Bryant Rosado Silva

Generalized Ważewski dendrites, generic subcontinua, and generic chains.

We say that a continuum $X$ is a hereditarily equivalent continuum if every non-degenerate subcontinuum of it is homeomorphic to $X$ . This concept is one of the main motivations behind the construction of the pseudo-arc. If considered in the hyperspace of continua of $X$ , denoted by $\operatorname{Cont}(X)$ , it means that

$\operatorname{Cont}(X) \setminus \operatorname{Fin}(X) = \{K \in \operatorname{Cont}(X) \ | \ K \simeq X\}.$

This is an open and dense set, hence comeager. Therefore, it is natural to ask if there exist other spaces that satisfy this weaker property of having such a collection of homeomorphic sets being comeager. We call these spaces generically hereditarily equivalent continua and have shown that the generalized Ważewski dendrites $W_M$ for $M \subseteq \{3, 4,\ldots, \infty\}$ are such spaces. Moreover, in the hyperspace of maximal order arcs of $W_M$ – $\operatorname{MOA}(W_M)$ , the chains having every non-degenerate element homeomorphic to $W_M$ make a comeager subset of the maximal order arcs.

Additionally, we consider the action of the homeomorphism group $\operatorname{Homeo}(W_M)$ on $\operatorname{Cont}(X)$ and on $\operatorname{MOA}(W_M)$ and describe a comeager orbit for each of them.

Algebra, Topology, and Dynamical Systems

A conference in honor of Dikran Dikranjan on the occasion of his 75th birthday and of Ivan Prodanov on the occasion of the 90th anniversary of his birth

Bryant Rosado Silva