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Analytic uniquely ergodic volume preserving maps on odd spheres. (English) Zbl 1336.37021

The aim of the paper under review is to construct examples of volume-preserving uniquely ergodic real-analytic diffeomorphisms on spheres of odd dimension. The construction is based on the so-called Anosov-Katok approximation-by-conjugation method.
More in details, let \(\mathbb S^{2n-1}\) be the Euclidean sphere. Let denote by \(C^\omega_\Delta\) the set of maps on \(\mathbb S^{2n-1}\) which extend holomorphically in the ball \(\{z\in \mathbb C^{2n}: \|z\|\leq \Delta\}\), \(\Delta>0\). A diffeomorphism \(f\) of the sphere is \(C^\omega_\Delta\) if all coordinates functions of both \(f\) and \(f^{-1}\) are \(C^\omega_\Delta\). If one can take \(\Delta=\infty\), the diffeomorphism \(f\) is called entire. Entire diffeomorphisms form a group, and its subgroup of entire diffeomorphisms preserving the Lebesgue measure \(\lambda\) is denoted by \(\text{End}(\mathbb S^{2n-1}, \lambda)\). A diffeomorphism of \(\mathbb S^{2n-1}\) is uniquely ergodic if it has only one invariant Borel probability measure. Finally, \(\mathbb S^{2n-1}\) admits a linear action of \(\mathbb S^1\) defined by the vector field whose expression in Euclidean coordinates is \(v(x_1,\ldots, x_{2n})=2\pi(x_2,-x_1,\ldots, x_{2n},-x_{2n-1})\). The flow of such an action is denoted by \((\varphi_t)\).
The main result of the paper is the following. For any \(t\in [0,1]\) and any \(\epsilon>0\) there exists a uniquely ergodic volume-preserving diffeomorphism \(f\in C^\omega_\Delta(\mathbb S^{2n-1})\) such that \(\|f-\varphi_t\|_\Delta<\epsilon\). Moreover, the diffeomorphism \(f\) is obtained as a limit (in the \(C^\omega_\Delta\)-norm) of entire maps of the form \(F_n=H_n\circ \varphi_{t_n}\circ H_n^{-1}\), with \(H_n\in \text{End}(\mathbb S^{2n-1}, \lambda)\).
Such a result extends to compact connected Lie groups with little modification.

MSC:

37C40 Smooth ergodic theory, invariant measures for smooth dynamical systems
32C99 Analytic spaces
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[1] D. V. Anosov, Existence of smooth ergodic flows on smooth manifolds.Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 518–545 (in Russian).Zbl 0314.58014 MR 0358863
[2] D. V. Anosov and A. Katok, New examples in smooth ergodic theory. Ergodic diffeomorphisms.Trans. Moscow Math. Soc. 23 (1970), 1–35.Zbl 0255.58007 MR 0370662 · Zbl 0255.58007
[3] A. Avìla, B. Fayad and A. Kocsard, On manifolds supporting distributionally uniquely ergodic diffeomorphisms.J. Differential Geometry, to appear; preprint,arXiv:1211.1519. · Zbl 1316.37015
[4] M. Brin, J. Feldman and A. Katok, Bernoulli diffeomorphisms and group extensions of dynamical systems with non-zero characteristic exponents.Ann. of Math. 113 (1981), 159–179.Zbl 0477.58021 MR 0604045 · Zbl 0477.58021
[5] B. Fayad, Analytic mixing reparametrizations of irrational flows.Ergodic Theory Dynam. Systems 22 (2002), 437–468.Zbl 1136.37307 MR 1898799 · Zbl 1136.37307
[6] B. Fayad andA. Katok, Constructions in elliptic dynamics.Ergodic Theory Dynam. Systems (Herman memorial issue)24 (2004), 1477–1520.Zbl 1089.37012 MR 2104594 · Zbl 1089.37012
[7] B. Fayad, M. Saprykina and A. Windsor, Non-standard smooth realizations of Liouville rotations.Ergodic Theory Dynam. Systems 27 (2007), no. 6, 1803–1818.Zbl 1127.37008 MR 2371596 · Zbl 1127.37008
[8] M. Gerber, Conditional stability and real analytic pseudo-Anosov maps.Mem. Amer. Math. Soc. 54 (1985), no. 321.Zbl 0572.58018 MR 0783000 · Zbl 0572.58018
[9] A. Katok, Ergodic perturbations of degenerate integrable Hamiltonian systems.Izv. Akad. Nauk. SSSR Ser. Math. 37 (1973), 539–576.Zbl 03495405 MR 0331425
[10] A. Katok, Bernoulli diffeomorphisms on surfaces,Ann. of Math. 110, (1979), 529–547. Zbl 0435.58021 MR 0554383 · Zbl 0435.58021
[11] A. Katok, Smooth non-Bernoulli K-automorphisms.Invent. Math. 61 (1980), 291–300. Zbl 0467.58016 MR 0592695
[12] J. Lewowitz and E. Lima de Sá, Analytic models of pseudo-Anosov maps.Ergodic Theory Dynam. Systems 6 (1986), 385–392.Zbl 0608.58035 MR 0863201
[13] D. Rudolph, Asymptotically Brownian skew products give non-loosely Bernoulli Kautomorphisms.Invent. Math. 91 (1988), 105–128.Zbl 0655.58034 MR 0918238 Received December 19, 2012 Bassam Fayad, Institut de Mathématiques de Jussieu, Université Paris Diderot-Université Pierre et Marie Curie, 75205 Paris Cedex 13, France E-mail: bassam@math.jussieu.fr Anatole Katok, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, U.S.A. E-mail: katok_a@math.psu.edu · Zbl 0655.58034
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