Dynamical Systems Seminar

In 1987 Del Junco introduced the notion of topological minimal self-joinings as a topological analog of minimal self-joinings from measurable dynamics. An infinite dynamical system (X,T) is said to be an $n$-fold topological minimal self-joining (TMSJ) if every $n$-tuple of points lying on different orbits is dense in $X^n$ for the diagonal action. Crucially, and in sharp contrast to the measurable analog, King showed in 1990 that there are no 4-fold TMSJs. In this presentation we will generalize Del Junco's definition, and find a bound for TMSJs on $\mathbb{Z}^d$ and the discrete Heisenberg group. To arrive at this result, we will look at the space of horoballs of groups, non-deterministic directions, and doubly minimal systems. This is joint work with Sebastián Donoso and Samuel Petite.