1 Bumps, blocks, and bubbles
Bubbles
Our goal here is to define the bubble relation on a linear order. Let \(X\) be a linear order.
Let \(f\colon X \to X\) an automorphism. For any \(x\in X\), the orbit of \(x\) under \(f\) is the set \(\{ f^n(x) : n \in \mathbb {Z}\} \).
Let \(f\colon X \to X\) an automorphism. For any \(x \in X\), the orbital of \(x\) under \(f\) is the convex closure of the orbit of \(x\) under \(f\).
A bump is an automorphism \(f\colon X \to X\) with exactly one non-singleton orbital.
A bounded bump is a bump whose unique non-singleton orbital is either strictly bounded below or strictly bounded above.
We define a relation \(\sim _b\) on \(X\) as follows: \(x \sim _b y\) if and only if there exists a bounded bump such that \(x\) and \(y\) are in its orbital or \(x = y\).
The relation \(\sim _b\) is reflexive and symmetric.
The definition is clearly reflexive and the definition is symmetric in \(x\) and \(y\).
The relation \(\sim _b\) is convex.
Let \(x\in X\). We want to show that the \(\sim _b\)-equivalence class of \(x\) is an interval. Let \(y\) and \(z\) be elements of \(X\) such that \(x {\lt} y {\lt} z\) and \(x \sim _b z\). It cannot be that \(x = z\) and so we must have that \(x,z \in O(f)\) for some bounded bump \(f\colon X \to X\). Since \(O(f)\) is an interval and \(x {\lt} y {\lt} z\), we have that \(y \in O(f)\). Thus, \(x\sim _b y\) and so we are done.
The relation \(\sim _b\) is transitive.
Let \(x\), \(y\), and \(z\) be elements of \(X\) such that \(x \sim _b y\) and \(y \sim _b z\). If \(x = y\) or \(y =z\), then we are done.