Beyond $0$ and $\mathrm{\infty }$: A solution to the Barge Entropy Conjecture

We prove the entropy conjecture of M. Barge from 1989: for every $r\in [0,\mathrm{\infty }]$ there exists a pseudo-arc homeomorphism $h$, whose topological entropy is $r$. Until now all pseudo-arc homeomorphisms with known entropy have had entropy $0$ or $\mathrm{\infty }$.