View Abstract
Abstract:Â
We will define and discuss the concept of \textit{group of rigidity} (associated with a collection of finitely many sequences). As we will see, groups of rigidity play an instrumental role in answering questions stemming from the theory of generic Lebesgue preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, and spectral theory. A simple statement which epitomizes the type of results that one can obtain with the help of groups of rigidity is the following:\\
For any $(b_1,...,b_\ell)\in \mathbb N^\ell$ one has that there is no vector $(a_1,...,a_\ell)\in \mathbb Z^\ell$ orthogonal to $(b_1,...,b_\ell)$ with some  $|a_j|=1$ if and only if there is an increasing sequence $(n_k)_{k\in \mathbb N}$ in $\mathbb N$ with the property that  for every $F\subseteq \{1,...,\ell\}$ there is a $\mu$-preserving transformation $T_F:[0,1]\rightarrow [0,1]$ ($\mu$ denotes the Lebesgue measure) such thatÂ
$$
\lim_{k\rightarrow\infty}\mu(A\cap T_F^{-b_jn_k}B)=\begin{cases}
\mu(A\cap B),\,\text{ if }j\in F,\\
\mu(A)\mu(B),\,\text{ if }j\not\in F,
  \end{cases}
$$
for every pair of measurable sets $A,B\subseteq [0,1]$. Part of this talk is based
on joint work with Vitaly Bergelson.