Let $f : X → X$ and $g : Y → Y$ be continuous functions. We say that $f$ and $g$ are topologically conjugate if there exists a homeomorphism $α : X → Y$ such that $$f∘α=α∘g$$.

A related idea is the notion of topological semi-conjugacy. We say that $f$ is topologically semi-conjugate to $g$ if there exists a continuous, surjective function $α$ such that $$f∘α=α∘g$$

Consider the family of maps: $f_{a,b}(x,y)$. Assuming that there exist a continuous function $u$ such that if $a<u(b)$ then the map $f_{a,b}$ is semiconjugate to another continuous map $h$. When $a=u(b)$ assuming that the map $f_{u(b),b}$ is conjugate to another continuous map $w$. Assuming also that the distance between the maps $h$ and $w$ is very small, i.e., $h$ is a very small perturbation (in the values of the parameter $b$) of the map $w$. Or one can assume that $u(b)-a$ is very small. Assuming also, that all these maps have finite topological entropy.

My question is: Can we deduce that the maps $f_{a,b}$ and $f_{u(b),b}$ are semicongugate or conjugate.