Abstract:
As we have proved in [11], the geodesic flows associated with the flat metrics on T2 minimize the polynomial entropy hpol. In this paper, we show that, among the geodesic flows that are Bott integrable and dynamically coherent, the geodesic flows associated with flat metrics are local strict minima for hpol. To this aim, we prove a graph property for invariant Lagrangian tori in near-integrable systems.
\Bibitem{Lab12}
\by Cl\'emence Labrousse
\paper Flat Metrics are Strict Local Minimizers for the Polynomial Entropy
\jour Regul. Chaotic Dyn.
\yr 2012
\vol 17
\issue 6
\pages 479--491
\mathnet{http://mi.mathnet.ru/rcd416}
\crossref{https://doi.org/10.1134/S1560354712060019}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3001095}
\zmath{https://zbmath.org/?q=an:1264.53077}
\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2012RCD....17..479L}
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This publication is cited in the following 20 articles:
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Bin Zhang, Deyu Meng, Dongmei Peng, Junjie Zhang, “Billingsley type theorem of Bowen polynomial entropy for fixed-point free flows”, Asian-European J. Math., 17:11 (2024)
Lei Liu, Xiao Yao Zhou, “Polynomial Entropy of Amenable Group Actions for Noncompact Sets”, Acta. Math. Sin.-English Ser., 39:7 (2023), 1351
Maša Ɖorić, Jelena Katić, “Polynomial Entropy of Induced Maps of Circle and Interval Homeomorphisms”, Qual. Theory Dyn. Syst., 22:3 (2023)
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Lei Liu, Cao Zhao, “Polynomial entropy of nonautonomous dynamical systems for noncompact sets”, Journal of Mathematical Analysis and Applications, 509:2 (2022), 125974
Liu L., Zhao C., “Polynomial Entropy of Subsets For Free Semigroup Actions”, J. Dyn. Control Syst., 2021
Peric M., “Polynomial Entropy of the Logistic Map”, Stud. Sci. Math. Hung., 58:2 (2021), 206–215
Fan Yu.-W., Fu L., Ouchi G., “Categorical Polynomial Entropy”, Adv. Math., 383 (2021), 107655
Cantat S., Paris-Romaskevich O., “Automorphisms of Compact Kahler Manifolds With Slow Dynamics”, Trans. Am. Math. Soc., 374:2 (2021), 1351–1389
Katic J., Peric M., “On the Polynomial Entropy For Morse Gradient Systems”, Math. Slovaca, 69:3 (2019), 611–624
Marco J.-P., “Entropy of Billiard Maps and a Dynamical Version of the Birkhoff Conjecture”, J. Geom. Phys., 124 (2018), 413–420
A. Artigue, D. Carrasco-Olivera, I. Monteverde, “Polynomial entropy and expansivity”, Acta Math. Hungar., 152:1 (2017), 140
Patrick Bernard, Clémence Labrousse, “An entropic characterization of the flat metrics on the two torus”, Geom Dedicata, 180:1 (2016), 187
Yun Zhao, Yakov Pesin, “Scaled Entropy for Dynamical Systems”, J Stat Phys, 158:2 (2015), 447
Stefan Rosemann, Konrad Schöbel, “Open problems in the theory of finite-dimensional integrable systems and related fields”, Journal of Geometry and Physics, 87 (2015), 396
Urs Frauenfelder, Clémence Labrousse, Felix Schlenk, “Slow volume growth for Reeb flows on spherizations and contact Bott–Samelson theorems”, J. Topol. Anal., 07:03 (2015), 407
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