Friday September 08, 2017 2:30 PM  3:30 PM Math Tower P131
 Tetsuo Ueda, Kyoto University
Some problems on the local structure of holomorphic mapsWe will discuss the relations between local holomorphic dynamics and structure of complex manifolds. Examples will be taken from Schroeder equation for of attracting fixed point, EcalleVoronin theory for parabolic fixed point in one variable and saddle or semiattracting fixed point in two variables.

Friday September 15, 2017 2:30 PM  3:30 PM Math Tower P131
 Tali Pinsky, TIFR
On the Lorenz flow and the modular surfaceThe talk addresses three famous, albeit fundamentally different, three dimensional flows: The geodesic flow on the modular surface, the chaotic Lorenz equations and the geometric Lorenz model. I will describe a new approach, potentially showing that these flows are orbit equivalent for the correct choice of parameters.
This will be an introductory talk.

Friday September 22, 2017 2:30 PM  3:30 PM Math Tower P131
 Jonguk Yang, Stony Brook University
Renormalization of Dissipative Hénon MapsSuppose that within the space of certain onedimensional dynamical systems, a suitably defined renormalization operator has a hyperbolic fixed point. By incorporating a nonlinear change of coordinates to the rescaling map, I will show that this renormalization operator can be extended to nearby twodimensional dynamical systems without losing its hyperbolicity.

Friday September 29, 2017 2:30 PM  3:30 PM Math Tower P131
 Nataliya Goncharuk, Cornell University
Complex rotation numbersGiven an analytic circle diffeomorphism f ∶ ℝ/ℤ → ℝ/ℤ and a complex number w, ℑw > 0, consider the quotient space of the annulus 0 < ℑz < ℑw, z ∈ ℂ/ℤ, by the action of f + w. This quotient space is a torus, and we can ask about its modulus. This modulus is called the *complex rotation number* of f + w.
The limit of the complex rotation number as w tends to w₀ ∈ ℝ/ℤ coincides with the rotation number of f + w₀, if this rotation number is irrational. In general, limit values of the complex rotation number on ℝ/ℤ form a bubbly picture in the upper halfplane: infinitely many bubbles (analytic curves) grow from rational points of the real axis. Bubbles are complex analogue to Arnold tongues.
In the talk, I'll give a survey of results on shapes of bubbles, with some proofs, and list open questions.
Mostly based on the joint work with X. Buff.

Friday October 06, 2017 2:30 PM  3:30 PM Math Tower P131
 Dmitry Jakobson, McGill University
Distances between closed geodesicsWe discuss upper and lower bounds for the size of gaps in the length spectrum of negatively curved manifolds. For manifolds with algebraic generators for the fundamental group, we establish the existence of exponential lower bounds for the gaps. On the other hand, we show that the existence of arbitrary small gaps is topologically generic: this is established both for surfaces of constant negative curvature, and for the space of negatively curved metrics. While arbitrary small gaps are topologically generic, it is plausible that the gaps are not too small for almost every metric; we discuss a result in this direction. Joint work with Dima Dolgopyat.

Friday October 13, 2017 2:30 PM  3:30 PM Math Tower P131
 Peter Makienko, Instituto de Matematicas UNAM
On the summability and instability for rational mapsIn this talk we discuss an approach to the Fatou conjecture
based on different types of summability of the spectrum of critical values.

Friday October 20, 2017 2:30 PM  3:30 PM Math Tower P131
 Yury Kudryashov, Cornell University
Bifurcations of planar vector fieldsIn 1985 V. Arnold stated six problems aimed to outline the future development of the global bifurcation theory in the plane. In particular, he conjectured that a generic finiteparametric family of vector fields on the twosphere is structurally stable. Recently Yulij
Ilyashenko, Ilya Schurov and I disproved this conjecture [1]. It turns out that 3parametric families of vector fields may have numerical invariants of (moderate) topological classification, and 6parametric families may have functional invariants.
I will discuss the main construction of [1], then I'll give a survey of the results obtained by Yu. Ilyashenko and his students N. Goncharuk, me, I. Schurov, N. Solodovnikov, V. Starichkova.
I will also discuss some ongoing research in this field that may lead to many independent numerical invariants of the 3parametric family of vector fields constructed in [1].
[1]: http://arxiv.org/abs/1506.06797

Friday October 27, 2017 2:30 PM  3:30 PM Math Tower P131
 Babak Modami, Stony Brook University
Renormalization in Teichmuller dynamicsAfter a brief review of Masur's criterion which is an example of the renormalization technique in Teichmuller dynamics, we show that an analogue of the criterion does not hold for the WeilPetersson geodesic flow. The talk is based on a joint work with Jeff Brock from 2015.

Friday November 03, 2017 2:30 PM  3:30 PM Math Tower P131
 Pierre Berger, Université Paris 13
Zoology in the Hénon family: twin babies and Milnor's swallowsWe study $C^{d,r}$Hénonlike families $(f_{a, b})_{a, b}$ with two parameters $(a,b)∈ \mathbb{R}^2$. We show the existence of an open set of parameters $(a,b)∈ D$, so that a renormalization chart conjugates an iterate of $f_{a, b}$ to a perturbation of $(x,y)→ ((x^2+c_1)^2+c_2,0)$. We prove that the map $(a,b)∈ D→ (c_1,c_2)$ is a $C^d$diffeomorphism ; as first numerically conjectured by Milnor in 1992.
Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{a, b}$ displays exactly two different renormalized Hénonlike maps which attract Lebesgue a.e. point with bounded forward orbit. A great freedom in the choice of the renormalized parameters enables us to deduce in particular the existence of a Hénon map with exactly $2$ sinks (an answer to a Question by Lyubich).
The proof is based on a generalization of puzzle pieces for Hénonlike maps, and on a generalization of both the affinelike formalism of PalisYoccoz and the cross map of Shilnikov. The distortion bounds enable $C^{d,r}$renormalizations without loss of regularity.

Friday November 10, 2017 2:30 PM  3:30 PM Math Tower P131
 Michael Yampolsky, University of Toronto
New results on renormalization and rigidity of analytic critical circle mapsI will review the renormalization theory of maps of the circle with critical points, and will present some recent results and applications, and discuss the problems which remain open.

Friday November 17, 2017 2:30 PM  3:30 PM Math Tower P131
 Enrique Pujals, IMPA
Dissipative diffeomorphisms of the disk with zero entropy: structure of periodic points and infinite renormalizationWe will discuss a class of volumecontracting surface diffeomorphisms (named strong dissipative diffeomorphisms) whose dynamics is intermediate between onedimensional dynamics and general surface dynamics.
For the particular case of the disk, we will consider the ones that have zero entropy, explaining their structure of periodic orbits and showing that they are infinitely renormalizable if they are in the boundary of zero entropy.
The talk is a joint works with Sylvain Crovisier and Charles Tresser.

Friday December 01, 2017 2:30 PM  3:30 PM Math Tower P131
 Sabya Mukherjee, Stony Brook University
TBATBA

