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The critical function for the semistandard map. (English) Zbl 0997.37500

Summary: For the semistandard map \(F(x,y)=(x+y+ie^{ix}, y+ie^{ix})\), we consider the critical function \(K_{\text{ss}}(\omega)\), defined as the radius of convergence of a series expansion of a complex invariant curve of rotation number \(\omega\), and show that \(\log K_{\text{ss}}(\omega)+2\sum q^{-1}_k\log q_{k+1}\) is bounded on the set of \(\omega\) where it is well defined, where \(\{q_k\}\) are the denominators of the convergents to the real number \(\omega\). We discuss the implications for critical functions for the standard map.

MSC:

37C55 Periodic and quasi-periodic flows and diffeomorphisms
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