Figure 1.
The action of the map $g_{\varepsilon}$ .
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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 |
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.
Keywords:
Smooth ergodic theory,
conjugation-approximation-method,
almost isometries,
weakly mixing diffeomorphisms,
measurable Riemannian metric.
Mathematics Subject Classification:
Primary: 37A05; Secondary: 37C40, 57R50, 53C99.
Citation:
Roland Gunesch, Philipp Kunde. Weakly mixing diffeomorphisms preserving a measurable Riemannian metric with prescribed Liouville rotation behavior. Discrete and Continuous Dynamical Systems,
2018, 38
(4)
: 1615-1655.
doi: 10.3934/dcds.2018067
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References:
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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. |
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R. Berndt, Einführung in Die Symplektische Geometrie, Vieweg, Braunschweig [u.a.], 1998. |
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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. |
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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. |
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B. Fayad, M. 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. |
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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. Fayad, M. 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 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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