Game ◽  
Set Of Points ◽  
Partially Hyperbolic Diffeomorphism

Let$f:M\rightarrow M$be a partially hyperbolic diffeomorphism with conformality on unstable manifolds. Consider a set of points with non-dense forward orbit:$E(f,y):=\{z\in M:y\notin \overline{\{f^{k}(z),k\in \mathbb{N}\}}\}$for some$y\in M$. Define$E_{x}(f,y):=E(f,y)\cap W^{u}(x)$for any$x\in M$. Following a method of Broderick, Fishman and Kleinbock [Schmidt’s game, fractals, and orbits of toral endomorphisms.Ergod. Th. & Dynam. Sys.31(2011), 1095–1107], we show that$E_{x}(f,y)$is a winning set for Schmidt games played on$W^{u}(x)$which implies that$E_{x}(f,y)$has Hausdorff dimension equal to$\dim W^{u}(x)$. Furthermore, we show that for any non-empty open set$V\subset M$,$E(f,y)\cap V$has full Hausdorff dimension equal to$\dim M$, by constructing measures supported on$E(f,y)\cap V$with lower pointwise dimension converging to$\dim M$and with conditional measures supported on$E_{x}(f,y)\cap V$. The results can be extended to the set of points with forward orbit staying away from a countable subset of$M$.

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