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Journal of Modern Dynamics

2008 , Volume 2 , Issue 2

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Partial hyperbolicity and ergodicity in dimension three
Federico Rodriguez Hertz, María Alejandra Rodriguez Hertz and Raúl Ures
2008, 2(2): 187-208 doi: 10.3934/jmd.2008.2.187 +[Abstract](586) +[PDF](357.9KB)
Abstract:
In [15] the authors proved the Pugh–Shub conjecture for partially hyperbolic diffeomorphisms with 1-dimensional center, i.e., stably ergodic diffeomorphisms are dense among the partially hyperbolic ones. In this work we address the issue of giving a more accurate description of this abundance of ergodicity. In particular, we give the first examples of manifolds in which all conservative partially hyperbolic diffeomorphisms are ergodic.
Algebraically periodic translation surfaces
Kariane Calta and John Smillie
2008, 2(2): 209-248 doi: 10.3934/jmd.2008.2.209 +[Abstract](472) +[PDF](385.0KB)
Abstract:
We develop an algebraic framework for studying translation surfaces. We study the Sah--Arnoux--Fathi-invariant and the collection of directions in which it vanishes. We show that these directions are described by a number field which we call the periodic direction field. We study the $J$-invariant of a translation surface, introduced by Kenyon and Smillie and used by Calta. We relate the $J$-invariant to the periodic direction field. For every number field $K\subset\ \mathbb R$ we show that there is a translation surface for which the periodic direction field is $K$. We study automorphism groups associated to a translation surface and relate them to the $J$-invariant. We relate the existence of decompositions of translation surfaces into squares with the total reality of the periodic direction field.
Spectral invariants in Lagrangian Floer theory
Rémi Leclercq
2008, 2(2): 249-286 doi: 10.3934/jmd.2008.2.249 +[Abstract](602) +[PDF](411.6KB)
Abstract:
Let $(M,\omega)$ be a symplectic manifold compact or convex at infinity. Consider a closed Lagrangian submanifold $L$ such that $\omega |_{\pi_2(M,L)}=0$ and $\mu|_{\pi_2(M,L)}=0$, where $\mu$ is the Maslov index. Given any Lagrangian submanifold $L'$, Hamiltonian isotopic to $L$, we define Lagrangian spectral invariants associated to the non zero homology classes of $L$, depending on $L$ and $L'$. We show that they naturally generalize the Hamiltonian spectral invariants introduced by Oh and Schwarz, and that they are the homological counterparts of higher order invariants, which we also introduce here, via spectral sequence machinery introduced by Barraud and Cornea. These higher order invariants are new even in the Hamiltonian case and carry strictly more information than the classical ones. We provide a way to distinguish them one from another and estimate their difference in terms of a geometric quantity.
Quantum ergodicity for products of hyperbolic planes
Dubi Kelmer
2008, 2(2): 287-313 doi: 10.3934/jmd.2008.2.287 +[Abstract](593) +[PDF](272.9KB)
Abstract:
For manifolds with geodesic flow that is ergodic on the unit tangent bundle, the Quantum Ergodicity Theorem implies that almost all Laplacian eigenfunctions become equidistributed as the eigenvalue goes to infinity. For a locally symmetric space with a universal cover that is a product of several upper half-planes, the geodesic flow has constants of motion so it cannot be ergodic. It is, however, ergodic when restricted to the submanifolds defined by these constants. Accordingly, we show that almost all eigenfunctions become equidistributed on these submanifolds.
Growth and mixing
Krzysztof Frączek and Leonid Polterovich
2008, 2(2): 315-338 doi: 10.3934/jmd.2008.2.315 +[Abstract](435) +[PDF](248.9KB)
Abstract:
Given a bi-Lipschitz measure-preserving homeomorphism of a finite dimensional compact metric measure space, consider the sequence of the Lipschitz norms of its iterations. We obtain lower bounds on the growth rate of this sequence assuming that our homeomorphism mixes a Lipschitz function. In particular, we get a universal lower bound which depends on the dimension of the space but not on the rate of mixing. Furthermore, we get a lower bound on the growth rate in the case of rapid mixing. The latter turns out to be sharp: the corresponding example is given by a symbolic dynamical system associated to the Rudin–Shapiro sequence
Ergodic properties of isoperimetric domains in spheres
Gerhard Knieper and Norbert Peyerimhoff
2008, 2(2): 339-358 doi: 10.3934/jmd.2008.2.339 +[Abstract](468) +[PDF](209.9KB)
Abstract:
Let $\varphi$ be a function on the unit tangent bundle of a compact manifold of negative curvature. We show that averages of $\varphi$ over subdomains of increasing spheres converge to the horospherical mean if these domains satisfy an isoperimetric condition. We apply this result to spherical means with continuous density and, by using relations between the horospherical mean and the Patterson-Sullivan measure, we derive some kind of mixing properties.
$C^1$-generic conservative diffeomorphisms have trivial centralizer
Christian Bonatti, Sylvain Crovisier and Amie Wilkinson
2008, 2(2): 359-373 doi: 10.3934/jmd.2008.2.359 +[Abstract](507) +[PDF](185.2KB)
Abstract:
We prove that the spaces of $C^1$ symplectomorphisms and of $C^1$ volume-preserving diffeomorphisms of connected manifolds contain residual subsets of diffeomorphisms whose centralizers are trivial.

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