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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: $f_{a,b}(x,y):ℝ²→ℝ²$ where $a$ and $b$ are real parameters. 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 $h_{a,b}$. When $a=u(b)$ assuming that the map $f_{u(b),b}$ is conjugate to another continuous map $w_{a,b}$. Assuming also that the distance between the maps $h_{a,b}$ and $w_{a,b}$ is very small, i.e., $h_{a,b}$ is a very small perturbation (in the values of the parameter $b$) of the map $w_{a,b}$. 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 semiconjugate or conjugate.

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  • $\begingroup$ Are $a, b$ both indexed by $\mathbb{R}$? Do $h, w$ depend on $a, b$? $\endgroup$ – user44191 Feb 19 at 16:06
  • $\begingroup$ @user44191: Yes. Corrected. $\endgroup$ – China Feb 19 at 16:08

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