报告人：Karoly Simon 教授（Budapest University of Technology and Economics）

时间： 2026年03月06日  9:30—11:45

           2026年03月13日  9:30—11:45

           2026年03月20日  9:30—11:45

地点：理科楼LA104

摘要： A self-affine Iterated Function System (IFS) is a finite list of contractions of the form $\mathcal{F}=\{f_i(x)=A_ix+t_i\}$, $x,t_i\in R^d$ and $A_i$ are invertible $d\times d$ matrices.  First, I give a general introduction to the classical results. Then I turn to recent developments in the case when we are on the plane, that is, below we always assume that  $d=2$.

We form a tuple $\mathcal{A}:=\{A_1,\dots,A_m\}$ from the lienar parts of the functions of $\mathcal{F}$. We say that this tuple  $\mathcal{A} $ is

-- $\mathcal{A} $ is   irreducible if there is no proper subspace of $R^2$, which is preserved by all matrices in  $\mathcal{A}$. Moreover,

-- $\mathcal{A} $ is  strongly irreducible if there is no finite collection $V_1,\dots,V_k$ of proper subspaces of $R^2$, whose union is preserved by all the matrices in $\mathcal{A}$.

I will mention the breakthrough results of the Bárány-Hochman-Rapaport and Hochman-Rapaport papers, which describe the dimension theory in the strongly irreducible case. Then I will review the dimension theory results in the reducible case. The main part of the mini-course will be a detailed analysis of the irreducible but not strongly irreducible situation, which is based on our very recent result, joint with D. Allen, A. Kaenmaki, D. Prokaj, and S. Trocheit.

邀请人：数学研究中心

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