# Mini-workshop in Dynamics

**Time: **
Thu 2021-10-07 10.00 - 16.00

**Location: **
Room 3721, Department of Mathematics

**Program:**

10:00-10:50 Jörg Schmeling: *On potentials with a logarithmic singularity.*

11:10-12:00 Tere Seara: *Non existence of small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting*

14:00-14:50 Konstantin Khanin: *On obstacles to C^2 rigidity for circle maps with singularities.*

15:10-16:00 Raphael Krikorian:* On the accumulation of non-split separtrices by invariant circles.*

**Titles and Abstracts**

Jörg Schmeling

Title: On potentials with a logarithmic singularity

Abstract: Potentials with a logarithmic singularity have besides a general interest a strong connection to number theory. We will explain that this kind of potentials on subshifts of finite type form a critical class that undergoes subtle phase transitions and that these phase transitions are not present for other potentials. We study the approximation speed of trajectories to the singularity and draw conclusions of the time averages with respect to such a potential. That gives insight into their equilibrium states and gives examples of random walks without drift but rare excursions times of arbitrary high average speed. This is based on joint work with A.H. Fan and W. Shen.

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Teresa Martinez-Seara

Title: Non existence of small breathers of nonlinear Klein-Gordon equations via exponentially small homoclinic splitting.

Abstract: Breathers are periodic in time and spatially localized solutions of evolutionary PDEs. They are known to exist for integrable PDEs such as the sine-Gordon equation, but are believed to be rare for general nonlinear PDEs. When the spatial dimension is equal to one, exchanging the roles of time and space variables, breathers can be interpreted as homoclinic solutions to steady solutions and thus arising from the intersections of the stable and unstable manifolds of the steady states. In this talk, we shall study small breathers of the nonlinear Klein-Gordon equation generated in an unfolding bifurcation as a pair of eigenvalues collide at the origin when a parameter (temporal frequency) varies. Due to the presence of the oscillatory modes, generally the finite dimensional stable and unstable manifolds do not intersect in the infinite dimensional phase space, but with an exponentially small distance (relative to the amplitude of the breather) in this singular perturbation problem of multiple time scales. Due to the exponential small splitting, classical perturbative techniques cannot be applied. We will explain how to obtain an asymptotic formula for the distance between the stable and unstable manifold of the steady solutions. This is a joint work with O. Gomide, M. Guardia and C. Zeng.

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Konstantin Khanin

Title: On obstacles to C^2 rigidity for circle maps with singularities.

Abstract: We shall present a general argument which explains why one cannot expect C^2 rigidity for circle maps with singularities. In the case of circle maps with breaks we show that for any irrational rotation number the conjugacy in general is not smoother than C^{2-δ} where delta is a universal constant which depends only on the size of a break. This result holds for maps which are arbitrarily smooth outside of their break points.

The talk is based on a joint work with Nataliya Goncharuk and Yuri Kydriashov.

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Raphael Krikorian

Title : On the accumulation of non-split separtrices by invariant circles

Abstract : A theorem by M.R. Herman,“Herman’s last geometric theorem”, asserts that if a smooth orientation and area preserving diffeomorphism f of the 2-plane R^2 (or the 2-cylinder R/Z \times R) admits a KAM curve C (a smooth invariant curve on which the dynamics of f is conjugated to a Diophantine translation) then C is accumulated by other KAM curves, the union of which covers a set of positive 2-dimensional Lebesgue measure in any neighborhood of C. In this talk we shall investigate whether such a phenomenon holds if, instead of being a KAM circle, the invariant set C is a (non-split) separatrix of a hyperbolic fixed point of f. This analysis might be useful for understanding symplectic diffeomorphisms with zero entropy or in the search of a smooth twist map admitting an isolated irrational invariant curve bounding two Birkhoff instability regions. The renormalization paradigm is an important element in our approach. This is a joint work with Anatole Katok.