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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, April 2017
1 
Special functions created by BorelLaplace transform of Hénon map
Volume 18, Number 0, Pages: 1  11, 2011
Chihiro Matsuoka
and Koichi Hiraide
Abstract
References
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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.

2 
Research announcement: The structure of groups with a quasiconvex hierarchy
Volume 16, Number 0, Pages: 44  55, 2009
Daniel T. Wise
Abstract
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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.

3 
A characterization of the concept of duality
Volume 14, Number 0, Pages: 42  59,
Abstract
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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 
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
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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].

6 
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.

7 
On subgroups of the Dixmier group and CalogeroMoser spaces
Volume 18, Number 0, Pages: 12  21, 2011
Yuri Berest,
Alimjon Eshmatov
and Farkhod Eshmatov
Abstract
References
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We describe the structure of the automorphism groups of algebras
Morita equivalent to the first Weyl algebra $ A_1(k) $.
In particular, we give a geometric presentation for these groups in terms of amalgamated products, using the BassSerre theory of groups acting on graphs. A key rôle in our approach is played by a transitive action of the automorphism group of the free algebra $ k< x, y>$ on the CalogeroMoser varieties $ \CC_n $ defined in [5]. In the end, we propose a natural extension of the Dixmier Conjecture
for $ A_1(k) $ to the class of Morita equivalent algebras.

8 
The Generalized WeinsteinMoser Theorem
Volume 14, Number 0, Pages: 20  29,
Abstract
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9 
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
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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.

10 
Hölder cocycles and ergodic integrals for translation flows on flat surfaces
Volume 17, Number 0, Pages: 34  42, 2010
Alexander I. Bufetov
Abstract
Full Text
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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].

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