By M. Aizenman (Chief Editor)

Articles during this volume:

1-33

Renormalization and Asymptotic enlargement of Dirac’s Polarized Vacuum

Philippe Gravejat, Mathieu Lewin and Éric Séré

35-49

Uniqueness of SRB Measures for Transitive Diffeomorphisms on Surfaces

F. Rodriguez Hertz, M. A. Rodriguez Hertz, A. Tahzibi and R. Ures

51-82

A Generalized illustration formulation for structures of Tensor Wave Equations

Arick Shao

83-118

A sequence of Algebras Generalizing the Octonions and Hurwitz-Radon Identity

Sophie Morier-Genoud and Valentin Ovsienko

119-163

Quasi-Normal Modes and Exponential strength Decay for the Kerr-de Sitter Black Hole

Semyon Dyatlov

165-186

Min- and Max-Entropy in countless Dimensions

Fabian Furrer, Johan Åberg and Renato Renner

187-191

Schrödinger Operators and the Zeros in their Eigenfunctions

Sol Schwartzman

193-228

Second Order Perturbation thought for Embedded Eigenvalues

J. Faupin, J. S. Møller and E. Skibsted

229-260

Energy Properness and Sasakian-Einstein Metrics

Xi Zhang

261-290

Burning vehicles in a Parking Lot

Jean Bertoin

291-331

Small BGK Waves and Nonlinear Landau Damping

Zhiwu Lin and Chongchun Zeng

333-364

Superconformal constructions on Generalized Calabi-Yau Metric Manifolds

Reimundo Heluani and Maxim Zabzine

365-380

Rapid Convergence to Frequency for Substitution Tilings of the Plane

José Aliste-Prieto, Daniel Coronel and Jean-Marc Gambaudo

381-417

Chiral Equivariant Cohomology of some extent: a primary Look

Andrew R. Linshaw

419-447

Genus Partition and Correlation features for Fermionic Vertex Operator Superalgebras I

Michael P. Tuite and Alexander Zuevsky

449-483

Instantons on Gravitons

Sergey A. Cherkis

485-509

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

Milivoje Lukic

511-563

From vulnerable to robust Coupling in ABJM Theory

Nadav Drukker, Marcos Mariño and Pavel Putrov

565-578

The Excitation Spectrum for Weakly Interacting Bosons

Robert Seiringer

579-615

The Quantum opposite Shannon Theorem according to One-Shot details Theory

Mario Berta, Matthias Christandl and Renato Renner

617-645

The Szegő Kernel on a Sewn Riemann Surface

Michael P. Tuite and Alexander Zuevsky

647-662

A pressure consequence for Extensions of Braided Tensor C*–Categories Derived from Compact Matrix Quantum Groups

Claudia Pinzari and John E. Roberts

663-694

The Quantum Double version with Boundary: Condensations and Symmetries

Salman Beigi, Peter W. Shor and Daniel Whalen

695-746

Large Violation of Bell Inequalities with Low Entanglement

M. Junge and C. Palazuelos

747-776

Current in Periodic Lorentz Gases with Twists

Hong-Kun Zhang

777-784

Lower Bounds for Nodal units of Eigenfunctions

Tobias H. Colding and William P. Minicozzi

785-803

Specifying Angular Momentum and heart of Mass for Vacuum preliminary info Sets

Lan-Hsuan Huang, Richard Schoen and Mu-Tao Wang

805-830

Faithful Squashed Entanglement

Fernando G. S. L. Brandão, Matthias Christandl and Jon Yard

831-851

Global options to the Ultra-Relativistic Euler Equations

B. D. Wissman

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**Additional info for Communications in Mathematical Physics - Volume 306**

**Example text**

Then using a finer analysis of laminations and Sard’s Theorem we prove transversal homoclinical intersection. Proof. Recall that Pμ and Pν (which we suppose are fixed points) comes from Katok’s closing lemma and as μ, ν are hyperbolic ergodic, both Pμ and Pν have non trivial homoclinic classes. Consequently there exist topological rectangles whose boundaries (∂ s and ∂ u ) consist of stable and unstable segments of Pμ , Pν . (See Fig. ) Choose two such rectangles Rμ , Rν such that Pμ , Pν respectively belong to the boundary of Rμ and Rν and Rμ ∩ Rν = ∅.

Given a hyperbolic periodic point p, let us define the ergodic homoclinic class of p, ( p), as the set of points x ∈ M such that (Fig. 1) Here W s (x) is the Pesin stable manifold of x, that is, 1 log d( f n (x), f n (y)) < 0 , n n→+∞ W s (x) = y ∈ M : lim sup and W u (x), is Pesin unstable manifold of x. For almost every point, Pesin stable and unstable manifolds are, indeed, immersed manifolds. 2) . It is clear that s ( p) and u ( p) are respectively s−saturated and u−saturated. 1. Let f : M → M be a C 1+α diffeomorphism over a compact manifold M and μ a hyperbolic SRB measure.

AMS Translations 71 (1952) 21. : A measure associated with Axiom A attractors. Amer. J. Math. 98, 619–654 (1976) 22. : Thermodynamic Formalism. Reading, MA: Addison Wesley, 1978 23. : Gibbs measure in ergodic theory. Russ. Math. Surv. 27, 21–69 (1972) 24. : Physical measures for partially hyperbolic surface endomorphisms. Acta Math. 194, 37–132 (2005) 25. : Analytic extensions of functions defined in closed sets. Transactions of American Mathematical Society 36(1), 63–89 (1934) 26. : What are SRB measures, and which dynamical systems have them?

### Communications in Mathematical Physics - Volume 306 by M. Aizenman (Chief Editor)

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