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Journal of Modern Dynamics

2009 , Volume 3 , Issue 1

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Dynamical coherence of partially hyperbolic diffeomorphisms of the 3-torus
Michael Brin , Dmitri Burago and  Sergey Ivanov
2009, 3(1): 1-11 doi: 10.3934/jmd.2009.3.1 +[Abstract](1103) +[PDF](143.7KB)
We show that partially hyperbolic diffeomorphisms of the 3-torus are dynamically coherent.
Absence of mixing for smooth flows on genus two surfaces
Dmitri Scheglov
2009, 3(1): 13-34 doi: 10.3934/jmd.2009.3.13 +[Abstract](35) +[PDF](338.7KB)
We prove that typical area-preserving flows with linearly isomorphic nondegenerate saddles on genus two surfaces are not mixing.
Weak mixing for logarithmic flows over interval exchange transformations
Corinna Ulcigrai
2009, 3(1): 35-49 doi: 10.3934/jmd.2009.3.35 +[Abstract](49) +[PDF](239.6KB)
We consider a class of special flows over interval exchange transformations which includes roof functions with symmetric logarithmic singularities. We prove that such flows are typically weakly mixing. As a corollary, minimal flows given by multivalued Hamiltonians on higher-genus surfaces are typically weakly mixing.
Maximizing orbits for higher-dimensional convex billiards
Misha Bialy
2009, 3(1): 51-59 doi: 10.3934/jmd.2009.3.51 +[Abstract](29) +[PDF](112.4KB)
The main result of this paper is that, in contrast to the 2D case, for convex billiards in higher dimensions, for every point on the boundary, and for every $n$, there always exist billiard trajectories developing conjugate points at the $n$-th collision with the boundary. We shall explain that this is a consequence of the following variational property of the billiard orbits in higher dimension. If a segment of an orbit is locally maximizing, then it can not pass too close to the boundary. This fact follows from the second variation formula for the length functional. It turns out that this formula behaves differently with respect to "longitudinal'' and "transverse'' variations.
Floer homology in disk bundles and symplectically twisted geodesic flows
Michael Usher
2009, 3(1): 61-101 doi: 10.3934/jmd.2009.3.61 +[Abstract](42) +[PDF](446.9KB)
We show that if $K: P\to\mathbb{R}$ is an autonomous Hamiltonian on a symplectic manifold $(P,\Omega)$ which attains a Morse-Bott nondegenerate minimum of 0 along a symplectic submanifold $M$ and if $c_1(TP)$↾M vanishes in real cohomology, then the Hamiltonian flow of $K$ has contractible periodic orbits with bounded period on all sufficiently small energy levels. As a special case, if the geodesic flow on T*M is twisted by a symplectic magnetic field form, then the resulting flow has contractible periodic orbits on all low energy levels. These results were proven by Ginzburg and Gürel when $\Omega$↾M is spherically rational, and our proof builds on their work; the argument involves constructing and carefully analyzing at the chain level a version of filtered Floer homology in the symplectic normal disk bundle to $M$.
Nearly continuous Kakutani equivalence of adding machines
Mrinal Kanti Roychowdhury and  Daniel J. Rudolph
2009, 3(1): 103-119 doi: 10.3934/jmd.2009.3.103 +[Abstract](38) +[PDF](188.8KB)
One says that two ergodic systems $(X,\mathcal F,\mu)$ and $(Y,\mathcal G,\nu)$ preserving a probability measure are evenly Kakutani equivalent if there exists an orbit equivalence $\phi: X\to Y$ such that, restricted to some subset $A\subseteq X$ of positive measure, $\phi$ becomes a conjugacy between the two induced maps $T_A$ and $S_{\phi(A)}$. It follows from the general theory of loosely Bernoulli systems developed in [8] that all adding machines are evenly Kakutani equivalent, as they are rank-1 systems. Recent work has shown that, in systems that are both topological and measure-preserving, it is natural to seek to strengthen purely measurable results to be "nearly continuous''. In the case of even Kakutani equivalence, what one asks is that the map $\phi$ and its inverse should be continuous on $G_\delta$ subsets of full measure and that the set $A$ should be within measure zero of being open and of being closed. What we will show here is that any two adding machines are indeed equivalent in this nearly continuous sense.
Anosov automorphisms of nilpotent Lie algebras
Tracy L. Payne
2009, 3(1): 121-158 doi: 10.3934/jmd.2009.3.121 +[Abstract](46) +[PDF](401.2KB)
Each matrix $A$ in $GL_n(Z)$ naturally defines an automorphism $f$ of the free $r$-step nilpotent Lie algebra $\frf_{n,r}$. We study the relationship between the matrix $A$ and the eigenvalues and rational invariant subspaces for $f$. We give applications to the study of Anosov automorphisms.

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