On conjugacy between natural extensions of 1-dimensional maps

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 $\underset{}{\lim }(D,F)$ 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.