Electronic Research Announcements

2010 , Volume 17

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Multifractal formalism derived from thermodynamics for general dynamical systems
Vaughn Climenhaga
2010, 17: 1-11 doi: 10.3934/era.2010.17.1 +[Abstract](272) +[PDF](195.7KB)
We show that under quite general conditions, various multifractal spectra may be obtained as Legendre transforms of functions $T$: $ \RR\to \RR$ arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map $f$ on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions $T$ be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. Applications include most previously known results, as well as some new ones.
Sharp weighted estimates for approximating dyadic operators
David Cruz-Uribe, SFO, José María Martell and Carlos Pérez
2010, 17: 12-19 doi: 10.3934/era.2010.17.12 +[Abstract](201) +[PDF](160.5KB)
We give a new proof of the sharp weighted $L^p$ inequality

$ |\|T\||_{L^p(w)} \leq C_{n,T}[w]_{A_p}^{\max(1,\frac{1}{p-1})}, $

where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner [15] to estimate the oscillation of dyadic operators.
   The method we use is flexible enough to obtain the sharp one-weight result for other important operators as well as a very sharp two-weight bump type result for $T$ as can be found in [5].

Theory of $(a,b)$-continued fraction transformations and applications
Svetlana Katok and Ilie Ugarcovici
2010, 17: 20-33 doi: 10.3934/era.2010.17.20 +[Abstract](199) +[PDF](834.7KB)
We study a two-parameter family of one-dimensional maps and the related $(a,b)$-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) 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 zero measure that we completely describe; (ii) 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 give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.
Hölder cocycles and ergodic integrals for translation flows on flat surfaces
Alexander I. Bufetov
2010, 17: 34-42 doi: 10.3934/era.2010.17.34 +[Abstract](223) +[PDF](173.1KB)
The main results announced in this note are an asymptotic expansion for ergodic integrals of translation flows on flat surfaces of higher genus (Theorem 1) and a limit theorem for such flows (Theorem 2). Given an abelian differential on a compact oriented surface, consider the space $\mathfrak B^+$ of Hölder cocycles over the corresponding vertical flow that are invariant under holonomy by the horizontal flow. Cocycles in $\mathfrak B^+$ are closely related to G.Forni's invariant distributions for translation flows [10]. Theorem 1 states that ergodic integrals of Lipschitz functions are approximated by cocycles in $\mathfrak B^+$ up to an error that grows more slowly than any power of time. Theorem 2 is obtained using the renormalizing action of the Teichmüller flow on the space $\mathfrak B^+$. A symbolic representation of translation flows as suspension flows over Vershik's automorphisms allows one to construct cocycles in $\mathfrak B^+$ explicitly. Proofs of Theorems 1, 2 are given in [5].
Scalar curvature and $Q$-curvature of random metrics
Yaiza Canzani, Dmitry Jakobson and Igor Wigman
2010, 17: 43-56 doi: 10.3934/era.2010.17.43 +[Abstract](185) +[PDF](219.8KB)
We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. We next consider analogous questions for the scalar curvature in dimension $n>2$, and for the $Q$-curvature of random Riemannian metrics.
Linear approximate groups
Emmanuel Breuillard, Ben Green and Terence Tao
2010, 17: 57-67 doi: 10.3934/era.2010.17.57 +[Abstract](180) +[PDF](202.0KB)
This is an informal announcement of results to be described and proved in detail in [3]. We give various results on the structure of approximate subgroups in linear groups such as $\SL_n(k)$. For example, generalizing a result of Helfgott (who handled the cases $n = 2$ and $3$), we show that any approximate subgroup of $\SL_n(\F_q)$ which generates the group must be either very small or else nearly all of $\SL_n(\F_q)$. The argument is valid for all Chevalley groups $G(\F_q)$. Extending work of Bourgain-Gamburd we also announce some applications to expanders, which will be proven in detail in [4].
Local rigidity of partially hyperbolic actions
Zhenqi Jenny Wang
2010, 17: 68-79 doi: 10.3934/era.2010.17.68 +[Abstract](134) +[PDF](192.4KB)
We prove the local differentiable rigidity of partially hyperbolic abelian algebraic high-rank actions on compact homogeneous spaces obtained from simple indefinite orthogonal and unitary groups. The conclusions are based on geometric Katok-Damjanovic way and progress towards computations of the Schur multipliers of these non-split groups.
Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows
Yong Fang, Patrick Foulon and Boris Hasselblatt
2010, 17: 80-89 doi: 10.3934/era.2010.17.80 +[Abstract](122) +[PDF](182.0KB)
In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin, and we prove new results of this smooth rigidity type for a class of flows.
    For a transversely symplectic uniformly quasiconformal $C^2$ Anosov flow on a compact Riemannian manifold we define the longitudinal KAM-cocycle and use it to prove a rigidity result: The joint stable/unstable subbundle is Zygmund-regular, and higher regularity implies vanishing of the KAM-cocycle, which in turn implies that the subbundle is Lipschitz-continuous and indeed that the flow is smoothly conjugate to an algebraic one. To establish the latter, we prove results for algebraic Anosov systems that imply smoothness and a special structure for any Lipschitz-continuous invariant 1-form.
    We obtain a pertinent geometric rigidity result: Uniformly quasiconformal magnetic flows are geodesic flows of hyperbolic metrics.
    Several features of the reasoning are interesting: The use of exterior calculus for Lipschitz-continuous forms, that the arguments for geodesic flows and infranilmanifoldautomorphisms are quite different, and the need for mixing as opposed to ergodicity in the latter case.
Fourier analysis methods in operator ergodic theory on super-reflexive Banach spaces
Earl Berkson
2010, 17: 90-103 doi: 10.3934/era.2010.17.90 +[Abstract](112) +[PDF](241.7KB)
On reflexive spaces trigonometrically well-bounded operators (abbreviated "twbo's'') have an operator-ergodic-theory characterization as the invertible operators $U$ whose rotates "transfer'' the discrete Hilbert averages $(C,1)$-boundedly. Twbo's permeate many settings of modern analysis, and this note treats advances in their spectral theory, Fourier analysis, and operator ergodic theory made possible by applying classical analysis techniques pioneered by Hardy-Littlewood and L.C. Young to the R.C. James inequalities for super-reflexive spaces. When the James inequalities are combined with spectral integration methods and Young-Stieltjes integration for the spaces $V_{p}(\mathbb{T}) $ of functions having bounded $p$-variation, it transpires that every twbo on a super-reflexive space $X$ has a norm-continuous $V_{p}(\mathbb{T}) $-functional calculus for a range of values of $p>1$, and we investigate the ways this outcome logically simplifies and simultaneously advances the structure theory of twbo's on $X$. In particular, on a super-reflexive space $X$ (but not on the general reflexive space) Tauberian-type theorems emerge which improve to their $(C,0) $ counterparts the $(C,1) $ averaging and convergence associated with twbo's.
Notes on monotone Lagrangian twist tori
Yuri Chekanov and Felix Schlenk
2010, 17: 104-121 doi: 10.3934/era.2010.17.104 +[Abstract](156) +[PDF](260.8KB)
We construct monotone Lagrangian tori in the standard symplectic vector space, in the complex projective space and in products of spheres. We explain how to classify these Lagrangian tori up to symplectomorphism and Hamiltonian isotopy, and how to show that they are not displaceable by Hamiltonian isotopies.
Curvature bounded below: A definition a la Berg--Nikolaev
Nina Lebedeva and Anton Petrunin
2010, 17: 122-124 doi: 10.3934/era.2010.17.122 +[Abstract](102) +[PDF](94.9KB)
We give a new characterization of spaces with nonnegative curvature in the sense of Alexandrov.
Optimally sparse 3D approximations using shearlet representations
Kanghui Guo and Demetrio Labate
2010, 17: 125-137 doi: 10.3934/era.2010.17.125 +[Abstract](145) +[PDF](228.1KB)
This paper introduces a new Parseval frame, based on the 3-D shearlet representation, which is especially designed to capture geometric features such as discontinuous boundaries with very high efficiency. We show that this approach exhibits essentially optimal approximation properties for 3-D functions $f$ which are smooth away from discontinuities along $C^2$ surfaces. In fact, the $N$ term approximation $f_N^S$ obtained by selecting the $N$ largest coefficients from the shearlet expansion of $f$ satisfies the asymptotic estimate

||$f-f_N^S$||$_2^2$ ≍ $N^{-1} (\log N)^2, as N \to \infty.$

Up to the logarithmic factor, this is the optimal behavior for functions in this class and significantly outperforms wavelet approximations, which only yields a $N^{-1/2}$ rate. Indeed, the wavelet approximation rate was the best published nonadaptive result so far and the result presented in this paper is the first nonadaptive construction which is provably optimal (up to a loglike factor) for this class of 3-D data.
    Our estimate is consistent with the corresponding 2-D (essentially) optimally sparse approximation results obtained by the authors using 2-D shearlets and by Candès and Donoho using curvelets.

The equivariant index theorem for transversally elliptic operators and the basic index theorem for Riemannian foliations
Jochen Brüning, Franz W. Kamber and Ken Richardson
2010, 17: 138-154 doi: 10.3934/era.2010.17.138 +[Abstract](179) +[PDF](418.2KB)
In this expository paper, we explain a formula for the multiplicities of the index of an equivariant transversally elliptic operator on a $G$-manifold. The formula is a sum of integrals over blowups of the strata of the group action and also involves eta invariants of associated elliptic operators. Among the applications is an index formula for basic Dirac operators on Riemannian foliations, a problem that was open for many years.
On almost Poisson commutativity in dimension two
Frol Zapolsky
2010, 17: 155-160 doi: 10.3934/era.2010.17.155 +[Abstract](129) +[PDF](139.1KB)
Consider the following question: given two functions on a symplectic manifold whose Poisson bracket is small, is it possible to approximate them in the $C^0$ norm by commuting functions? We give a positive answer in dimension two, as a particular case of a more general statement which applies to functions on a manifold with a volume form. This result is based on a lemma in the spirit of geometric measure theory. We give some immediate applications to function theory and the theory of quasi-states on surfaces with area forms.

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