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Let $\phi(x)$ be continuous and differentiable everywhere and its automorphisms be bijections $\gamma_{n}$ such that $\phi(\gamma_{n}(x))=\phi(x)$. If the order of $Aut(\phi)$ is infinite, would that mean that $\phi(x)$ is periodic?

I can think of an example of the top of my head: $\phi(x)=sin(x)$ with automorphisms of the form $\gamma_{n}(x)=x+2\pi n$. Could this be applied to any general function $\phi$?

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  • $\begingroup$ What is, in this case, an automorphism of $\phi$? A diffeomorphism $\gamma$ of the domain of $\phi$ such that $\phi \circ \gamma = \phi$? $\endgroup$ – Daniel Fischer Nov 7 '17 at 16:08
  • $\begingroup$ Yes, the pdf that im reading describes it as a bijection $\alpha$ such that $\phi(\alpha x)=\phi(x)$ $\endgroup$ – aleden Nov 7 '17 at 16:09
  • $\begingroup$ If there are no continuity requirements on $\alpha$, it would suffice that some value is attained by $\phi$ infinitely often. $\endgroup$ – Daniel Fischer Nov 7 '17 at 16:11
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    $\begingroup$ If we don't have any further condition on $\gamma$, then it's wrong. Let $\phi : \mathbb{R} \to \mathbb{R}, x \mapsto x^2$. Then for any $a \in \mathbb{R}$ the function $\gamma: \mathbb{R} \to \mathbb{R}, \gamma(x) = -x$ for $x \in \{a, -a\}$ and $\gamma(x) = x$ for $x \notin \{a, -a\}$ is such an automorphism. $\endgroup$ – S. M. Roch Nov 7 '17 at 16:12
  • $\begingroup$ How would $\phi(x)=x^2$ have an infinite amount of automorphisms? $\endgroup$ – aleden Nov 7 '17 at 16:17
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If we don't have any further condition on $\gamma$, then it's wrong. Let $\phi : \mathbb{R} \to \mathbb{R}, x \mapsto x^2$. Then for every $a > 0$ the function $$\gamma: \mathbb{R} \to \mathbb{R},\hspace{3mm} x \mapsto \begin{cases}-a & x \in \{a, -a\} \\ x & x \notin \{a, -a\}\end{cases}$$

is such an automorphism. Hence we have infinitely many automorphisms. But $\phi$ isn't periodic, so this is a counterexample.

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  • $\begingroup$ Just wondering, if we restrict the bijections to a certain form would it be possible to prove periodicity? $\endgroup$ – aleden Nov 7 '17 at 16:29

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