Mating is an operation to construct a rational map f from two polynomials, which are not in conju... more Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of postcritical ray-equivalence classes for the modified mating, but it is enough to implement the pullback map for the formal mating. The proof combines local estimates and the Selinger extension of the Thurston Algorithm to augmented Teichmüller space. The latter is illustrated with several examples of canonical obstructions and canonical strata, including a relation between matings of conjugate polynomials and ...
Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries n... more Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries no invariant line fields and that the topological conjugacy is equivalent to quasi-conformal conjugacy in this case.
In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding ... more In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic 1 Mandelbrot set M 1 has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M. The landing of the (pre)-periodic ray R c θ at the (pre)-periodic point z(c) is locally stable at c when Q l c (z) is repelling and 0 is not in the orbit of z. In particular the landing is globally stable in any connected open set for which Q l c (z) remains repelling and 0 does not enter the forward orbit of R c θ. Definition 1.2. A q-cycle under Q k c of external rays R c 0 ,. .. , R c q−1 co-landing on a common k-periodic point z and numbered in the counter clockwise order around z defines combinatorial rotation number p/q, (p, q) = 1 iff Q k c (R j) = R (j+p) mod q .
Journal of Difference Equations and Applications, 2010
In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in... more In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in super-attracting basins. It is used to construct parabolic puzzles similar to Yoccoz puzzles, but modified near the parabolic fixed point. As an application, we show that any non-renormalizable quadratic rational map in the connectedness locus M1 of Per1(1) has locally connected Julia set.
Proceedings of the London Mathematical Society, 2016
In this paper we prove existence of matings between a large class of renormalizable cubic polynom... more In this paper we prove existence of matings between a large class of renormalizable cubic polynomials with one fixed critical point and another cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part towards a conjecture by Tan Lei, stating that all (cubic) Newton maps can be described as matings or captures.
We prove that the boundary of any bounded Fatou component for a polynomial is a Jordan curve, exc... more We prove that the boundary of any bounded Fatou component for a polynomial is a Jordan curve, except maybe for Siegel disks. * Research partially supported by the ANR ABC and the NSF of China denotes the connected component containing U of the filled Julia set K(f ), then
Herman proved the presence of critical points on the boundary of Siegel disks of unicritical poly... more Herman proved the presence of critical points on the boundary of Siegel disks of unicritical polynomials under some diophantine condition now called the Herman condition. We extend this result to polynomials with two critical points. arXiv:1111.4629v1 [math.DS]
In this article, we study the hyperbolic components of Mc-Mullen maps. We show that the boundarie... more In this article, we study the hyperbolic components of Mc-Mullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the boundary of the unbounded hyperbolic component. This is a dynamical analogue of McMullen's theorem that cusps are dense on the Bers' boundary of Teichmüller space.
In Celebration of John Milnor's 80th Birthday, 2014
This article deals with the question of local connectivity of the Julia set of polynomials and ra... more This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.
Journal of Difference Equations and Applications, 2010
In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in... more In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in super-attracting basins. It is used to construct parabolic puzzles similar to Yoccoz puzzles, but modified near the parabolic fixed point. As an application, we show that any non-renormalizable quadratic rational map in the connectedness locus M1 of Per1(1) has locally connected Julia set.
Carrots for dessert is the title of a section of the paper 'On polynomiallike mappings', [DH]. In... more Carrots for dessert is the title of a section of the paper 'On polynomiallike mappings', [DH]. In that section Douady and Hubbard define a notion of dyadic carrot fields of the Mandelbrot set and more generally for Mandelbrot like families (for a precise statement see below). They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M ′ of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set M1 shrink to points when the denominator diverge to ∞.
Conformal Geometry and Dynamics of the American Mathematical Society, 2010
The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749-8... more The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749-841]. This nest, once combined with the KL-Lemma, has proven to be a powerful machinery, leading to several important advancements in the field of holomorphic dynamics. We give here a presentation of the KSS nest in terms of tableau. This is an effective language invented by Branner and Hubbard to deal with the complexity of the dynamics of puzzle pieces. We show, in a typical situation, how to make the combination between the KSS nest and the KL-Lemma. One consequence of this is the recently proved Branner-Hubbard conjecture. Our estimates here can be used to give an alternative proof of the rigidity property.
In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding ... more In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic 1 Mandelbrot set M 1 has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M.
Mating is an operation to construct a rational map f from two polynomials, which are not in conju... more Mating is an operation to construct a rational map f from two polynomials, which are not in conjugate limbs of the Mandelbrot set. When the Thurston Algorithm for the unmodified formal mating is iterated in the case of postcritical identifications, it will diverge to the boundary of Teichmüller space, because marked points collide. Here it is shown that the colliding points converge to postcritical points of f , and the associated sequence of rational maps converges to f as well, unless f is of type (2, 2, 2, 2). So to compute f , it is not necessary to encode the topology of postcritical ray-equivalence classes for the modified mating, but it is enough to implement the pullback map for the formal mating. The proof combines local estimates and the Selinger extension of the Thurston Algorithm to augmented Teichmüller space. The latter is illustrated with several examples of canonical obstructions and canonical strata, including a relation between matings of conjugate polynomials and ...
Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries n... more Non-renormalizable Newton maps are rigid. More precisely, we prove that their Julia set carries no invariant line fields and that the topological conjugacy is equivalent to quasi-conformal conjugacy in this case.
In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding ... more In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic 1 Mandelbrot set M 1 has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M. The landing of the (pre)-periodic ray R c θ at the (pre)-periodic point z(c) is locally stable at c when Q l c (z) is repelling and 0 is not in the orbit of z. In particular the landing is globally stable in any connected open set for which Q l c (z) remains repelling and 0 does not enter the forward orbit of R c θ. Definition 1.2. A q-cycle under Q k c of external rays R c 0 ,. .. , R c q−1 co-landing on a common k-periodic point z and numbered in the counter clockwise order around z defines combinatorial rotation number p/q, (p, q) = 1 iff Q k c (R j) = R (j+p) mod q .
Journal of Difference Equations and Applications, 2010
In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in... more In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in super-attracting basins. It is used to construct parabolic puzzles similar to Yoccoz puzzles, but modified near the parabolic fixed point. As an application, we show that any non-renormalizable quadratic rational map in the connectedness locus M1 of Per1(1) has locally connected Julia set.
Proceedings of the London Mathematical Society, 2016
In this paper we prove existence of matings between a large class of renormalizable cubic polynom... more In this paper we prove existence of matings between a large class of renormalizable cubic polynomials with one fixed critical point and another cubic polynomial having two fixed critical points. The resulting mating is a Newton map. Our result is the first part towards a conjecture by Tan Lei, stating that all (cubic) Newton maps can be described as matings or captures.
We prove that the boundary of any bounded Fatou component for a polynomial is a Jordan curve, exc... more We prove that the boundary of any bounded Fatou component for a polynomial is a Jordan curve, except maybe for Siegel disks. * Research partially supported by the ANR ABC and the NSF of China denotes the connected component containing U of the filled Julia set K(f ), then
Herman proved the presence of critical points on the boundary of Siegel disks of unicritical poly... more Herman proved the presence of critical points on the boundary of Siegel disks of unicritical polynomials under some diophantine condition now called the Herman condition. We extend this result to polynomials with two critical points. arXiv:1111.4629v1 [math.DS]
In this article, we study the hyperbolic components of Mc-Mullen maps. We show that the boundarie... more In this article, we study the hyperbolic components of Mc-Mullen maps. We show that the boundaries of all hyperbolic components are Jordan curves. This settles a problem posed by Devaney. As a consequence, we show that cusps are dense on the boundary of the unbounded hyperbolic component. This is a dynamical analogue of McMullen's theorem that cusps are dense on the Bers' boundary of Teichmüller space.
In Celebration of John Milnor's 80th Birthday, 2014
This article deals with the question of local connectivity of the Julia set of polynomials and ra... more This article deals with the question of local connectivity of the Julia set of polynomials and rational maps. It essentially presents conjectures and questions.
Journal of Difference Equations and Applications, 2010
In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in... more In this paper, we define a notion of parabolic ray which is analogous to external/internal ray in super-attracting basins. It is used to construct parabolic puzzles similar to Yoccoz puzzles, but modified near the parabolic fixed point. As an application, we show that any non-renormalizable quadratic rational map in the connectedness locus M1 of Per1(1) has locally connected Julia set.
Carrots for dessert is the title of a section of the paper 'On polynomiallike mappings', [DH]. In... more Carrots for dessert is the title of a section of the paper 'On polynomiallike mappings', [DH]. In that section Douady and Hubbard define a notion of dyadic carrot fields of the Mandelbrot set and more generally for Mandelbrot like families (for a precise statement see below). They remark that such carrots are small when the dyadic denominator is large, but they do not even try to prove a precise such statement. In this paper we formulate and prove a precise statement of asymptotic shrinking of dyadic Carrot-fields around M. The same proof carries readily over to show that the dyadic decorations of copies M ′ of the Mandelbrot set M inside M and inside the parabolic Mandelbrot set M1 shrink to points when the denominator diverge to ∞.
Conformal Geometry and Dynamics of the American Mathematical Society, 2010
The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749-8... more The KSS nest is a sophisticated choice of puzzle pieces given in [Ann. of Math. 165 (2007), 749-841]. This nest, once combined with the KL-Lemma, has proven to be a powerful machinery, leading to several important advancements in the field of holomorphic dynamics. We give here a presentation of the KSS nest in terms of tableau. This is an effective language invented by Branner and Hubbard to deal with the complexity of the dynamics of puzzle pieces. We show, in a typical situation, how to make the combination between the KSS nest and the KL-Lemma. One consequence of this is the recently proved Branner-Hubbard conjecture. Our estimates here can be used to give an alternative proof of the rigidity property.
In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding ... more In this paper we develop a combinatorial analytic encoding of the Mandelbrot set M. The encoding is implicit in Yoccoz' proof of local connectivity of M at any Yoccoz parameter, i.e. any at most finitely renormalizable parameter for which all periodic orbits are repelling. Using this encoding we define an explicit combinatorial analytic modelspace, which is sufficiently abstract that it can serve as a go-between for proving that other sets such as the parabolic 1 Mandelbrot set M 1 has the same combinatorial structure as M. As an immediate application we use here the combinatorial-analytic model to reprove that the dyadic veins of M are arcs and that more generally any two Yoccoz parameters are joined by a unique ruled (in the sense of Douady-Hubbard) arc in M.
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