May 02, 2022
Monday

09:15 AM  09:30 AM


Welcome

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video


 Abstract
 
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09:30 AM  10:30 AM


Spiraling Domains in Dimension 2
Xavier Buff (Université de Toulouse III (Paul Sabatier))

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
I will present work in progress with Jasmin Raissy. We study the dynamics of polynomials maps of C^2 which are tangent to the identity at some fixed point. Our goal is to prove that there exist such maps for which the basin of attraction of the fixed point has infinitely many fixed connected components. This should be the case for the map (x,y)>(x+y^2+2x^2y,y+x^2+2y^2x)
 Supplements


10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Real Quadratic Maps
John Milnor (Institute for Mathematical Sciences)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Real quadratic rational maps with real critical points form one of the simplest and most classical dynamical systems. There has been real progress in recent years, especially due to work of Khashayar Filom and Kevin Pilgrim; but there are still open problems.
Joint work with Araceli Bonifant and Scott Sutherland
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Spectrum of the Laplacian for the Basilica Group and Renormalization
NguyenBac Dang (Université ParisSaclay)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
In this talk, based on an ongoing work with Eric Bedford, Rostislav Grigorchuk and Mikhail Lyubich, I will present how the spectrum of the Laplacian on the Basilica Schreier graphs is related to the iteration of the rational map and to the pullback of a particular line.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Complex Rotation Numbers and Renormalization
Nataliia Goncharuk (University of Toronto, Mississauga)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
The complex rotation number (suggested by V.Arnold) is an invariant related to the dynamics of an analytic circle diffeomorphism $f$. Complex rotation numbers give rise to a nice fractal set "bubbles" analogous to classical Arnold's tongues.
I will give a survey on complex rotation numbers and list open problems. Also, I will explain how the renormalization operator makes the "bubbles" selfsimilar and controls their sizes.
 Supplements



May 03, 2022
Tuesday

09:30 AM  10:30 AM


Dimension of Expanding Maps
Volodymyr Nekrashevych (Texas A & M University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Expanding covering maps (e.g., subhyperbolic rational functions restricted to their Julia sets) are encoded (uniquely determined up to topological conjugacy) by the associated iterated monodromy groups. We will discuss how dimension of the space can be deducted directly from the structure of the iterated monodromy groups, and how algebraic properties of the group are related to the dimension.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:00 AM  12:00 PM


Stable Marked Points in Holomorphic Dynamics
Thomas Gauthier (Université ParisSaclay)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
A celebrated theorem of McMullen states that if an algebraic family of rational maps is Jstable, then it has to be a family of Lattès maps or an isotrivial family. DeMarco generalized partially this result proving that, if a pair $(f,a)$ constituted of a marked point and an algebraic family of rational maps is stable, i.e. the family of iterates of $a$ is a normal family of functions of the parameter, then it is either stably preperiodic, or isotrivial.
In this talk, I will explain how to adapt this result to families of endomorphisms of higher dimensional projective spaces together with a marked point. If time allows, I will give an arithmetic interpretation of this result. This is a joint work with Gabriel Vigny.
 Supplements


12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Irreducibility of Gleason polynomials Implies Irreducibility of Per_n
Rohini Ramadas (University of Warwick)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Per_n is a punctured (nodal) Riemann surface parametrizing quadratic rational maps with an nperiodic critical point; its irreducibility over C is an open question. The Gleason polynomial G_n is the polynomial whose roots are {c such that 0 is nperiodic under z^2+c}; its irreducibility over Q is an open question. I will discuss very recent workinprogress, finding a smooth Qrational point “at infinity” on Per_n, and using this to conclude that if G_n is irreducible over Q, then Per_n is irreducible over C. This talk will also include joint work with Rob Silversmith.
 Supplements



03:00 PM  03:30 PM


Afternoon Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


MLC Along the Real Line
Dzmitry Dudko (Stony Brook University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
The problem of local connectivity of the Mandelbrot set goes back to the 80s, and is now closely linked to the Renormalization Theory of quadratic polynomials. A key task is to establish a priori bounds (compactness) for the quadraticlike renormalization operator. Working in the near degenerate regime, we prove such bounds for maps with real combinatorics. As a consequence, real combinatorial classes are singletons on the real line. We also obtain a uniform control of the shapes of realsymmetric copies of the Mandelbrot set, as well as their universal scaling properties. Joint work in progress with Jeremy Kahn and Misha Lyubich.
 Supplements




May 04, 2022
Wednesday

09:00 AM  10:00 AM


Topogical Entropy of Rational Maps (Over any Metrized Field)
Charles Favre (École Polytechnique)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Gromov, Dinh and Sibony gave an upperbound for the topological entropy of any rational map on a projective complex variety. In a joint work with T. T. Truong and J. Xie, we recently extended this bound to arbitrary complete metrized nonArchimedean fields. We also related maps over such fields for which the topological entropy vanishes to the notion of good reduction.
 Supplements



10:05 AM  11:05 AM


OutoftheBox Rigidity via Box Mappings
Kostiantyn Drach (Institute of Science and Technology Austria)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
In holomorphic dynamics, complex box mappings arise as first return maps to wellchosen collections of disks. They generalize the classical notion of polynomiallike mapping, as we allow for infinitely many disk components in the domain and finitely many components in the range (while the restriction of the map to each domain component is still a branched covering). In this way, one can talk about generalized renormalization of a given map every time this map restricts to a nontrivial box mapping (even if the map is nonrenormalizable in the classical DouadyHubbard sense). This concept turned out to be extremely useful for tackling diverse problems in natural families of maps.
In our talk, we will discuss rigidity properties of complex box mappings, e.g. quasiconformal rigidity, shrinking of puzzle pieces, etc. This will give us a toolbox of useful results. We will then show how to apply the toolbox almost as a “black box” to conclude similar results in those families of rational maps which are renormalizable in this generalized sense. We will illustrate the success of our strategy with several examples, including polynomials with Siegel disks of bounded type rotation number (work in progress) and Newton maps of polynomials (in both cases, polynomials are of arbitrary degree). The talk is based on joint work with several people, including the participants of the MSRI semester D. Schleicher, S. van Strien, and J. Yang.
 Supplements


11:05 AM  11:30 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
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11:30 AM  12:30 PM


Special Parameters in Spaces of Meromorphic Maps
Anna Miriam Benini (Università di Parma)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
In this talk we will explore parameter spaces of meromorphic maps with finitely many singular values. We will list various types of parameters for which the orbits of one or more singular values have peculiar characteristics, for example, are preperiodic, converge to a parabolic cycle, or are truncated after a certain number of iterates. We will look at the relations between such parameters and show that several of these types of parameters are dense in the bifurcation locus. To do so we will need to implement new strategies with respect to the rational case. This is joint work with N. Fagella and M. Astorg.
 Supplements




May 05, 2022
Thursday

09:30 AM  10:30 AM


The Modular Mandelbrot Set
Luciana Luna Anna Lomonaco (Institute of Pure and Applied Mathematics (IMPA))

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
The Modular Mandelbrot set is the connectedness locus of a family of (2:2) correspondences (introduced by Bullett and Penrose in 1994). We show that these correspondences are matings between the modular group and the family of quadratic rational maps P_A(z)=z+1/z+A, and that there exists a dynamical homeomorphism between the modular Mandelbrot set and the parabolic Mandelbrot set (this last being the connectedness locus of the family P_A(z), and is itself homeomorphic to the classical Mandelbrot set by a result of Petersen and Roesch). The talk is based on joint work with Shaun Bullett.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Nice Quasidisks and Prime Orbit Counting
Juan RiveraLetelier (University of Rochester)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
For a complex rational map, a nice set is somewhat analogous to a Yoccoz puzzle for polynomials. For a nonuniformly hyperbolic map, we show how to construct nice sets that are quasidisks. These are used to establish a prime orbit theorem with a power savings error term.
This is a joint work with Zhiqiang Li.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


A Spectral Gap for the Transfer Operator on Complex Projective Spaces
Fabrizio Bianchi (Université de Lille)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
We study the transfer (PerronFrobenius) operator on Pk(C) induced by a generic holomorphic endomorphism and a suitable continuous weight. We prove the existence of a unique equilibrium state and we introduce various new invariant functional spaces, including a dynamical Sobolev space, on which the action of f admits a spectral gap. This is one of the most desired properties in dynamics. It allows us to obtain a list of statistical properties for the equilibrium states such as the equidistribution of points, speed of convergence, Kmixing, mixing of all orders, exponential mixing, central limit theorem, BerryEsseen theorem, local central limit theorem, almost sure invariant principle, law of iterated logarithms, almost sure central limit theorem and the large deviation principle. Most of the results are new even in dimension 1 and in the case of constant weight function, i.e., for the operator f_*. Our construction of the invariant functional spaces uses ideas from pluripotential theory and interpolation between Banach spaces. This is a joint work with TienCuong Dinh.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


On the Limits of Quasiconformal Deformation of Rational Maps
Mitsuhiro Shishikura (Kyoto University)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
We consider the quasiconformal deformation of rational maps supported on Fatou components so that the annuli are stretched. The limit is described by a piecewise linear map on a tree, and inverse construction will be discussed.
 Supplements



May 06, 2022
Friday

09:30 AM  10:30 AM


When do Two Rational Functions have Locally Biholomorphic Julia Sets?
Romain Dujardin (Sorbonne Université)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
We address the following question, whose interest was recently renewed by problems arising in arithmetic dynamics and goes back to classical work of Fatou and Julia: under which conditions does there exist a local biholomorphism between the Julia sets of two given onedimensional rational maps? In particular we find criteria ensuring that such a local isomorphism is induced by an algebraic correspondence. This is joint work with Charles Favre and Thomas Gauthier.
 Supplements



10:30 AM  11:00 AM


Break

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



11:00 AM  12:00 PM


Deformation Space Analogies between Kleinian Reflection Groups and Rational Maps
Sabyasachi Mukherjee (Tata Institute of Fundamental Research)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
We will describe an explicit correspondence between certain Kleinian reflection groups and certain antiholomorphic rational maps acting on the Riemann sphere. This correspondence has several dynamical and parameter space consequences. To illustrate these, we will discuss some striking similarities between the deformation spaces of these two classes of conformal dynamical systems, including an analogue of Thurston’s compactness theorem for antiholomorphic rational maps and relations between the global topology of the corresponding deformation spaces. Based on joint work with Russell Lodge and Yusheng Luo.
 Supplements



12:00 PM  02:00 PM


Lunch

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



02:00 PM  03:00 PM


Invariant Currents for Surface Maps with Transcendental First Dynamical Degree
Jeff Diller (University of Notre Dame)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
Let f:X>X be a rational map in complex dimension two. Assuming the first dynamical degree of f exceeds 1, there is a standard way to construct a dynamically natural positive closed current invariant under pullback. The construction requires, however, that f be `algebraically stable.' In this talk I'll focus on some recent examples where the dynamical degree is transcendental and construct invariant currents in a situation where algebraic stability is unachievable.
 Supplements


03:00 PM  03:30 PM


Afternoon Tea

 Location
 MSRI: Atrium
 Video


 Abstract
 
 Supplements



03:30 PM  04:30 PM


Bifurcation Loci of Families Finite Type Meromorphic Maps
Núria Fagella (University of Barcelona)

 Location
 MSRI: Simons Auditorium, Online/Virtual
 Video

 Abstract
In this talk we will discuss parameter spaces of natural families of transcendental meromorphic maps of finite type,
parametrized over a complex manifold. In this setting, a new type of bifurcation arises for which a periodic cycle can disappear to infinity along a parameter curve. We shall relate these type of bifurcation parameters, with those for which an asymptotic value is a prepole (virtual cycle parameters), and use these relations to study stability of Julia sets ($J$stability), concluding that $J$stable parameters form an open and dense subset of the parameter space. All our theorems hold for general finite type maps in the sense of Epstein, satisfying certain conditions.
This is joint work with Anna Miriam Benini and Matthieu Astorg.
 Supplements


04:30 PM  06:30 PM


Reception

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