Bounded orbits of certain diagonalizable flows on $SL_{n}(R)/SL_{n}(Z)$

Abstract

Abstract. We prove that the set of points that have bounded orbits under certain diagonalizable flows is a hyperplane absolute winning subset of SLn(R)/SLn(Z).

The topological entropy of non-dense orbits and generalized Schmidt games

“…by (16) and (18). This implies that B u (z jr +1 , n jr +1 , ) intersects at most one I k by the choice of c and L. So there are at most (a + b)r many I k with different k satisfying (18) and (19).…”

“…by (16). This implies that f k (B u (z 1 , n 1 , )) intersects at most one connected component of W u ( f k (z 1 )) ∩ (c) by the choice of c and L. In other words, for each 0 ≤ k < (a + b)r , there is at most one I k intersecting with B u (z 1 , n 1 , ).…”

“…The tree-like family will be constructed by considering possible moves for Bob at each turn and Alice's corresponding countermoves in generalized Schmidt games. Bob may begin the game by choosing a Bowen ball B u (z 1 , n 1 , ) with n 1 satisfying (16). Alice can choose a Bowen ball B u (z 1 , n 1 , ) ⊂ B u (z 1 , n 1 , ).…”

“…There is a well-known connection between Diophantine approximation and the bounded orbits of non-quasiunipotent flows on homogeneous spaces (see, e.g., [18]). From the dynamical point of view, there are also many applications of Schmidt games to the study of bounded orbits in homogeneous dynamics or related dynamics (see, e.g., [4,5,11,16,21] and many others). Recently, Kleinbock and Weiss developed a type of modified Schmidt game in [20] and utilized it to Non-dense orbits and generalized Schmidt games 501 prove a conjecture of Margulis concerning the Hausdorff dimension of the bounded orbits of non-quasiunipotent flows on homogeneous spaces in [21], which strengthens the results in [19].…”

Nondense orbits on homogeneous spaces and applications to geometry and number theory

“…Later this property was upgraded by Broderick, Fishman and Simmons [5] to an even stronger hyperplane absolute winning property (abbreviated as HAW). See [4,25], as well as §2.1, for definitions and discussion, and [24,20,2,36,39,40,18,1,15] for other recent results involving winning properties of exceptional sets in dynamical systems. We point out that one of the important advantages of this strengthening is the fact that a countable intersection of winning (resp., HAW) sets is also winning (resp., HAW).…”