# American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

All Issues

## Journal of Modern Dynamics

October 2008 , Volume 2 , Issue 4

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2008, 2(4): 541-580 doi: 10.3934/jmd.2008.2.541 +[Abstract](3887) +[PDF](436.8KB)
Abstract:
We present the first known nontrivial topological obstructions to the existence of partially hyperbolic diffeomorphisms. In particular, we show that there are no partially hyperbolic diffeomorphisms on the 3-sphere. More generally, we show that for a partially hyperbolic diffeomorphism of a 3-mani-fold with an Abelian fundamental group, the induced action in the first homology group is partially hyperbolic. This improves the results of [4] by dropping the assumption of dynamical coherence.
2008, 2(4): 581-627 doi: 10.3934/jmd.2008.2.581 +[Abstract](1774) +[PDF](544.0KB)
Abstract:
In this paper we discuss a coding and the associated symbolic dynamics for the geodesic flow on Hecke triangle surfaces. We construct an explicit cross-section for which the first-return map factors through a simple (explicit) map given in terms of the generating map of a particular continued-fraction expansion closely related to the Hecke triangle groups. We also obtain explicit expressions for the associated first return times.
2008, 2(4): 629-643 doi: 10.3934/jmd.2008.2.629 +[Abstract](1490) +[PDF](143.0KB)
Abstract:
Given a diffeomorphism of the interval, we consider the uniform norm of the derivative of its $n$-th iteration. We get a sequence of real numbers called the growth sequence. Its asymptotic behavior is an invariant which naturally appears both in smooth dynamics and in the geometry of the diffeomorphism group. We find sharp estimates for the growth sequence of a given diffeomorphism in terms of the modulus of continuity of its derivative. These estimates extend previous results of Polterovich--Sodin and Borichev.
2008, 2(4): 645-700 doi: 10.3934/jmd.2008.2.645 +[Abstract](1497) +[PDF](584.7KB)
Abstract:
Let $L$ be a hyperbolic automorphism of $\mathbb T^d$, $d\ge3$. We study the smooth conjugacy problem in a small $C^1$-neighborhood $\mathcal U$ of $L$.

The main result establishes $C^{1+\nu}$ regularity of the conjugacy between two Anosov systems with the same periodic eigenvalue data. We assume that these systems are $C^1$-close to an irreducible linear hyperbolic automorphism $L$ with simple real spectrum and that they satisfy a natural transitivity assumption on certain intermediate foliations.

We elaborate on the example of de la Llave of two Anosov systems on $\mathbb T^4$ with the same constant periodic eigenvalue data that are only Hölder conjugate. We show that these examples exhaust all possible ways to perturb a $C^{1+\nu}$ conjugacy class without changing any periodic eigenvalue data. Also we generalize these examples to majority of reducible toral automorphisms as well as to certain product diffeomorphisms of $\mathbb T^4$ $C^1$-close to the original example.
2008, 2(4): 701-718 doi: 10.3934/jmd.2008.2.701 +[Abstract](1510) +[PDF](212.7KB)
Abstract:
On a compact Kähler manifold, there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator, and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace--Beltrami operator. Because of the high degree of symmetry, the Laplace--Beltrami operator on forms can not be quantum ergodic. We show that, after taking these symmetries into account, quantum ergodicity holds for the Laplace--Beltrami operator and for the Spin$^\cbb$-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved Kähler manifolds of odd complex dimension.
2008, 2(4): 719-740 doi: 10.3934/jmd.2008.2.719 +[Abstract](1447) +[PDF](294.8KB)
Abstract:
This paper presents hyperbolic rank-rigidity results for nonpositively curved spaces of rank 1. Let $M$ be a compact, rank-1 manifold with nonpositive sectional curvature and suppose that along every geodesic in $M$ there is a parallel vector field making curvature $-a^2$ with the geodesic direction. We prove that $M$ has constant curvature equal to $-a^2$ if $M$ is odd-dimensional or if $M$ is even-dimensional and has sectional curvature pinched as follows: $-\Lambda^2 < K < -\lambda^2$ where $\lambda/\Lambda >.93$. When $-a^2$ is the upper curvature bound this gives a shorter proof of the hyperbolic rank-rigidity theorem of Hamenstädt, subject to the pinching condition in even dimension; in all other cases it is a new result. We also present a rigidity result using only an assumption on maximal Lyapunov exponents in direct analogy with work done by Connell. The proof uses the dynamics of the frame flow, developed by Brin for negatively curved manifolds; portions of his work are adapted here for use in the nonpositively curved, rank-1 situation.

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