Consider the standard map. Might it happen that for some nonzero parameter value $K$ and for some positive integer $q$ that there exist an infinite number of periodic orbits having period $q $ I would also be interested in the same question for "generic" twist maps. (I was motivated to ask this question by reading the section on zeta functions from Smale's famous "Differentiable Dynamical Systems" which was reprinted in the most recent issue of the Bulletin of the AMS.)
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asked Jun 30, 2018 at 22:49
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$\begingroup$ Isn’t the map (and its iterates) real analytic? This should prevent accumulation of fixed points of a single power, I think. $\endgroup$– Anthony QuasCommented Jul 1, 2018 at 0:30
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$\begingroup$ A real analytic map F of a real analytic surface can have a curve's worth of fixed points, or points of any given period $q$. Indeed the standard map for K = 0 has this property for every period $q$, including $q=1$. But your question is nice. I guess that there are three possibilities for the solution set to $F^q (p) = p$: a finite set (possibly empty) , an analytic curve (possiby singular), or the whole surface. $\endgroup$– Richard MontgomeryCommented Jul 1, 2018 at 22:45
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