Title: Limit theorems for time dependent expanding dynamical systems

Speaker: Yeor Hafuta – Ohio State University

Abstract: Some of the results like the Berry-Esseen theorem and moderate deviations principle hold true for general sequences of maps when the variance of the underlying partial sums grows faster than n^{2/3}, while other results such as the local central limit theorem hold true for certain classes of random not necessarily stationary transformations. The results also include a certain type of stability theorem in a complex version of the sequential Rulle-Perron-Frobenius theorem, which yields that the variance grows linearly fast when the underlying maps are close enough to a single expanding map.

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Title: On the dimension drop conjecture for diagonal flows on the space of lattices

Speaker: Shahriar Mirzadeh – Michigan State University

Abstract: (see attached pdf for better formatting): Consider the set of points in a homogeneous space X=G/Γ whose gt-orbit misses a fixed open set. It has measure zero if the flow is ergodic. It has been conjectured that this set has Hausdorff dimension strictly smaller than the dimension of X. This conjecture is proved when X is compact or when has real rank 1.

In this talk we will prove the conjecture for probably the most important example of the higher rank case namely: G=SLm+n(R), Γ=SLm+n(Z)and gt=\diag(et/m,⋯,et/m,e−t/n,⋯,e−t/n) We can also use our main result to produce new applications to Diophantine approximation. This project is joint work with Dmitry Kleinbock.

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Pdf of slides available here