Itamar Bellaïche | Searcharxiv

arXiv2026

Determining the Winner in Alternating-Move Games

We provide a criterion for determining the winner in two-player win-lose alternating-move games on trees, in terms of the Hausdorff dimension of the target set. We focus our study on special cases, including the Gale-Stewart game on the complete binary tree and a family of Schmidt games, generalizing a result of Schmidt from Hilbert spaces to arbitrary complete metric spaces. Building on the Hausdorff dimension games originally introduced by Das, Fishman, Simmons, and Urbański, which provide a game-theoretic approach for computing Hausdorff dimensions, we employ a generalized family of these games to obtain lower bounds on the Hausdorff dimensions of target sets whenever Player I can guarantee a win.

math.DS↗

arXiv2024

Ultrametrizable spaces are homeomorphic to clade spaces of pruned trees

This paper demonstrates that every ultrametric space is homeomorphic to a clade space of a pruned tree, i.e., a subspace of a tree's canopy. Furthermore, it characterizes several topological properties of ultrametrizable spaces through the features of their representing trees. This approach suggests that topological properties of ultrametrizable spaces should be studies via the study of naturally ordered pruned trees.

math.GN↗