# Natural extensions of unimodal maps: virtual sphere homeomorphisms and prime ends of basin boundaries

## TL;DR

This paper characterizes the prime ends of natural extensions of unimodal maps with high entropy and constructs sphere homeomorphisms that serve as virtually sphere homeomorphisms for these extensions, especially for tent maps.

## Contribution

It provides a complete description of prime ends for natural extensions of unimodal maps and constructs associated sphere homeomorphisms, including generalized pseudo-Anosov maps for post-critically finite cases.

## Key findings

- Prime ends of Barge-Martin embeddings are fully described.
- Constructed sphere homeomorphisms are factors of natural extensions with controlled fibers.
- Identified generalized pseudo-Anosov maps for dense parameter sets in tent maps.

## Abstract

Let $\{f_t\colon I\to I\}$ be a family of unimodal maps with topological entropies $h(f_t)>\frac12\log 2$, and ${\widehat{f}}_t\colon{\widehat{I}}_t\to{\widehat{I}}_t$ be their natural extensions, where ${\widehat{I}}_t=\varprojlim(I,f_t)$. Subject to some regularity conditions, which are satisfied by tent maps and quadratic maps, we give a complete description of the prime ends of the Barge-Martin embeddings of ${\widehat{I}}_t$ into the sphere. We also construct a family $\{\chi_t\colon S^2\to S^2\}$ of sphere homeomorphisms with the property that each $\chi_t$ is a factor of ${\widehat{f}}_t$, by a semi-conjugacy for which all fibers except one contain at most three points, and for which the exceptional fiber carries no topological entropy: that is, unimodal natural extensions are virtually sphere homeomorphisms. In the case where $\{f_t\}$ is the tent family, we show that $\chi_t$ is a generalized pseudo-Anosov map for the dense set of parameters for which $f_t$ is post-critically finite, so that $\{\chi_{t}\}$ is the completion of the unimodal generalized pseudo-Anosov family introduced in [21].