Homeomorphisms of the Cantor set (“dust”) play a fundamental role in topology, dynamical systems, and descriptive set theory, where they are studied from different perspectives. Recently, various properties of so-called fence-like objects have attracted attention. These include the Lelek fan (from topology), the hairy Cantor set and Cantor bouquet (from dynamical systems), and the Fraïssé fence (from model theory). Several recent works investigate both the structure of these spaces and the dynamics of homeomorphisms defined on them.

In this work, we develop a general technique that allows one to transfer—or lift—the dynamics of a given homeomorphism of the Cantor set to a homeomorphism of a fence of the types described above.