Manfred Einsiedler
Badly approximable systems of affine forms, fractals, and Schmidt games.
Tue, 12/15/2009 - 16:01 — SomNambulAuthors: Manfred Einsiedler, Jimmy Tseng
Subjects: Dynamical Systems
AbstractA badly approximable system of affine forms is determined by a matrix and a
vector. We show Kleinbock's conjecture for badly approximable systems of affine
forms: for any fixed vector, the set of badly approximable systems of affine
forms is winning (in the sense of Schmidt games) even when restricted to a
fractal (from a certain large class of fractals). In addition, we consider
fixing the matrix instead of the vector where an analog statement holds.Entropy and escape of mass for $SL(3,Z)\SL(3,R)$.
Thu, 12/03/2009 - 16:01 — SomNambulAuthors: Manfred Einsiedler, Shirali Kadyrov
Subjects: Dynamical Systems
AbstractWe study the relation between measure theoretic entropy and escape of mass
for the case of a singular diagonal flow on the moduli space of
three-dimensional unimodular lattices.Diophantine approximations on fractals.
Tue, 08/18/2009 - 11:53 — SomNambulAuthors: Manfred Einsiedler, Lior Fishman, Uri Shapira
Subjects: Dynamical Systems
AbstractWe exploit dynamical properties of diagonal actions to derive results in
Diophantine approximations. In particular, we prove that the continued fraction
expansion of almost any point on the middle third Cantor set (with respect to
the natural measure) contains all finite patterns (hence is well approximable).
Similarly, we show that for a variety of fractals in [0,1]^2, possessing some
symmetry, almost any point is not Dirichlet improvable (hence is well
approximable) and has property C (after Cassels). We then settle by similar
methods a conjecture of M.
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