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Journal of Modern Dynamics

2010 , Volume 4 , Issue 4

Special issue dedicated to Jan Boman
on the occasion of his 75th birthday

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New cases of differentiable rigidity for partially hyperbolic actions: Symplectic groups and resonance directions
Zhenqi Jenny Wang
2010, 4(4): 585-608 doi: 10.3934/jmd.2010.4.585 +[Abstract](1372) +[PDF](295.5KB)
Abstract:
We prove the local differentiable rigidity of generic partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from split symplectic Lie groups. We also give examples of rigidity for nongeneric actions on compact homogeneous spaces obtained from SL$(2n,\RR)$ or SL$(2n,\CC)$. The conclusions are based on the geometric approach by Katok--Damjanovic and a progress towards computations of the generating relations in these groups.
Ratner's property and mild mixing for special flows over two-dimensional rotations
Krzysztof Frączek and Mariusz Lemańczyk
2010, 4(4): 609-635 doi: 10.3934/jmd.2010.4.609 +[Abstract](301) +[PDF](324.7KB)
Abstract:
We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)dxdy\ne 0$ or $\int_{\T^2}f_y(x,y)dxdy\ne 0 $. Such flows are shown to be always weakly mixing and never partially rigid. It is proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy the so-called weak Ratner property. As a consequence, such flows turn out to be mildly mixing.
Structure of attractors for $(a,b)$-continued fraction transformations
Svetlana Katok and Ilie Ugarcovici
2010, 4(4): 637-691 doi: 10.3934/jmd.2010.4.637 +[Abstract](275) +[PDF](1290.6KB)
Abstract:
We study a two-parameter family of one-dimensional maps and related $(a,b)$-continued fractions suggested for consideration by Don Zagier. We prove that the associated natural extension maps have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional Lebesgue zero measure that we completely describe. We show that the structure of these attractors can be "computed'' from the data $(a,b)$, and that for a dense open set of parameters the Reduction theory conjecture holds, i.e., every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.
Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory
Maxime Zavidovique
2010, 4(4): 693-714 doi: 10.3934/jmd.2010.4.693 +[Abstract](294) +[PDF](285.8KB)
Abstract:
In this article, following [29], we study critical subsolutions in discrete weak KAM theory. In particular, we establish that if the cost function $c: M \times M\to \R$ defined on a smooth connected manifold is locally semiconcave and satisfies twist conditions, then there exists a $C^{1,1}$ critical subsolution strict on a maximal set (namely, outside of the Aubry set). We also explain how this applies to costs coming from Tonelli Lagrangians. Finally, following ideas introduced in [18] and [26], we study invariant cost functions and apply this study to certain covering spaces, introducing a discrete analog of Mather's $\alpha$ function on the cohomology.
Infinite translation surfaces with infinitely generated Veech groups
Pascal Hubert and Gabriela Schmithüsen
2010, 4(4): 715-732 doi: 10.3934/jmd.2010.4.715 +[Abstract](356) +[PDF](249.0KB)
Abstract:
We study infinite translation surfaces which are $\ZZ$-covers of finite square-tiled surfaces obtained by a certain two-slit cut and paste construction. We show that if the finite translation surface has a one-cylinder decomposition in some direction, then the Veech group of the infinite translation surface is either a lattice or an infinitely generated group of the first kind. The square-tiled surfaces of genus two with one zero provide examples for finite translation surfaces that fulfill the prerequisites of the theorem.
Survival of infinitely many critical points for the Rabinowitz action functional
Jungsoo Kang
2010, 4(4): 733-739 doi: 10.3934/jmd.2010.4.733 +[Abstract](304) +[PDF](174.1KB)
Abstract:
In this paper, we show that if Rabinowitz Floer homology has infinite dimension, there exist infinitely many critical points of a Rabinowitz action functional even though it could be non-Morse. This result is proved by examining filtered Rabinowitz Floer homology.

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