April 2018, 38(4): 1615-1655. doi: 10.3934/dcds.2018067

Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior

1. 

University of Education Vorarlberg, Liechtensteinerstrasse 33 - 37, 6800 Feldkirch, Austria

2. 

University of Hamburg, Department of Mathematics, Bundesstrasse 55, 20146 Hamburg, Germany

Received  November 2015 Revised  October 2017 Published  January 2018

We show that on any smooth compact connected manifold of dimension $m≥2$ admitting a smooth non-trivial circle action $\mathcal{S} = \left\{S_t\right\}_{t ∈ \mathbb{R}}$, $S_{t+1}=S_t$, the set of weakly mixing $C^{∞}$-diffeomorphisms which preserve both a smooth volume $ν$ and a measurable Riemannian metric is dense in ${{\mathcal{A}}_{\alpha }}\left( M \right)={{\overline{\left\{ h\circ {{S}_{\alpha }}\circ {{h}^{-1}}:h\in \text{Dif}{{\text{f}}^{\infty }}\left( M,\nu \right) \right\}}}^{{{C}^{\infty }}}}$ for every Liouville number $α$. The proof is based on a quantitative version of the approximation by conjugation-method with explicitly constructed conjugation maps and partitions.

Citation: Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1615-1655. doi: 10.3934/dcds.2018067
References:
[1]

D. V. Anosov and A. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obsc., 23 (1970), 3-36.

[2]

M. Benhenda, Non-standard smooth realization of translations on the torus, J. Mod. Dyn., 7 (2013), 329-367. doi: 10.3934/jmd.2013.7.329.

[3]

R. Berndt, Einführung in Die Symplektische Geometrie, Vieweg, Braunschweig [u.a.], 1998.

[4]

G. M. Constantine and T. H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520. doi: 10.1090/S0002-9947-96-01501-2.

[5]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520. doi: 10.1017/S0143385703000798.

[6]

B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Scient. Ecole. Norm. Sup., 42 (2009), 193-219. doi: 10.24033/asens.2093.

[7]

B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Scient. Ecole. Norm. Sup.(4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.

[8]

B. FayadM. Saprykina and A. Windsor, Nonstandard smooth realizations of Liouville rotations, Ergodic Theory Dynam. Systems, 27 (2007), 1803-1818.

[9]

R. Gunesch and A. Katok, Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure, Discrete Contin. Dynam. Systems, 6 (2000), 61-88.

[10]

P. R. Halmos, Lectures on Ergodic Theory, Japan Math Soc., Tokyo, 1956.

[11]

P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.

[12]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge [u.a.], 1995.

[13]

B. Hasselblatt and A. Katok, Principal Structures In: B. Hasselblatt und A. Katok (Editors) Handbook of Dynamical Systems, Band 1A. Elsevier, Amsterdam [u.a.], 2002.

[14]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, American Math. Soc., Providence, 1997.

[15]

P. Kunde, Real-analytic weak mixing diffeomorphism preserving a measurable Riemannian metric, Ergodic Theory & Dynam. Systems 37, no. 37 (2017), 1547-1569. doi: 10.1017/etds.2015.125.

[16]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.

[17]

H. Omori, Infinite Dimensional Lie Transformation Groups, Springer, Berlin [u.a.], 1974.

[18]

B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.

[19]

M. D. Sklover, Classical dynamical systems on the torus with continuous spectrum, Izv. Vys. Ucebn. Zaved. Mat., 1967 (1967), 113-124.

show all references

References:
[1]

D. V. Anosov and A. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms, Trudy Moskov. Mat. Obsc., 23 (1970), 3-36.

[2]

M. Benhenda, Non-standard smooth realization of translations on the torus, J. Mod. Dyn., 7 (2013), 329-367. doi: 10.3934/jmd.2013.7.329.

[3]

R. Berndt, Einführung in Die Symplektische Geometrie, Vieweg, Braunschweig [u.a.], 1998.

[4]

G. M. Constantine and T. H. Savits, A multivariate Faa di Bruno formula with applications, Trans. Amer. Math. Soc., 348 (1996), 503-520. doi: 10.1090/S0002-9947-96-01501-2.

[5]

B. Fayad and A. Katok, Constructions in elliptic dynamics, Ergodic Theory Dynam. Systems, 24 (2004), 1477-1520. doi: 10.1017/S0143385703000798.

[6]

B. Fayad and R. Krikorian, Herman's last geometric theorem, Ann. Scient. Ecole. Norm. Sup., 42 (2009), 193-219. doi: 10.24033/asens.2093.

[7]

B. Fayad and M. Saprykina, Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary, Ann. Scient. Ecole. Norm. Sup.(4), 38 (2005), 339-364. doi: 10.1016/j.ansens.2005.03.004.

[8]

B. FayadM. Saprykina and A. Windsor, Nonstandard smooth realizations of Liouville rotations, Ergodic Theory Dynam. Systems, 27 (2007), 1803-1818.

[9]

R. Gunesch and A. Katok, Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure, Discrete Contin. Dynam. Systems, 6 (2000), 61-88.

[10]

P. R. Halmos, Lectures on Ergodic Theory, Japan Math Soc., Tokyo, 1956.

[11]

P. R. Halmos, Measure Theory, D. Van Nostrand Company, Inc., New York, N. Y., 1950.

[12]

B. Hasselblatt and A. Katok, Introduction to the Modern Theory of Dynamical Systems, Cambridge University Press, Cambridge [u.a.], 1995.

[13]

B. Hasselblatt and A. Katok, Principal Structures In: B. Hasselblatt und A. Katok (Editors) Handbook of Dynamical Systems, Band 1A. Elsevier, Amsterdam [u.a.], 2002.

[14]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, American Math. Soc., Providence, 1997.

[15]

P. Kunde, Real-analytic weak mixing diffeomorphism preserving a measurable Riemannian metric, Ergodic Theory & Dynam. Systems 37, no. 37 (2017), 1547-1569. doi: 10.1017/etds.2015.125.

[16]

J. Moser, On the volume elements on a manifold, Trans. Amer. Math. Soc., 120 (1965), 286-294. doi: 10.1090/S0002-9947-1965-0182927-5.

[17]

H. Omori, Infinite Dimensional Lie Transformation Groups, Springer, Berlin [u.a.], 1974.

[18]

B. O'Neill, Semi-Riemannian Geometry, Academic Press, New York, 1983.

[19]

M. D. Sklover, Classical dynamical systems on the torus with continuous spectrum, Izv. Vys. Ucebn. Zaved. Mat., 1967 (1967), 113-124.

Figure 1.  The action of the map $g_{\varepsilon}$.
Figure 2.  The action of the map $g_{a,b,\varepsilon}$.
Figure 3.  The map $\psi_\mu$ has the useful property of rotating several small cuboids individually while being the identity outside of a neighborhood of them.
Figure 4.  The map $\phi_n$ is constructed as concatenation of a stretch map $C_\lambda$, a rotation $\varphi$, the map $\psi_\mu$ mentioned before, and $C_\lambda^{-1}$ (the inverse of the stretch map). The map thus constructed has the very useful property of stretching a cuboid (illustrated here by the underlying grey rectangle) in one direction (similar to what a hyperbolic map would do), yet it is almost an isometry on all of the smaller cuboids (illustrated here by black squares with letters). In particular, a partition element $\hat{I} \in \eta_n$ (the leftmost grey rectangle) is mapped to a set that has size almost 1 in one of its coordinates.
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