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WORKSHOP: FRACTALS IN GEOMETRY AND DYNAMICS

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WHERE: GRADUATE CENTER, CUNY, ROOM 6417

WHEN: FRIDAY, JANUARY 20, 2017

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SPEAKERS:

MARIO BONK, UCLA**Title:** Latt\`es maps and their dynamics

**Abstract:** Latt\`es maps have a long history going back to more than a hundred years. In complex dynamics they often appear as exceptional cases among generic rational maps. Among other things they are distinguished by their associated orbifolds, their measure-theoretic properties, or their combinatorial expansion rate. In my talk I will give a survey on this subject and present some open problems.

GUY C. DAVID, NYU**Title:** Lipschitz differentiability and rigidity for convex-cocompact actions on rank-one symmetric spaces

**Abstract:** We discuss a recent theorem of the speaker and Kyle Kinneberg concerning rigidity for convex-cocompact actions on non-compact rank-one symmetric spaces, which generalizes a result of Bonk and Kleiner from real hyperbolic space. A key step in the proof studies Cheeger's Lipschitz differentiability spaces and their behavior when embedded in Carnot groups, and so we will also discuss this theory.

ZHIQIANG LI, SBU**Title:** Prime orbit theorems and rational maps

**Abstract:** Periodic orbits play an important role in the study of dynamical systems. In resemblance to the classical Prime Number Theorem in number theory and its relation to the Riemann Hypothesis, it is a natural problem to investigate precise asymptotes for the number of (primitive) periodic orbits as well as the error terms. Such results, known as Prime Orbit Theorems, have been established in many dynamical systems thanks to the works of W. Parry, M. Pollicott, V. Baladi, D. Dolgopyat, C. Liverani, L. Stoyanov, G. A. Margulis, A. Avila, S. Gou\"ezel, J. C. Yoccoz, M. Tsujii, and many others.

In this talk, we are going to introduce a brief history of such results, focusing mainly on the works of F. Naud, H. Oh, and D. Winter on hyperbolic rational maps. We are going to discuss the main ideas used to obtain such results. If time permits, we are going to discuss how to extend such results to a class of non-hyperbolic rational maps known as (rational) expanding Thurston maps. This is a work-in-progress joint with T. Zheng.

RUSSELL LODGE, SBU**Title:** Invariant combinatorics from self-similar groups

**Abstract:** Bartholdi and Nekrashevych's celebrated solution to the twisted rabbit problem has led to a flourishing interaction between self-similar group theory and holomorphic dynamics. I will show how some important combinatorial structures for postcritically finite rational maps (and beyond) can be characterized in terms of group theory. For instance, given a non-postcritical point z_0 in a repelling cycle, group theory determines which postcritical Fatou components contain z_0 in their closure. This result has further implications for the global dynamics of W. Thurston's pullback map on Teichmueller space, and moreover gives a partial answer to Pilgrim's conjecture on the dynamics of multicurves under pullback by a postcritically finite rational map.

MISHA LYUBICH, SBU**Title:** Pacman Renormalization and scaling of the Mandelbrot set at Siegel points

**Abstract:** In the 1980s Branner and Douady discovered a surgery relating various limbs of the Mandelbrot set. We put this surgery in the framework of ``Pacman Renormalization Theory" that combines features of quadratic-like and Siegel renormalizations. Siegel renormalization periodic points (constructed by McMullen in the 1990s) can be promoted to pacman renormalization periodic points. We prove that these periodic points are hyperbolic with one-dimensional unstable manifold. As a consequence, we obtain the scaling laws for the centers of satellite components of the Mandelbrot set near the corresponding Siegel parameters. It is a joint work with Dima Dudko and Nikita Selinger.

REMUS RADU, SBU**Title:** Siegel disks for complex Henon maps

**Abstract:** We look at the family of complex Henon maps which have a semi-indifferent fixed point with eigenvalues w1 and w2, where |w1|<1 and w2=exp(2 pi i t) and t is a Brjuno number. These maps have a Siegel disk and we are interested in the regularity properties of its boundary. When t is the golden mean and the Jacobian is small enough, we show, using hyperbolicity of golden mean renormalization of dissipative Henon-like maps, that the boundary of the Siegel disk is homeomorphic to a circle. This is joint work with D. Gaidashev and M. Yampolsky.

RALUCA TANASE, SBU**Title:** Hedgehogs in higher dimensions

**Abstract:** Hedgehogs in dimension one were introduced by Perez-Marco in the '90s to study linearization properties and dynamics of holomorphic univalent germs of (C, 0) with a neutral fixed point. In this talk we discuss hedgehogs and their dynamics for germs of holomorphic diffeomorphisms of (C^n, 0) with a fixed point at the origin with exactly one neutral eigenvalue. This is based on joint work with T. Firsova, M. Lyubich, and R. Radu.

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SCHEDULE:

Light refreshments available in the Math Lounge, Rm. 4214, between 8:30 am and 2:30 pm.

09:00-10:00 MISHA LYUBICH

10:15-10:45 REMUS RADU

11:00-11:30 RALUCA TANASE

11:45-12:15 ZHIQIANG LI

12:15-02:30 LUNCH BREAK

02:30-03:00 GUY C. DAVID

03:15-03:45 RUSSELL LODGE

04:00-05:00 MARIO BONK

For questions please contact SERGIY MERENKOV (smerenkov@ccny.cuny.edu)