Randomized trial assesses orbit frequency of polynomial iterations on the unit circle, suggesting implications for rational functions.
We use a recent result of Orevkov and Pakovich [‘On intersection of lemniscates of rational functions’, Arnold J. Math. 11 (2025), 7–26] to obtain a bound on the frequency with which elements in an orbit of polynomial iterations can fall on the unit circle.
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Igor E. Shparlinski (2026) studied this question.