ISSN 1842-6298 (electronic), 1843-7265 (print)
Volume 20 (2025), 375 -- 380
This work is licensed under a Creative Commons Attribution 4.0 International License.
ON THE USE OF THE SCHWARZIAN DERIVATIVE IN REAL ONE-DIMENSIONAL DYNAMICS
Felipe Correa and Bernardo San Martín
Abstract. In the study of properties within one-dimensional dynamics, the negative Schwarzian derivative condition has been shown to be very useful. However, this condition may seem somewhat arbitrary, as it is not inherently a dynamical condition, except for the fact that it is preserved under iteration. In this brief work, we show that the negative Schwarzian derivative condition is not arbitrary in any sense but is instead strictly related to the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which plays a key role in the proof of Singer’s Theorem.
2020 Mathematics Subject Classification: 3702, 37C25, 37C75
Keywords: Schwarzian derivative; Minimum Principle; One-dimensional dynamics.
References
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D. Singer, Stable orbits and bifurcation of maps of the interval, SIAM J. Appl. Math. 35 (1978), 260--267. MR494306. Zbl 0391.58014.
Felipe Correa
Departamento de Matemáticas, Universidad de Antofagasta.
Av. Angamos 601, Antofagasta, Chile.
e-mail: felipe.correa@uantof.cl
Bernardo San Martín
Departamento de Matemáticas, Universidad Católica del Norte.
Av. Angamos 0610, Antofagasta, Chile.
e-mail: sanmarti@ucn.cl
https://www.utgjiu.ro/math/sma