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Free to readers and authors, Electronic Research Announcements in Mathematical Sciences rapidly publishes announcements of significant advances in all branches of mathematics and short complete papers of original research (up to about 15 journal pages). Research announcements are an opportunity for lucid exposition of ideas and context unburdened by technical detail. All articles should be designed to communicate their contents to a broad mathematical audience and must meet high standards for mathematical content and clarity. After review and acceptance by the entire Editorial Board, articles enter production for immediate publication.
ERA is a continuation of Electronic Research Announcements of the AMS published by the American Mathematical Society from 1995 to the middle of 2007. After over two decades of leading this journal, Svetlana Katok became Founding Editor Emerita in January 2017.

TOP 10 Most Read Articles in ERAMS, May 2017
1 
Research announcement: The structure of groups with a quasiconvex hierarchy
Volume 16, Number 0, Pages: 44  55, 2009
Daniel T. Wise
Abstract
Full Text
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Let $G$ be a wordhyperbolic group with a quasiconvex hierarchy.
We show that $G$ has a finite index subgroup $G'$ that embeds as a
quasiconvex subgroup of a rightangled Artin group.
It follows that every quasiconvex subgroup of $G$ is a virtual retract,
and is hence separable.
The results are applied to certain 3manifold and onerelator groups.

2 
A characterization of the concept of duality
Volume 14, Number 0, Pages: 42  59,
Abstract
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3 
Special functions created by BorelLaplace transform of Hénon map
Volume 18, Number 0, Pages: 1  11, 2011
Chihiro Matsuoka
and Koichi Hiraide
Abstract
References
Full Text
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We present a novel class of functions that can describe the stable and unstable manifolds of the Hénon map. We propose an algorithm to construct these functions by using the BorelLaplace transform. Neither linearization nor perturbation is applied in the construction, and the obtained functions are exact solutions of the Hénon map. We also show that it is possible to depict the chaotic attractor of the map by using one of these functions without explicitly using the properties of the attractor.

4 
Multifractal formalism derived from thermodynamics for general dynamical systems
Volume 17, Number 0, Pages: 1  11, 2010
Vaughn Climenhaga
Abstract
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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.

5 
Theory of $(a,b)$continued fraction transformations and applications
Volume 17, Number 0, Pages: 20  33, 2010
Svetlana Katok
and Ilie Ugarcovici
Abstract
Full Text
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We study a twoparameter family of onedimensional 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 Cantorlike set of onedimensional 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.

6 
The Generalized WeinsteinMoser Theorem
Volume 14, Number 0, Pages: 20  29,
Abstract
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7 
The equivariant index theorem for
transversally elliptic operators and the basic index theorem for Riemannian
foliations
Volume 17, Number 0, Pages: 138  154, 2010
Jochen Brüning,
Franz W. Kamber
and Ken Richardson
Abstract
References
Full Text
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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.

8 
A method for the study of whiskered quasiperiodic and
almostperiodic solutions in finite and infinite dimensional
Hamiltonian systems
Volume 16, Number 0, Pages: 9  22, 2009
Ernest Fontich,
Rafael de la Llave
and Yannick Sire
Abstract
Full Text
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We describe a method to study the existence of
whiskered quasiperiodic solutions in Hamiltonian
systems.
The method applies to finite dimensional systems
and also to lattice systems and to PDE's including
some ill posed ones.
In coupled map lattices, we can also
construct solutions of infinitely many frequencies
which do not vanish asymptotically.

9 
Sharp weighted estimates for approximating dyadic operators
Volume 17, Number 0, Pages: 12  19, 2010
David CruzUribe, SFO,
José María Martell
and Carlos Pérez
Abstract
Full Text
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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}{p1})}, $
where $T$ is the Hilbert transform, a Riesz transform, the
BeurlingAhlfors 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 oneweight
result for other important operators as well as a very sharp
twoweight bump type result for $T$ as can be found in
[5].

10 
Linear approximate groups
Volume 17, Number 0, Pages: 57  67, 2010
Emmanuel Breuillard,
Ben Green
and Terence Tao
Abstract
References
Full Text
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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 BourgainGamburd we also announce some applications to expanders, which will be proven in detail in [4].

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