Connectedness loci of complex polynomials: beyond the Mandelbrot set

Connectedness loci of complex polynomials: beyond the Mandelbrot set Sabyasachi Mukherjee Stony Brook University TIFR, June 2016

Contents 1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint with Hiroyuki Inou) 4 Proof ideas

1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint with Hiroyuki Inou) 4 Proof ideas

Holomorphic dynamics Holomorphic dynamics is the study of iteration of holomorphic maps (polynomials/rational maps/entire functions or more generally, complex analytic maps on complex manifolds).

Holomorphic dynamics Holomorphic dynamics is the study of iteration of holomorphic maps (polynomials/rational maps/entire functions or more generally, complex analytic maps on complex manifolds). Founders: Pierre Fatou and Gaston Julia (around 1920).

Holomorphic dynamics Holomorphic dynamics is the study of iteration of holomorphic maps (polynomials/rational maps/entire functions or more generally, complex analytic maps on complex manifolds). Founders: Pierre Fatou and Gaston Julia (around 1920). Around 1980, Benoit Mandelbrot drew a computer picture of what is now known as the Mandelbrot set.

Holomorphic dynamics Holomorphic dynamics is the study of iteration of holomorphic maps (polynomials/rational maps/entire functions or more generally, complex analytic maps on complex manifolds). Founders: Pierre Fatou and Gaston Julia (around 1920). Around 1980, Benoit Mandelbrot drew a computer picture of what is now known as the Mandelbrot set. Adrien Douady and John Hubbard proved fundamental theorems which paved the way for subsequent work on parameter spaces of holomorphic maps.

The dynamical plane Let f be a monic centered polynomial of degree d.

The dynamical plane Let f be a monic centered polynomial of degree d. Here are some dynamically defined sets that we would be interested in:

The dynamical plane Let f be a monic centered polynomial of degree d. Here are some dynamically defined sets that we would be interested in: The set of all points whose orbits converge to is called the basin of infinity A (f ).

The dynamical plane Let f be a monic centered polynomial of degree d. Here are some dynamically defined sets that we would be interested in: The set of all points whose orbits converge to is called the basin of infinity A (f ). The set of all points which remain bounded under all iterations of f is called the filled Julia set K(f ).

The dynamical plane Let f be a monic centered polynomial of degree d. Here are some dynamically defined sets that we would be interested in: The set of all points whose orbits converge to is called the basin of infinity A (f ). The set of all points which remain bounded under all iterations of f is called the filled Julia set K(f ). The boundary of the filled Julia set is called the Julia set J(f ).

The dynamical plane Let f be a monic centered polynomial of degree d. Here are some dynamically defined sets that we would be interested in: The set of all points whose orbits converge to is called the basin of infinity A (f ). The set of all points which remain bounded under all iterations of f is called the filled Julia set K(f ). The boundary of the filled Julia set is called the Julia set J(f ). Some examples of filled Julia sets.

The parameter space Let F be a family of monic centered polynomials of degree d depending holomorphically on some parameters. We can define the connectedness locus of F as:

The parameter space Let F be a family of monic centered polynomials of degree d depending holomorphically on some parameters. We can define the connectedness locus of F as: Definition The connectedness locus of F is defined as C(F) = {f F : K(f ) is connected}.

The parameter space Let F be a family of monic centered polynomials of degree d depending holomorphically on some parameters. We can define the connectedness locus of F as: Definition The connectedness locus of F is defined as C(F) = {f F : K(f ) is connected}. Example: The connectedness locus of degree 2 monic centered polynomials is the Mandelbrot set. The Mandelbrot set.

Topology of the Mandelbrot set: ray landing, and the lamination model Landing of rational parameter rays of the Mandelbrot set.

Topology of the Mandelbrot set: small copies via renormalization A baby Mandelbrot set (homeomorphic to the original one).

Higher dimensional parameter spaces Understanding the parameter spaces of polynomials of higher degree is harder. The parameter space of cubic polynomials already offers more challenges.

Higher dimensional parameter spaces Understanding the parameter spaces of polynomials of higher degree is harder. The parameter space of cubic polynomials already offers more challenges. Up to affine conjugation, any cubic polynomial can be written as g a,b (z) = z 3 3az + b; a, b C.

Higher dimensional parameter spaces Understanding the parameter spaces of polynomials of higher degree is harder. The parameter space of cubic polynomials already offers more challenges. Up to affine conjugation, any cubic polynomial can be written as g a,b (z) = z 3 3az + b; a, b C. We will restrict to the region {a < 0, b R}.

Higher dimensional parameter spaces Understanding the parameter spaces of polynomials of higher degree is harder. The parameter space of cubic polynomials already offers more challenges. Up to affine conjugation, any cubic polynomial can be written as g a,b (z) = z 3 3az + b; a, b C. We will restrict to the region {a < 0, b R}. The two critical orbits {g n a,b (± a)} n of g a,b are symmetric w.r.t. the real line.

Renormalization in real cubics Left: Blow-up of a suitable region in the parameter plane of real cubic polynomials. Right: The dynamical plane of a real cubic polynomial with real-symmetric critical orbits.

Renormalization in real cubics Left: Blow-up of a suitable region in the parameter plane of real cubic polynomials. Right: The dynamical plane of a real cubic polynomial with real-symmetric critical orbits. Via renormalization, one can define a map from this black object (on the left) to the connectedness locus of quadratic anti-holomorphic polynomials.

1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint with Hiroyuki Inou) 4 Proof ideas

The tricorn Up to affine conjugation, any quadratic anti-holomorphic polynomial can be written as f c (z) = z 2 + c; c C.

The tricorn Up to affine conjugation, any quadratic anti-holomorphic polynomial can be written as f c (z) = z 2 + c; c C. The tricorn is the connectedness locus of this family: T = {c C : K( z 2 + c) is connected}.

The tricorn Up to affine conjugation, any quadratic anti-holomorphic polynomial can be written as f c (z) = z 2 + c; c C. The tricorn is the connectedness locus of this family: T = {c C : K( z 2 + c) is connected}. The tricorn.

Parameter rays of the tricorn Theorem (Nakane) The tricorn is full, compact, and connected. Moreover, there is a dynamically natural diffeomorphism from C \ T onto C \ D.

Parameter rays of the tricorn Theorem (Nakane) The tricorn is full, compact, and connected. Moreover, there is a dynamically natural diffeomorphism from C \ T onto C \ D. This theorem allows us to define external parameter rays of the tricorn as pre-images of radial lines in C \ D under the map Φ.

Parameter rays of the tricorn Theorem (Nakane) The tricorn is full, compact, and connected. Moreover, there is a dynamically natural diffeomorphism from C \ T onto C \ D. This theorem allows us to define external parameter rays of the tricorn as pre-images of radial lines in C \ D under the map Φ. Question: Do the external parameter rays of the tricorn always land?

Baby tricorns Like the Mandelbrot set, the tricorn also contains infinitely many small copies of itself (renormalization, again).

Baby tricorns Like the Mandelbrot set, the tricorn also contains infinitely many small copies of itself (renormalization, again). A baby tricorn.

Baby tricorns Like the Mandelbrot set, the tricorn also contains infinitely many small copies of itself (renormalization, again). A baby tricorn. Question: Are baby tricorns homeomorphic to the original tricorn?

1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint with Hiroyuki Inou) 4 Proof ideas

Non-landing parameter rays Theorem There are infinitely many parameter rays that non-trivially accumulate on the boundary of T.

Non-landing parameter rays Theorem There are infinitely many parameter rays that non-trivially accumulate on the boundary of T. Left: Ray landing for the Mandelbrot set. Right: Ray wiggling for the tricorn.

Umbilical cord wiggling Theorem Let H be a non-real hyperbolic component of odd period of T. Then its umbilical cord wiggles. In particular, the tricorn is not path connected.

Umbilical cord wiggling Theorem Let H be a non-real hyperbolic component of odd period of T. Then its umbilical cord wiggles. In particular, the tricorn is not path connected. Wiggly umbilical cords destroy path connectedness.

Discontinuity of straightening Theorem 1 The straightening map from any baby tricorn in the real cubic locus to the original tricorn is discontinuous.

Discontinuity of straightening Theorem 1 The straightening map from any baby tricorn in the real cubic locus to the original tricorn is discontinuous. 2 Let H be a hyperbolic component of odd period (other than 1) of T. Then the straightening map from the baby tricorn based at H to T is discontinuous.

1 Background 2 Antiholomorphic Dynamics 3 Main Theorems (joint with Hiroyuki Inou) 4 Proof ideas

Ray wiggling Landing of a parameter ray implies that the corresponding dynamical ray projects to a round circle in the repelling Ecalle cylinder.

Ray wiggling Landing of a parameter ray implies that the corresponding dynamical ray projects to a round circle in the repelling Ecalle cylinder.

Ray wiggling Landing of a parameter ray implies that the corresponding dynamical ray projects to a round circle in the repelling Ecalle cylinder. The projection of the basin of infinity onto the repelling Ecalle cylinder is a conformal annulus bounded by fractal structures (part of the Julia set) from above and below, and the projection of the dynamical ray is the core curve of this conformal annulus. The core curve is a round circle if and only if the annulus is symmetric with respect to this round circle.

Ray wiggling We show that except in a very special situation, such a miracle never happens.

Ray wiggling We show that except in a very special situation, such a miracle never happens. Thus, we have:

Ray wiggling We show that except in a very special situation, such a miracle never happens. Thus, we have: Theorem There are infinitely many parameter rays that non-trivially accumulate on the boundary of T.

Umbilical cord wiggling

Umbilical cord wiggling Lemma (A local consequence) If the umbilical cord of a hyperbolic component of T lands, then the landing point has a real-symmetric parabolic germ.

A local to global principle Analytic continuation yields a polynomial correspondence.

A local to global principle Analytic continuation yields a polynomial correspondence. Now one can use the work of Ritt on decomposition of polynomials with respect to composition to conclude that: Theorem The parabolic germ of f c is real-symmetric if and only if c is real.

Discontinuity of straightening

Discontinuity of straightening Continuity of the straightening map would imply that it is a homeomorphism onto its range (injective, compact domain).

Discontinuity of straightening Continuity of the straightening map would imply that it is a homeomorphism onto its range (injective, compact domain). Then the inverse of the straightening map would send a landing curve to a wiggly curve!

Discontinuity of straightening Continuity of the straightening map would imply that it is a homeomorphism onto its range (injective, compact domain). Then the inverse of the straightening map would send a landing curve to a wiggly curve! Theorem (Discontinuity of Straightening) Let H k be a hyperbolic component of odd period k 1 of T. Then the straightening map from the baby baby tricorn based at H k to T is discontinuous.

Thank you!