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Journal of Modern Dynamics

2013 , Volume 7 , Issue 3

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Nonstandard smooth realization of translations on the torus
Mostapha Benhenda
2013, 7(3): 329-367 doi: 10.3934/jmd.2013.7.329 +[Abstract](240) +[PDF](583.5KB)
Abstract:
Let $M$ be a smooth compact connected manifold of dimension greater than two, on which there exists a free (modulo zero) smooth circle action that preserves a positive smooth volume. In this article, we construct volume-preserving diffeomorphisms on $M$ that are metrically isomorphic to ergodic translations on the torus of dimension greater than two, where one given coordinate of the translation is an arbitrary Liouville number. To obtain this result, we determine sufficient conditions on translation vectors of the torus that allow us to explicitly construct the sequence of successive conjugacies in Anosov--Katok's method, with suitable estimates of their norm.
Some virtually abelian subgroups of the group of analytic symplectic diffeomorphisms of a surface
John Franks and Michael Handel
2013, 7(3): 369-394 doi: 10.3934/jmd.2013.7.369 +[Abstract](278) +[PDF](278.0KB)
Abstract:
We show that if $M$ is a compact oriented surface of genus $0$ and $G$ is a subgroup of Symp$^\omega_\mu(M)$ that has an infinite normal solvable subgroup, then $G$ is virtually abelian. In particular the centralizer of an infinite order $f \in$ Symp$^\omega_\mu(M)$ is virtually abelian. Another immediate corollary is that if $G$ is a solvable subgroup of Symp$^\omega_\mu(M)$ then $G$ is virtually abelian. We also prove a special case of the Tits Alternative for subgroups of Symp$^\omega_\mu(M)$.
Winning games for bounded geodesics in moduli spaces of quadratic differentials
Jonathan Chaika, Yitwah Cheung and Howard Masur
2013, 7(3): 395-427 doi: 10.3934/jmd.2013.7.395 +[Abstract](308) +[PDF](304.6KB)
Abstract:
We prove that the set of bounded geodesics in Teichmüller space is a winning set for Schmidt's game. This is a notion of largeness in a metric space that can apply to measure $0$ and meager sets. We prove analogous closely related results on any Riemann surface, in any stratum of quadratic differentials, on any Teichmüller disk and for intervals exchanges with any fixed irreducible permutation.
Modified Schmidt games and a conjecture of Margulis
Dmitry Kleinbock and Barak Weiss
2013, 7(3): 429-460 doi: 10.3934/jmd.2013.7.429 +[Abstract](259) +[PDF](323.2KB)
Abstract:
We prove a conjecture of G.A. Margulis on the abundance of certain exceptional orbits of partially hyperbolic flows on homogeneous spaces by utilizing a theory of modified Schmidt games, which are modifications of $(\alpha,\beta)$-games introduced by W. Schmidt in mid-1960s.
Ergodic properties of $k$-free integers in number fields
Francesco Cellarosi and Ilya Vinogradov
2013, 7(3): 461-488 doi: 10.3934/jmd.2013.7.461 +[Abstract](254) +[PDF](801.6KB)
Abstract:
Let $K/\mathbf{Q}$ be a degree-$d$ extension. Inside the ring of integers $\mathscr O_K$ we define the set of $k$-free integers $\mathscr F_k$ and a natural $\mathscr O_K$-action on the space of binary $\mathscr O_K$-indexed sequences, equipped with an $\mathscr O_K$-invariant probability measure associated to $\mathscr F_k$. We prove that this action is ergodic, has pure point spectrum, and is isomorphic to a $\mathbf Z^d$-action on a compact abelian group. In particular, it is not weakly mixing and has zero measure-theoretical entropy. This work generalizes the work of Cellarosi and Sinai [J. Eur. Math. Soc. (JEMS) 15 (2013), no. 4, 1343--1374] that considered the case $K=\mathbf{Q}$ and $k=2$.

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