This paper deals with the fractional Sobolev spaces W s;p . We analyze the relations among some of their possible denitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

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Hitchhiker's guide to the fractional Sobolev spaces

Controlling chaos

This paper deals with the fractional Sobolev spaces W^[s,p]. We analyze the relations among some of their possible definitions and their role in the trace theory. We prove continuous and compact embeddings, investigating the problem of the extension domains and other regularity results. Most of the results we present here are probably well known to the experts, but we believe that our proofs are original and we do not make use of any interpolation techniques nor pass through the theory of Besov spaces. We also present some counterexamples in non-Lipschitz domains.

Symplectic maps are the discrete-time analog of Hamiltonian motion. They arise in many applications including accelerator, chemical, condensed-matter, plasma, and fluid physics. Twist maps correspond to Hamiltonians for which the velocity is a monotonic function of the canonical momentum. Twist maps have a Lagrangian variational formulation. One-parameter families of twist maps typically exhibit the full range of possible dynamics-from simple or integrable motion to complex or chaotic motion. One class of orbits, the minimizing orbits, can be found throughout this transition; the properties of the minimizing orbits are discussed in detail. Among these orbits are the periodic and quasiperiodic orbits, which form a scaffold in the phase space and constrain the motion of the remaining orbits. The theory of transport deals with the motion of ensembles of trajectories. The variational principle provides an efficient technique for computing the flux escaping from regions bounded by partial barriers formed from minimizing orbits. Unsolved problems in the theory of transport include the explanation for algebraic tails in correlation functions, and its extension to maps of more than two dimensions.

Symplectic maps, variational principles, and transport

Variational models for image segmentation have many applications, but can be slow to compute. Recently, globally convex segmentation models have been introduced which are very reliable, but contain TV-regularizers, making them difficult to compute. The previously introduced Split Bregman method is a technique for fast minimization of L1 regularized functionals, and has been applied to denoising and compressed sensing problems. By applying the Split Bregman concept to image segmentation problems, we build fast solvers which can out-perform more conventional schemes, such as duality based methods and graph-cuts. The convex segmentation schemes also substantially outperform conventional level set methods, such as the Chan-Vese level set-based segmentation algorithm. We also consider the related problem of surface reconstruction from unorganized data points, which is used for constructing level set representations in 3 dimensions. The primary purpose of this paper is to examine the effectiveness of "Split Bregman" techniques for solving these problems, and to compare this scheme with more conventional methods.

/pdf/geometric-applications-of-the-split-bregman-method-1kurc04wl7.pdf

Geometric Applications of the Split Bregman Method: Segmentation and Surface Reconstruction

We give a proof of a KAM theorem on existence of invariant tori with a Diophantine rotation vector for Hamiltonian systems. The method of proof is based on the use of the geometric properties of Hamiltonian systems which, in particular, do not require the Hamiltonian system either to be written in action-angle variables or to be a perturbation of an integrable one. The proposed method is also useful to compute numerically invariant tori for Hamiltonian systems. We also prove a translated torus theorem in any number of degrees of freedom.

/pdf/kam-theory-without-action-angle-variables-3djxz1a0v9.pdf

KAM Theory without action--angle variables

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

We give a new proof of the fact that the eigenvalues at corresponding periodic orbits forms a complete set of invariants for the smooth conjugacy of low dimensional Anosov systems. We also show that, if a homeomorphism conjugating two smooth low dimensional Anosov systems is absolutely continuous, then it is as smooth as the maps. We furthermore prove generalizations of these facts for non-uniformly hyperbolic systems as well as extensions and counterexamples in higher dimensions.

/pdf/smooth-conjugacy-and-s-r-b-measures-for-uniformly-and-non-1s7n3av9jw.pdf

Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems

In two previous papers [J. Differential Equations, 228 (2006), pp. 530–579; Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1261–1300] we have developed fast algorithms for the computations of invariant tori in quasi‐periodic systems and developed theorems that assess their accuracy. In this paper, we study the results of implementing these algorithms and study their performance in actual implementations. More importantly, we note that, due to the speed of the algorithms and the theoretical developments about their reliability, we can compute with confidence invariant objects close to the breakdown of their hyperbolicity properties. This allows us to identify a mechanism of loss of hyperbolicity and measure some of its quantitative regularities. We find that some systems lose hyperbolicity because the stable and unstable bundles approach each other but the Lyapunov multipliers remain away from 1. We find empirically that, close to the breakdown, the distances between the invariant bundles and the Lyapun...

/pdf/a-parameterization-method-for-the-computation-of-invariant-56d70mieyh.pdf

A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi‐Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity

We prove that if two germs of diffeomorphisms preserving a voiume, symplectic, or contact structure are tangent to a high enough order and the linearization is hyperbolic, it is possible to find a smooth change of variables that sends one into the other and which, moreover, preserves the same geometric structure. This result is a geometric version of Sternberg’s linearization theorem, which we recover as a particular case. An analogous result is also proved for flows.

R. de la Llave

Papers

KAM Theory without action--angle variables

A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Rigorous results

Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems

A Parameterization Method for the Computation of Invariant Tori and Their Whiskers in Quasi‐Periodic Maps: Explorations and Mechanisms for the Breakdown of Hyperbolicity

Cohomology equations near hyperbolic points and geometric versions of sternberg linearization theorem