On conjugacy between natural extensions of 1-dimensional maps

TL;DR

This paper demonstrates that for any nondegenerate dendrite, there exist topologically mixing maps whose natural extensions are conjugate, leading to homeomorphic inverse limits, including the pseudo-arc, extending previous work on dendrite maps.

Contribution

It proves the existence of conjugate natural extensions for dendrite maps and shows these can be chosen independently of the dendrite, extending to multiple dendrites.

Findings

Natural extensions of dendrite maps can be conjugate to those of interval maps.

Inverse limits of these maps can be homeomorphic to the pseudo-arc.

The results generalize to multiple dendrites.

Abstract

We prove that for any nondegenerate dendrite $D$ there exist topologically mixing maps $F : D \to D$ and $f : [0, 1] \to [0, 1]$, such that the natural extensions (aka shift homeomorphisms) $\sigma_F$ and $\sigma_f$ are conjugate, and consequently the corresponding inverse limits are homeomorphic. Moreover, the map $f$ does not depend on the dendrite $D$, and can be selected so that the inverse limit $\underleftarrow{\lim} (D, F)$ is homeomorphic to the pseudo-arc. The result extends to any finite number of dendrites. Our work is motivated by, but independent of, the recent result of the first and third author on conjugation of Lozi and H\'enon maps to natural extensions of dendrite maps.