Where: Kirwan Hall 1311

Speaker: Marco Lenci (Universita' di Bologna) - http://www.dm.unibo.it/~lenci/

Abstract: In the first part of the talk, I will give some background on the

question of mixing for dynamical systems preserving an infinite

measure (a.k.a. 'infinite mixing'). Then I will recall and discuss the

definitions of 'infinite-volume mixing' that I have introduced in

recent years, with a survey on some examples of dynamical systems

which verify or do not verify such definitions. Among these examples

there will be one-dimensional intermittent maps, the subject of recent

work with C. Bonanno and P. Giulietti.

In the second part of the talk, I will better state the results for

the intermittent maps: they comprise a class of expanding maps of

[0,1] with a 'strongly neutral' fixed point in 0 and a class of

expanding maps of the real line with strongly neutral fixed point at

infinity. I will give a sketch of how some of the definitions of

infinite-volume mixing are proved or disproved. Finally I will show

how one property, called global-local mixing, entails certain limit

theorems for our intermittent maps.

Where: Kirwan Hall 1311

Speaker: Rodrigo Trevino (UMD) - http://trevino.cat

Abstract: The Frenkel-Kontorova model was first proposed in the 1930's to describe the structure and dynamics of a crystal lattice in the vicinity of a dislocation core, and by now has found many uses outside of solid state physics. Viewed from a dynamical systems point of view, it exhibits a lot of rich behavior tied to all sorts of great theories (e.g. KAM theory and Aubry-Mather theory) and fundamental open questions (e.g. Lyapunov exponents for the standard map).

I will talk about this model in the setting where the crystal is aperiodic. In this setting, most of the dynamics are no longer available, but some tools developed to study the (periodic) classical model are still useful. I will talk about how one of them in particular, the so-called anti-integrable limit, is useful to find ground states (also known as equilibrium configurations). No background on the model will be assumed.

Where: Kirwan Hall 1311

Speaker: Peter Nandori (UMD) - http://math.umd.edu/~pnandori/

Abstract: We consider a special flow over a mixing map with some hyperbolicity.

In case the roof function is square integrable, we find a set of conditions, under which the flow is mixing and also satisfies the local limit theorem. In case the roof function is non-integrable, we identify another set of conditions that imply Krickeberg mixing. The most important condition is the local limit theorem for the underlying map. We check that the conditions are satisfied for some examples, such as Axiom A flows, Sinai billiards, geometric Lorenz attractors (finite measure case) and suspensions over Pomeau-Manneville maps (finite and infinite measure cases). The talk is based on joint work with Dmitry Dolgopyat.

Where: Kirwan Hall 1308

Speaker: Semyon Dyatlov (MIT) - http://math.mit.edu/~dyatlov/

Abstract: Let $M$ be a nonelementary convex co-compact hyperbolic surface. It is well-known that the Selberg zeta function $Z_M(s)$ has a zero at $s=\delta$ and no zeroes to the right of it, where $0<\delta<1$ is the dimension of the limit set. I will show that $M$ has a spectral gap of some size $\varepsilon>0$ depending only on $\delta$, that is $Z_M(s)$ has only finitely

many zeroes in $\{\Re s > \delta-\varepsilon\}$. An application is an asymptotic counting formula for lengths of closed geodesics with remainder of relative size $O(e^{-\varepsilon t})$.

The key ingredient of the proof is a Fourier decay bound for the Patterson-Sullivan measure on the limit set. This bound relies on the nonlinearity of the transformations generating the corresponding group as well as bounds on exponential sums which follow from the discretized sum-product theorem. The Fourier decay bound implies a fractal uncertainty principle for the limit set, which in turn gives the gap. This talk will include an introduction to transfer operators on Schottky groups, which are used throughout the proofs.

This talk is based on joint works with Jean Bourgain and Maciej Zworski.

Where: Kirwan Hall 1311

Speaker: Vadim Kaloshin (UMD) - https://www.math.umd.edu/~vkaloshi/

Abstract: M. Kac popularized the question `Can you hear the shape of a drum?’. Mathematically, consider a bounded planar domain $\Omega$ and

the associated Dirichlet problem

$$

\Delta u+\lambda^2 u=0, u|_{\partial \Omega}=0.

$$

The set of $\lambda$’s such that this equation has a solution, denoted

$\mathcal L(\Omega)$ is called the Laplace spectrum of $\Omega$.

Does Laplace spectrum determines $\Omega$? In general, the answer is negative.

Consider the billiard problem inside $\Omega$. Call the length spectrum the closure

of the set of perimeters of all periodic orbits of the billiard. Due to deep properties of

the wave trace function, generically, the Laplace spectrum determines the length

spectrum. In the space of strictly convex axis-symmetric domains we shall discuss

Sarnak's question whether one can deform such a domain without changing

its spectrum. This is based joint works with J. De Simoi, A. Figalli and Q. Wei.

Where: Kirwan Hall 1311

Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/

Abstract:

Where: Kirwan Hall 1311

Speaker: Alex Blumenthal (UMD) - http://math.umd.edu/~alexb123/

Abstract: The Chirikov standard map $F_L$ is a prototypical example of a one-parameter family of volume-preserving maps for which one anticipates chaotic behavior on a non-negligible (positive-volume) subset of phase space for a large set of parameters. Analysis in this direction is notoriously difficult, and it remains an open question whether this chaotic region, the stochastic sea, has positive Lebesgue measure for any value of L.

I will discuss two related results on a more tractable version of this problem. The first is a kind of ‘finite-time mixing estimate, indicating that for large L and on a suitable timescale, the map $F_L$ is strongly mixing. The second pertains to statistical properties of compositions of standard maps with increasing parameter L: when the parameter L increases at a sufficiently fast polynomial rate, we obtain asymptotic decay of correlations estimates, a Strong Law, and a CLT, all for Holder observables.

Where: Kirwan Hall 1308

Speaker: Jianlu Zhang (UMD) -

Abstract: For the Restricted Circular Planar 3-Body Problem, we show that there exists a full dimensional open set $U$ in phase space independent of the mass ratio $\mu$, where the set of initial points which lead to collision is $O(\mu ^{1/20} )$ dense as $\mu \rightarrow 0$.

Where: Kirwan Hall 1311

Speaker: Zhihong Jeff Xia (Northwestern) - http://www.math.northwestern.edu/~xia/

Abstract: It is believed and conjectured that, for a typical Hamiltonian system, every hyperbolic periodic point has a homoclinic point. This is indeed the case in many situations. However, for geodesic flow and billiards, the usual perturbation techniques are no longer available, since there is no local perturbation. We will show that for geodesic flows on two-sphere and for convex billiards, it is still true. The proof uses prime ends and relies more on global analysis of stable and unstable manifolds, rather than perturbation techniques.

Where: Kirwan Hall 1311

Speaker: Fumihiko Nakamura (Hokkaido University ) -

Abstract: The non-expanding piecewise linear map, known as the Nagumo-Sato (NS)

model, is described as $S_{\alpha,\beta}(x)=\alpha x+\beta ({\rm

mod} 1)$, where $0<\alpha,\beta<1$.

The NS model corresponds to a

special case of Caianiello's model, and it describes the simplified

dynamics of a single neuron. It is known that the system shows

periodic behavior of the trajectory for almost every (alpha,

beta). Moreover, the map has one discontinuous point when

$\alpha+\beta>1$, and this leads to a complicated structure for a

periodicity of the NS model. This structure is called a Farey

structure which relates to regions in which $S_{\alpha,\beta}$ has a

periodic point. An important feature of the this structure is that

there exists a region in which $S_{\alpha,\beta}$ has a periodic point

with period (m+n) between the region with period m and n. [1]

In this talk, we will explain above properties for the NS model first,

and next consider a perturbed dynamical system in which noise is

applied to the NS model, $x_{t+1}=S_{\alpha,\beta}(x_t) +\xi_t

({\rm mod}1)$ for $0<\alpha,\beta<1$, where ${\xi_t}$ are

independent random variables each having same density g satisfying

${\rm supp}{g}=[0,\theta]$ with $\theta >0$. We discuss asymptotic

properties for the Markov operator[2] corresponding to this perturbed

model. Especially, we focus on the properties of ``asymptotic

periodicity'' and ``asymptotic stability''. Then I will introduce our

main result which tells us which the Markov operator corresponding to

the perturbed NS model has asymptotic periodicity or asymptotic

stability for almost all (alpha,beta) and theta. [3]

The topics of this talk are based on following references:

[1] F.Nakamura, Periodicity of non-expanding piecewise linear maps and

effects of random noises, Dynamical Systems, 30(2015), 450-467.

[2] A. Lasota and M. C. Mackey, Chaos, fractals, and noise: stochastic

aspects of dynamics, Springer Science and Business Media, 2013.

[3] F. Nakamura, Asymptotic behavior of non-expanding piecewise linear

maps in the presence of random noise, (accepted to DCDS-B).

Where: Kirwan Hall 1311

Speaker: Behrang Forghani (University of Connecticut) - https://sites.google.com/site/behrangforghani/

Abstract: TBA

Where: Kirwan Hall 1311

Speaker: (Francoise Pene) - http://lmba.math.univ-brest.fr/perso/francoise.pene/