On zero entropy homeomorphisms of the pseudo-arc

TL;DR

This paper investigates zero entropy interval maps called crooked maps, showing the existence of uncountably many non-conjugate examples with various fixed point configurations, and characterizes their relation to the identity.

Contribution

It demonstrates the abundance and diversity of zero entropy crooked maps, providing classifications based on fixed points and their relation to the identity.

Findings

Uncountably many non-conjugate zero entropy crooked maps with different fixed point sets.

Existence of uncountably many zero entropy crooked maps with exactly two fixed points.

Characterization of crooked maps relative to the identity diagonal.

Abstract

In this paper we study interval maps $f$ with zero topological entropy that are crooked; i.e. whose inverse limit with $f$ as the single bonding map is the pseudo-arc. We show that there are uncountably many pairwise non-conjugate zero entropy crooked interval maps with different sets of fixed points. We also show that there are uncountably many zero entropy crooked maps that are pairwise non-conjugate and have exactly two fixed points. Furthermore, we provide a characterization of crooked interval maps that are under or above the identity diagonal.