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Volume 1, 2007

Journal of Modern Dynamics

2007 , Volume 1 , Issue 1

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Open problems in dynamics and related fields
Alexander Gorodnik
2007, 1(1): 1-35 doi: 10.3934/jmd.2007.1.1 +[Abstract](33) +[PDF](308.5KB)
The paper discusses a number of open questions,which were collected during the AIM workshop “Emerging applications of measure rigidity”. The main emphasis is made on the rigidity problems in the theory of dynamical systems and their connections with Diophantine approximation, arithmetic geometry, and quantum chaos.
On the cohomological equation for nilflows
Livio Flaminio and  Giovanni Forni
2007, 1(1): 37-60 doi: 10.3934/jmd.2007.1.37 +[Abstract](34) +[PDF](231.8KB)
Let $X$ be a vector field on a compact connected manifold $M$. An important question in dynamical systems is to know when a function $g: M\to \mathbb{R}$ is a coboundary for the flow generated by $X$, i.e., when there exists a function $f: M\to \mathbb{R}$ such that $Xf=g$. In this article we investigate this question for nilflows on nilmanifolds. We show that there exists countably many independent Schwartz distributions $D_n$ such that any sufficiently smooth function $g$ is a coboundary iff it belongs to the kernel of all the distributions $D_n$.
The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory
David Mieczkowski
2007, 1(1): 61-92 doi: 10.3934/jmd.2007.1.61 +[Abstract](39) +[PDF](288.0KB)
We make use of representation theory to study the first smooth almost-cohomology of some higher-rank abelian actions by parabolic operators. First, let $N$ be the upper-triangular group of $SL(2,\mathbb{C})$, $\Gamma$ any lattice and $\pi = L^2(SL(2,\mathbb{C})$/$\Gamma)$ the usual left-regular representation. We show that the first smooth almost-cohomology group $H_a^1(N, \pi)$ $H_a^1(SL(2,\mathbb{C}) , \pi)$. In addition, we show that the first smooth almost-cohomology of actions of certain higher-rank abelian groups $A$ acting by left translation on $(SL(2,\mathbb{R}) \times G)$/$\Gamma$ trivialize, where $G = SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ and $\Gamma$ is any irreducible lattice. The abelian groups $A$ are generated by various mixtures of the diagonal and/or unipotent generators on each factor. As a consequence, for these examples we prove that the only smooth time changes for these actions are the trivial ones (up to an automorphism).
Entropy is the only finitely observable invariant
Donald Ornstein and  Benjamin Weiss
2007, 1(1): 93-105 doi: 10.3934/jmd.2007.1.93 +[Abstract](46) +[PDF](139.5KB)
Our main purpose is to present a surprising new characterization of the Shannon entropy of stationary ergodic processes. We will use two basic concepts: isomorphism of stationary processes and a notion of finite observability, and we will see how one is led, inevitably, to Shannon's entropy. A function $J$ with values in some metric space, defined on all finite-valued, stationary, ergodic processes is said to be finitely observable (FO) if there is a sequence of functions $S_{n}(x_{1},x_{2},...,x_{n})$ that for all processes $\mathcal{X}$ converges to $J(\mathcal{X})$ for almost every realization $x_{1}^{\infty}$ of $\mathcal{X}$. It is called an invariant if it returns the same value for isomorphic processes. We show that any finitely observable invariant is necessarily a continuous function of the entropy. Several extensions of this result will also be given.
A dichotomy between discrete and continuous spectrum for a class of special flows over rotations
Bassam Fayad and  A. Windsor
2007, 1(1): 107-122 doi: 10.3934/jmd.2007.1.107 +[Abstract](40) +[PDF](144.2KB)
We provide sufficient conditions on a positive function so that the associated special flow over any irrational rotation is either weak mixing or $L^2$-conjugate to a suspension flow. This gives the first such complete classification within the class of Liouville dynamics. This rigidity coexists with a plethora of pathological behaviors.
Measure rigidity beyond uniform hyperbolicity: invariant measures for cartan actions on tori
Boris Kalinin and  Anatole Katok
2007, 1(1): 123-146 doi: 10.3934/jmd.2007.1.123 +[Abstract](38) +[PDF](229.1KB)
We prove that every smooth action $\a$ of $\mathbb{Z}^k,k\ge 2$, on the $(k+1)$-dimensional torus whose elements are homotopic to corresponding elements of an action $\a_0$ by hyperbolic linear maps preserves an absolutely continuous measure. This is the first known result concerning abelian groups of diffeomorphisms where existence of an invariant geometric structure is obtained from homotopy data.
    We also show that both ergodic and geometric properties of such a measure are very close to the corresponding properties of the Lebesgue measure with respect to the linear action $\a_0$.
The Hopf argument
Yves Coudène
2007, 1(1): 147-153 doi: 10.3934/jmd.2007.1.147 +[Abstract](40) +[PDF](87.4KB)
Let $T$ be a measure-preserving transformation of a metric space $X$. Assume $T$ is conservative and $X$ can be covered by a countable family of open sets, each of finite measure. Then any eigenfunction is invariant with respect to the stable foliation of $T$.

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