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A dynamical system is called Devaney chaotic is it is (i) transitive, (ii) periodic points are dense, and (iii) the system depends sensitively on initial conditions.

My question is if Devaney chaos is maintained by topological semi-conjugacy.

More precisely:

Let $f\colon X\to X$ and $g\colon Y\to Y$ be continuous and $h\colon Y\to X$ be a continuous surjection such that $h\circ g=f\circ h$. Suppose that $g$ is Devaney chaotic on $Y$. Does it then follow that $f$ is also Devaney chaotic on $X$?

Mark McClure
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Salamo
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    I suppose the answer is yes. The key observation is that $$h\circ g = f\circ h\implies h\circ g^n = f^n\circ h.$$ Thus $h$ maps orbits of $g$ to orbits of $f$. The reverse, however, is not true, i.e. you can construct an example with $f$ chaotic but $g $ not chaotic. – Mark McClure Mar 04 '22 at 13:52
  • @MarkMcClure I am not sure if condition (iii) is preserved, see https://math.stackexchange.com/questions/380930/proving-semi-conjugacy-preserves-chaotic-behavior?rq=1 – Salamo Mar 05 '22 at 11:25
  • Condition (iii) is preserved, as it follows from (i) and (ii), see J. Banks, et.al.: On Devaney's Definition of Chaos, Amer. Math. Monthly 99 (1992), 332-334. – Gerd Mar 09 '22 at 16:43

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