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Real-analytic AbC constructions on the torus

Published online by Cambridge University Press:  25 January 2018

SHILPAK BANERJEE  and
PHILIPP KUNDE
SHILPAK BANERJEE
Affiliation:
The Pennsylvania State University, Department of Mathematics, University Park, PA 16802, USA email banerjee.shilpak@gmail.com
PHILIPP KUNDE
Affiliation:
University of Hamburg, Mathematics, Bundesstrasse 55, Hamburg 20146, Germany email philipp.kunde@math.uni-hamburg.de
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Abstract

In this article we demonstrate a way to extend the AbC (approximation by conjugation) method invented by Anosov and Katok from the smooth category to the category of real-analytic diffeomorphisms on the torus. We present a general framework for such constructions and prove several results. In particular, we construct minimal but not uniquely ergodic diffeomorphisms and non-standard real-analytic realizations of toral translations.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 10 , October 2019 , pp. 2643 - 2688
Copyright
© Cambridge University Press, 2018 

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References

Anosov, D. V. and Katok, A.. New examples in smooth ergodic theory. Trans. Moscow Math. Soc. 23 (1970), 1–35.Google Scholar
Banerjee, S.. Symbolic representation of real-analytic AbC systems. Preprint, 2017, arXiv:1705.05079.Google Scholar
Banerjee, S.. Non-standard real-analytic realization of some rotations of the circle. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1369–1386.Google Scholar
Benhenda, M.. Non-standard smooth realization of shifts on the torus. J. Mod. Dyn. 7(3) (2013), 329–367.Google Scholar
Bramham, B.. Pseudo-rotations with sufficiently Liouvillean rotation number are C 0 -rigid. Invent. Math. 2 (2015), 561–580.Google Scholar
Fathi, A. and Herman, M. R.. Existence de difféomorphismes minimaux. Dynamical Systems (Warsaw, 1977), Vol. I (Astérisque, 49) . Société Mathématique de France, Paris, 1977, pp. 37–59.Google Scholar
Fayad, B. R.. Analytic mixing reparametrizations of irrational flows. Ergod. Th. & Dynam. Sys. 22(2) (2002), 437–468.Google Scholar
Fayad, B. R. and Katok, A.. Constructions in elliptic dynamics. Ergod. Th. & Dynam. Sys. 24(5) (2004), 1477–1520.Google Scholar
Fayad, B. R. and Katok, A.. Analytic uniquely ergodic volume preserving maps on odd spheres. Comment. Math. Helv. 89(4) (2014), 963–977.Google Scholar
Fayad, B. R. and Saprykina, M.. Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary. Ann. Sci. Éc. Norm. Supér. (4) 38(3) (2005), 339–364.Google Scholar
Fayad, B. R., Saprykina, M. and Windsor, A.. Non-standard smooth realizations of Liouville rotations. Ergod. Th. & Dynam. Sys. 27(6) (2007), 1803–1818.Google Scholar
Furstenberg, H.. Strict ergodicity and transformation of the torus. Amer. J. Math. 83 (1961), 573–601.Google Scholar
Gunesch, R. and Katok, A.. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete Contin. Dyn. Syst. 6(1) (2000), 61–88.Google Scholar
Gunesch, R. and Kunde, P.. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouvillean rotation behavior. Discrete Contin. Dyn. Syst. A 38(4) (2018), 1615–1655.Google Scholar
Hasselblatt, B. and Katok, A.. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, Cambridge, 1995.Google Scholar
Katok, A.. Ergodic perturbations of degenerate integrable Hamiltonian systems. Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576 (in Russian).Google Scholar
Kolmogorov, A. N.. Théorie générale des systèmes dynamiques et mécanique classique. Proc. Int. Congr. Mathematicians (Amsterdam, 1954). Vol. 1. Eds. Erven, P. and Noordhoff, N. V.. North-Holland, Groningen, 1957, pp. 315–333 (in French).Google Scholar
Kunde, P.. Weakly mixing diffeomorphisms with ergodic derivative extension. Preprint.Google Scholar
Kunde, P.. Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions. J. Mod. Dyn. 10 (2016), 439–481.Google Scholar
Kunde, P.. Real-analytic weak mixing diffeomorphisms preserving a measurable Riemannian metric. Ergod. Th. & Dynam. Sys. 37(5) (2017), 1547–1569.Google Scholar
Nemytskii, V. V. and Stepanov, V. V.. Qualitative Theory of Differential Equations. Princeton University Press, Princeton, NJ, 1960.Google Scholar
Petersen, K.. Ergodic Theory. Cambridge University Press, Cambridge, 1983.Google Scholar
Saprykina, M.. Analytic non-linearizable uniquely ergodic diffeomorphisms on 𝕋2 . Ergod. Th. & Dynam. Sys. 23(3) (2003), 935–955.Google Scholar
Walters, P.. Ergodic Theory – Introductory Lectures. Springer, Berlin, 1975.Google Scholar
Windsor, A.. Minimal but not uniquely ergodic diffeomorphisms. Smooth Ergodic Theory and its Applications. American Mathematical Society, Providence, RI, 2001.Google Scholar