Challenges Rebirth M Universality Polynomial Julia sets with positive measure Why bother? Quasiconformal NILF Measure 0? Measure 0 Dimension 2 Measure > 0? The plan Xavier Buff & Arnaud Ch´ eritat ” ” Universit´ e Paul Sabatier (Toulouse III) Thanks ` A la m´ emoire d’Adrien Douady 1 / 16
Challenges At the end of the 1920’s, after the root works of Fatou and Julia Challenges Rebirth on the iteration of rational maps, there remained important open M Universality questions. Here is a selection: Why bother? Quasiconformal NILF Measure 0? Measure 0 Dimension 2 Measure > 0? The plan ” ” Thanks 2 / 16
Challenges At the end of the 1920’s, after the root works of Fatou and Julia Challenges Rebirth on the iteration of rational maps, there remained important open M Universality questions. Here is a selection: Why bother? Quasiconformal – Can a rational map have a wandering Fatou components? NILF Measure 0? Measure 0 Dimension 2 Measure > 0? The plan ” ” Thanks 2 / 16
Challenges At the end of the 1920’s, after the root works of Fatou and Julia Challenges Rebirth on the iteration of rational maps, there remained important open M Universality questions. Here is a selection: Why bother? Quasiconformal – Can a rational map have a wandering Fatou components? NILF Measure 0? – Examples of rational maps were known, for which the Julia set Measure 0 Dimension 2 J is the whole Riemann sphere. The others have a Julia set of Measure > 0? empty interior. But what about their measure? The plan ” ” Thanks 2 / 16
Challenges At the end of the 1920’s, after the root works of Fatou and Julia Challenges Rebirth on the iteration of rational maps, there remained important open M Universality questions. Here is a selection: Why bother? Quasiconformal – Can a rational map have a wandering Fatou components? NILF Measure 0? – Examples of rational maps were known, for which the Julia set Measure 0 Dimension 2 J is the whole Riemann sphere. The others have a Julia set of Measure > 0? empty interior. But what about their measure? The plan ” ” – Fatou asked whether, in the set of rational maps of given Thanks degree d � 2 , those that are hyperbolic form a dense subset 1 . The same question holds in the set of polynomials. All these questions are already difficult for degree 2 polynomials. 1 it is known to be an open subset 2 / 16
Rebirth In the 80s, computers helped revive the subject. Mandelbrot Challenges Rebirth drawed the notion of fractals. J.H. Hubbard investigated M Universality Newton’s method and got Adrien to enter in the field. This was Why bother? the birth of the holomorphic Dynamics school. Quasiconformal NILF Measure 0? Measure 0 Dimension 2 Measure > 0? The plan ” ” Thanks 3 / 16
Rebirth In the 80s, computers helped revive the subject. Mandelbrot Challenges Rebirth drawed the notion of fractals. J.H. Hubbard investigated M Universality Newton’s method and got Adrien to enter in the field. This was Why bother? the birth of the holomorphic Dynamics school. Quasiconformal NILF Measure 0? After having investigated elaborate questions on infinite Measure 0 Dimension 2 dimensional Banach algebraic varieties, Adrien told his colleagues Measure > 0? he would focus on iterating z 2 + c . The plan ” ” Thanks 3 / 16
Rebirth In the 80s, computers helped revive the subject. Mandelbrot Challenges Rebirth drawed the notion of fractals. J.H. Hubbard investigated M Universality Newton’s method and got Adrien to enter in the field. This was Why bother? the birth of the holomorphic Dynamics school. Quasiconformal NILF Measure 0? After having investigated elaborate questions on infinite Measure 0 Dimension 2 dimensional Banach algebraic varieties, Adrien told his colleagues Measure > 0? he would focus on iterating z 2 + c . The plan ” ” The family P c : z �→ z 2 + c Thanks – looks simple and useless in its aspect – is very complicated in the facts – and universal 3 / 16
The Mandelbrot set Dichotomy : Challenges Rebirth – J ( P c ) connected ⇐ ⇒ c ∈ M M – J ( P c ) Cantor oterwise Universality Why bother? Quasiconformal NILF The boundary ∂M is the Measure 0? bifurcation locus of the dynam- Measure 0 Dimension 2 ics, i.e. the set of parameters c Measure > 0? where the Julia set do not vary The plan ” continuously with respect to c . ” Thanks Theorem. MLC = ⇒ Fatou 2 (Douady, Hubbard) If the Mandelbrot set is locally connected, then the set of c such that P c is hyperbolic is dense in C . 4 / 16
Universality Challenges Rebirth M Universality Why bother? Quasiconformal copies of the bound- Quasiconformal ary ∂M are found in every neigh- NILF borhoood of every point of most bi- Measure 0? Measure 0 furcation loci, including ∂M itself. Dimension 2 Measure > 0? The plan ” Douady and Hubbard ex- ” plained this with their theory Thanks of polynomial-like maps. 5 / 16
Why bother? The Julia set is the place where a given rational map is chaotic, Challenges Rebirth and one may wonder whether there is a non-zero probability that M Universality a randomly chosen point may belong to the locus of chaos. Why bother? Quasiconformal NILF Measure 0? Measure 0 Dimension 2 Measure > 0? The plan ” ” Thanks 6 / 16
Why bother? The Julia set is the place where a given rational map is chaotic, Challenges Rebirth and one may wonder whether there is a non-zero probability that M Universality a randomly chosen point may belong to the locus of chaos. Why bother? Quasiconformal NILF Julia sets of positive measure were known in other settings: Measure 0? Measure 0 ■ Indeed there are rational maps whose Julia set is the whole Dimension 2 Measure > 0? Riemann sphere. Mary rees proved there can be lots of The plan them. ” ” Thanks ■ Transcendental entire maps: McMullen proved that in the sine family, the Julia sets always have positive measure, even when their interior is empty. 6 / 16
Quasiconformal methods At the end of the 80s, quasiconformal method were introduced. Challenges Rebirth These powerful methods allowed new progress, among which: M Universality – The end of the classification of the connected components of Why bother? Quasiconformal the Fatou sets, with Sullivan’s proof that there are no wandering NILF Measure 0? components . Measure 0 Dimension 2 – Shishikura’s optimal sharpening of Fatou’s inequality : a Measure > 0? degree d rational map has at most 2 d − 2 non repelling cycles. The plan ” ” – An equivalent formulation of Fatou’s conjecture (Ma˜ ne, Sad, Thanks Sullivan) Fatou 2 ⇐ ⇒ NILF : the density of hyperbolicity for degree 2 polynomials is equivalent to “No degree 2 polynomial has an invariant line field”. 7 / 16
No invariant line field Challenges Fatou 2 ⇐ ⇒ NILF (Ma˜ ne, Sad, Sullivan): the density of Rebirth hyperbolicity for degree 2 polynomials is equivalent to “No M Universality degree 2 polynomial has an invariant line field”. Why bother? Quasiconformal An invariant line field is an element µ ∈ L ∞ ( C ) with values in NILF S 1 ∪ { 0 } almost everywhere (a.e.), such that Measure 0? Measure 0 Dimension 2 µ ( z ) = µ ( P ( z )) P ′ ( z ) Measure > 0? P ′ ( z ) a.e., The plan ” ” such that the support of µ is contained in the Julia set, and such Thanks that µ is not vanishing a.e. 8 / 16
No invariant line field Challenges Fatou 2 ⇐ ⇒ NILF (Ma˜ ne, Sad, Sullivan): the density of Rebirth hyperbolicity for degree 2 polynomials is equivalent to “No M Universality degree 2 polynomial has an invariant line field”. Why bother? Quasiconformal An invariant line field is an element µ ∈ L ∞ ( C ) with values in NILF S 1 ∪ { 0 } almost everywhere (a.e.), such that Measure 0? Measure 0 Dimension 2 µ ( z ) = µ ( P ( z )) P ′ ( z ) Measure > 0? P ′ ( z ) a.e., The plan ” ” such that the support of µ is contained in the Julia set, and such Thanks that µ is not vanishing a.e. Such a line field requires a Julia set with non zero Lebesgue measure. 8 / 16
The measure zero conjecture So if the Fatou 2 conjecture fails, then there is a Julia set with Challenges Rebirth positive measure (the converse likely does not hold). M Universality The hope was then that every Julia set of a P c has Lebesgue Why bother? Quasiconformal measure equal to 0 , which would have proved Fatou 2 , whence NILF Measure 0? the measure zero conjecture and its generalization: Measure 0 Dimension 2 ■ Every degree 2 polynomial has a Julia set of measure 0 . Measure > 0? The plan ” ■ (gnrlz.) Every degree d � 2 rational map has a Julia set ” either equal to the Riemann sphere or of measure 0 . Thanks 9 / 16
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