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Consider the tent map defined on $[0, 1]$: $$f(x) = 2x, \; 0<x<\frac{1}{2} \;\; and \;\; f(x) = 2(1-x), \; \frac{1}{2} \leq x \leq 1.$$

I showed that $f$ is topological conjugate with $F(x) = 4x(1-x)$.

Are there other quadratic functions $F_\alpha(x) = \alpha x(1-x)$ for which $f$ is topological conjugate?.

Can someone give me a hint?

Thank you!

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  • $\begingroup$ Hint: presumably $f$ is onto, although you mention no domain... $\endgroup$ – John B Nov 23 '16 at 0:16
  • $\begingroup$ $f$ and $F_\alpha$ are both onto. I don't understand how to conclude if they are topologically conjugate or not (when $\alpha \neq 4$). Can you give me more details? Thank you! $\endgroup$ – g.pomegranate Nov 23 '16 at 16:02
  • $\begingroup$ If if you have two topologically conjugate maps, both are onto or both are not onto. Do you see why? Just look at the conjugacy. That's why $\alpha=4$ is special (for your question). $\endgroup$ – John B Nov 23 '16 at 16:46

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