# NYC Dynamics Seminar at CUNY & Yeshiva University

NYC Dynamics Seminar at CUNY & Yeshiva University is a research level seminar with a broad agenda aimed at research mathematicians and graduate students whose interests include various aspects of the modern theory of dynamical systems and related topics in analysis, geometry, number theory and possibly other subjects. Its aim is to supplement more specialized seminars in the NYC area and provide a meeting place and a venue for discussions for mathematicians associated with various universities and colleges in the NYC metropolitan area working or interested in dynamical systems. The seminar will primarily feature speakers from outside the area specially invited for this purpose as well as mathematicians visiting various NYC universities. Core financial support for the seminar is provided by the Center for Mathematical Sciences at Yeshiva University. Support from other institutions who contribute to funding visits of seminar speakers will be acknowledged.

Organizers:
* Anatole Katok will be at Penn State for the semester.
Locations:
• Yeshiva University, 215 Lexington Ave, Room 506. The entrance to the building is on the southeast corner of 33rd St and Lexington Ave.
• CUNY Graduate Center, 365 5th Ave. Rooms will likely vary from talk to talk.

Both locations require participants to present some form of identification to the security to be signed in.

Meeting Times: We plan to meet roughly every other Wednesday at 5pm for the Fall, 2016 semester. We will alternate locations between the CUNY Graduate Center and Yeshiva University. The typical lecture time will be 1 hour, followed by a short period for questions or discussions.

## Currently Scheduled Talks for Fall 2016:

### Tuesday, September 13th, 5:30pmRichard MontgomeryUC Santa Cruz

The Hyperbolic Geometry and three and four -body problems Yeshiva University, 215 Lexington Ave, Room 506. Observe the unusual day and time.

We begin by deriving the hyperbolic plane with its geodesic flow as the ‘scale plus symmetry reduction’ of a three-body problem in the Euclidean plane. The potential is homogeneous of degree -2, with denominator being the area squared of the triangle whose vertices are the three masses. The reduction method involves the Jacobi-Maupertuis metric in a central way and was used in my paper Putting Hyperbolic pants on a three body problem'. We will also review some aspects of that paper and some surprises from work of Connor Jackman on an analogous problem arising in the 1/r*r 4-body problem.

### Friday, September 16, 2pmMaciej CapinskiAGH University of Science and Technology, Krakow, Poland

Beyond the Melnikov method: a computer assisted approach CUNY Graduate Center, 365 Fifth Avenue, Room 5417. Observe the unusual date, time and location. This seminar is formally in the Complex Analysis and Dynamics Seminar.

We present a Melnikov type approach for establishing transversal intersections of stable/unstable manifolds of perturbed normally hyperbolic invariant manifolds. In our approach, we do not need to know the explicit formulas for the homoclinic orbits prior to the perturbation. We also do not need to compute any integrals along such homoclinics. All needed bounds are established using rigorous computer assisted numerics. Lastly, and most importantly, the method establishes intersections for an explicit range of parameters, and not only for perturbations that are ‘small enough’, as is the case in the classical Melnikov approach.

### Wednesday, September 28th, 5pmMark LeviPenn State

Gyroscopes and Gaussian curvature CUNY Graduate Center, 365 5th Ave, Room C198. (This room is in the basement of the building.)

I will describe a recent observation which shows that Gaussian curvature plays a key role in the gyroscopic effect, and in particular in the the dynamics of the Lagrange top. In addition, I will show some other examples exhibiting unexpected magnetic—like or gyroscopic—like effects. This talk is based on joint work with Graham Cox.

### Wednesday, October 5th, 5pmEnrique PujalsIMPA and CUNY Graduate Center

On the $$C^r$$-typicality of coexistence of infinitely many sinks CUNY Graduate Center, 365 5th Ave, The Science Center on the 4th floor.

In the seventies, Newhouse proved that nearby a smooth surfaces diffeomorphism exhibiting an homoclinic tangency there exist open set of diffeos such that the coexistence of infinitely many attractors (from now on property P) is generic. In an other words, property P is typical from a topological point of view.

It is natural to wonder if such property is typical whenever is considered parametric families (meaning that P is satisfied for a parameter set of positive Lebesgue measure and sometimes called Kolmogorov & Arnold $$C^r$$-typicality).

Recently, P. Berger shown that there are open set of smooth parametric families of surfaces endomorphisms such that generically such families exhibit the property P for all parameter.

In a joint work with P. Berger and S. Crovisier we extend that results showing that nearby a smooth surfaces endomorphisms exhibiting a bicycle (a coexistence of a tangency and a heterodimensional cycle) there open set of maps such that any generic family has an open and dense set of parameter displaying property P.

In the talk, we will focus on the new tools developed (for instances, recasting parametric hyperbolic sets as hyperbolic dynamics on the space of jets) and we will discuss the $$C^\infty$$ case.

### Wednesday, November 2nd, 5pmAlbert GranadosTechnical University of Denmark

Invariant Manifolds in energy harvesting piezoelectric oscillators Yeshiva University, 215 Lexington Ave, Room 506.

When perturbed with a small periodic forcing, two (or more) coupled conservative oscillators can exhibit instabilities: trajectories that become unstable while accumulating `unbounded'' energy from the source. This is known as Arnold diffusion, and such phenomenon could be extremely useful in energy harvesting systems, whose aim is to capture as much energy as possible from a source.

In this talk we consider an energy harvesting system based on two piezoelectric oscillators. When forced to oscillate, for instance when driven by a small periodic vibration, such oscillators create an electrical current which charges an accumulator (a capacitor or a battery). Unfortunately, such oscillators are not conservative, as they are not perfectly elastic (they are damped). Moreover, the piezoelectric coupling effects slows them down. However, we will discuss how to apply some of the techniques from Arnold diffusion theory to benefit the accumulation of energy in such devices.

### Wednesday, November 9th, 5pmBehrang ForghaniUniversity of Connecticut

Abramov's formula for random walks on groups. CUNY Graduate Center, 365 5th Ave, The Science Center on the 4th floor.

Given a random walk on a countable group, any Markov stopping time gives rise to a new random walk on the same group. We will show that the asymptotic entropy (rate of escape) of such transformations are equal to the asymptotic entropy (rate of escape) of the original random walk times the expectation of the stopping time. This fact is an analogue of the Abramov formula from ergodic theory. The proof is based on the fact that the Poisson boundaries of these random walks are the same.