报告人：Michal Rams教授（波兰科学院数学研究所）

报告题目一：Mass transference principle for sets of arbitrary shapes

报告摘要：In this talk I'll present a joint work with Henna Koivusalo, in which we generalized the Beresnevich and Velani's Mass Transference Principle. The MTP says that when we have a family of balls $(B_i)_i$ in the euclidean space such that their limsup set has full Lebesgue measure then we can give a lower bound for the Hausdorff dimension of the set $\limsup E_i$, where each $E_i$ is a ball with the same center as $B_i$ but with the radius $r(E_i)=r(B_i)^a$ for some fixed $a>1$. This result was first generalized by Wang, Wu, and Xu, who allowed the sets $E_i$ to be allipsoids. We can now give a bound when $E_i$ are arbitrary open sets contained in corresponding $B_i$'s.

报告时间：2019年11月13日（星期三）下午2:30-4:30

报告地点：科技楼南楼605

报告题目二：Birkhoff and Lyapunov spectra for planar affine iterated function systems

报告摘要：The Birkhoff spectrum is a calculation of the size (in the sense of Hausdorff dimension or topological entropy) of the set of points for which the Birkhoff average of some fixed potential $\varphi$ takes a prescribed value. The Lyapunov spectrum is an analogical calculation but for Lyapunov exponent instead of Birkhoff average. Both kinds of spectra are well understood in the conformal situation. However, for nonconformal systems we do not know much.

After a breakthrough result of Barany, Hochman, and Rapaport, who calculated the Hausdorff dimension of the limit set for a very general family of affine iterated function systems on a plane, we can now calculate the Birkhoff and Lyapunov spectra for such systems (except for some portion of the boundary). This is a joint work with Balazs Barany, Thomas Jordan, and Antti Kaenmaki.