# Rearrangements with supporting Trees, Isomorphisms and Combinatorics of coloured dyadic Intervals

## TL;DR

This paper characterizes a class of rearrangements with supporting trees, demonstrating their boundedness and isomorphic properties on large subspaces of L^p, and explores combinatorial strategies involving colored dyadic intervals.

## Contribution

It introduces a new class of rearrangements with supporting trees and analyzes their boundedness and isomorphic behavior on L^p spaces, along with combinatorial game strategies.

## Key findings

- Rearrangements with supporting trees have bounded vector-valued extensions.
- Large subspaces of L^p are isomorphic under these rearrangements.
- A combinatorial game with colored dyadic intervals is analyzed for strategic outcomes.

## Abstract

We determine a class of rearrangements that admit a supporting tree. This condition implies that the associated rearrangement operator has a bounded vector valued extension. We show that there exists a large subspace of $L^p$ on which a bounded rearrangement operator acts as an isomorphism. The combinatorial issues of these problems give rise to a two-person game, to be played with colored dyadic intervals. We determine winning strategies for each of the players.