Lusternik - Schnirelman theory and dynamics

In this paper we study a new topological invariant \Cat(X,\xi), where $X$ is a finite polyhedron and $\xi\in H^1(X;\R)$ is a real cohomology class. \Cat(X,\xi) is defined using open covers of $X$ with certain geometric properties; it is a generalization of the classical Lusternik -- Schnirelman category. We show that \Cat(X,\xi) depends only on the homotopy type of $(X,\xi)$. We prove that \Cat(X,\xi) allows to establish a relation between the number of equilibrium states of dynamical systems and their global dynamical properties (such as existence of homoclinic cycles and the structure of the set of chain recurrent points). In the paper we give a cohomological lower bound for \Cat(X,\xi), which uses cup-products of cohomology classes of flat line bundles with monodromy described by complex numbers, which are not Dirichlet units.Comment: 20 page

Double logarithms resummation in exclusive processes : the surprising behavior of DVCS

Double logarithms resummation has been much studied in inclusive as well as exclusive processes. The Sudakov mechanism has often be the crucial tool to exponentiate potentially large contributions to amplitudes or cross-sections near phase-space boundaries. We report on a recent work where a very different pattern emerges : the DVCS quark coefficient function C(x,\xi) develops, near the particular point x=\xi, a non-alternate series in \alpha_s^n log^{2n}(x-\xi) which may be resummed in a cosh[K sqrt \alpha_s log(x-\xi)] factor. This result is at odds with the known result for the corresponding coefficient function for the pion transition form factor near the end point C(z) although they are much related through a z -> x/\xi correspondence.Comment: 9 pages, 1 figure, Presented at the Low x workshop, May 30 - June 4 2013, Rehovot and Eilat, Israe

Fractional differentiability for solutions of nonlinear elliptic equations

We study nonlinear elliptic equations in divergence form $${\operatorname{div}}{\mathcal A}(x,Du)={\operatorname{div}}G.$$ When ${\mathcal A}$ has linear growth in $Du$, and assuming that $x\mapsto{\mathcal A}(x,\xi)$ enjoys $B^\alpha_{\frac{n}\alpha, q}$ smoothness, local well-posedness is found in $B^\alpha_{p,q}$ for certain values of $p\in[2,\frac{n}{\alpha})$ and $q\in[1,\infty]$. In the particular case ${\mathcal A}(x,\xi)=A(x)\xi$, $G=0$ and $A\in B^\alpha_{\frac{n}\alpha,q}$, $1\leq q\leq\infty$, we obtain $Du\in B^\alpha_{p,q}$ for each $p<\frac{n}\alpha$. Our main tool in the proof is a more general result, that holds also if ${\mathcal A}$ has growth $s-1$ in $Du$, $2\leq s\leq n$, and asserts local well-posedness in $L^q$ for each $q>s$, provided that $x\mapsto{\mathcal A}(x,\xi)$ satisfies a locally uniform $VMO$ condition