# Continuous self-maps of the plane are semiconjugate or conjugate?

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 $$a then the map $$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.

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