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Solving open problems in arithmetic dynamicsPartial resolution of Morton-Vivaldi’s conjecture 
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Chaotic behavior such as that observed in weather systems can be studied by approximating time as a discrete variable. Parabolic parameters are key quantities that govern chaos in such discretized settings; they are roots of polynomials with rational coefficients known as Delta factors. Recent progress on the number-theoretic properties of Delta factors is presented in this work. Except for trivial cases, it has long remained unresolved whether Delta factors become polynomials with integer coefficients after specific transformations, or whether they admit nontrivial factorizations. By uncovering a new connection between parabolic parameters and Euler’s totient function from number theory, this study provides a breakthrough on these problems. The research area linking complex dynamics and arithmetic properties of numbers is known as arithmetic dynamics and has developed rapidly over the past two decades. We aim to further elucidate number-theoretic phenomena arising in dynamical systems through a bidirectional use of methods and insights from both fields.
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[2] V. Huguin, “Unicritical Polynomial Maps with Rational Multipliers,” Conformal Geometry and Dynamics, Volume 25, pp. 79-87, 2021.
[3] J. Koizumi, Y. Murakami, K. Sano, K. Takehira, “Irreducibility of polynomials defining parabolic parameters of period 3,” Acta. Arithmetica, Volume 221, pp. 253-270, 2025.
[4] P. Morton, F. Vivaldi, “Bifurcations and discriminants for polynomial maps,” Nonlinearity 8, No.4, pp. 571-584, 1995.
[5] Y. Murakami, K. Sano, K. Takehira, “Arithmetic properties of multiplier polynomials for certain polynomial maps,” preprint, arXiv:2403.17315v3, 2024, under review.
Kaoru Sano, NTT Institute for Fundamental Mathematics/ Computing Theory Research Group, Media Information Laboratory