Modified Schmidt games and a conjecture of Margulis

We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.


General set-up.
Let (X, µ, F ) be a dynamical system, where • X is a metric space If µ is ergodic, almost all F -orbits are dense.
Specifically, for a subset Z of X, denote by E(F, Z) the set of points of X with F -orbits avoiding Z, i.e.
Also, if X is noncompact, consider All of these are sets of measure zero.

Examples.
If g t is an ergodic translation of a torus, then all orbits are dense ⇒ E(F, Z) = ∅ for all Z More generally, if X = G/Γ, where G is a connected Lie group and Γ ⊂ G a lattice, and g t is unipotent, it follows from the work of Ratner and Dani-Margulis that the set z∈X E(F, {x}) of all nondense orbits is a countable union of proper submanifolds of X (singular sets).
On the other hand, for partially hyperbolic flows the situation is different: the sets E(F, Z) are very big (although still of measure zero). Say that Y ⊂ X In what follows we will assume that g t is partially hyperbolic and let F = {g t | t ≥ 0}.
(i.e. there are many bounded orbits avoiding Z) Unfortunately, the methods of proof of called expanding horospherical with respect to F

(H-orbits = unstable leaves)
That is, it is proved that for any x ∈ X, the sets are thick in H.

6
Sketch of proof of Theorem 1.2. Fix z ∈ X and T > 0; then hx ∈ E(F, {z}) ⇐⇒ the trajectory Now one can choose a small enough ball V ⊂ H such that for any y ∈ X, the intersection of g T V g −T y with Z T consists of at most one point.
Then inside V one can construct a Cantor subset of large dimension whose points will avoid a small neighborhood of Z T . a neighborhood of a point, we need to avoid the complement of a big compact K ⊂ X.
and a ball V ⊂ H is not too small, then mixing of the g t -action on X will force most of g T V g −T y to come back to K, and a similar Cantor set construction can be carried out. W wins if the point of intersection is (α, β)-winning for all β > 0, and winning if it is α-winning for some α > 0. 10 The following results were proved by Schmidt: is a sequence of α-winning sets for some α ⇒ ∩ ∞ k=1 S k is also an α-winning set.
is thick.

11
Then in 1985 S.G. Dani proved Dani showed that whenever G has R-rank 1, for any x ∈ G/Γ, the set • the sets h ∈ H hx ∈ E(F, ∞) of bounded F -orbits are also winning, BUT the rules of the game in which they are winning have to be modified (adjusted according to F ).
[more difficult, by using a precise description of compact subsets of G/Γ instead of mixing]