Monodromy and Hénon mappings
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2 Monodromy and Hénon mappings
3 Monodromy and Hénon mappings John Hubbard joint work with Sarah Koch Chris Lipa
4 Acknowledgements John Smillie Eric Bedford Ralph Oberste-Vorth Ricardo Oliva Suzanne Hruska Zin Arai
5 Acknowledgements John Smillie Eric Bedford Ralph Oberste-Vorth Ricardo Oliva Suzanne Hruska Zin Arai The pictures are due to Ben Hinkle,
6 Acknowledgements John Smillie Eric Bedford Ralph Oberste-Vorth Ricardo Oliva Suzanne Hruska Zin Arai The pictures are due to Ben Hinkle, and especially Karl Papadantonakis
7 The formula for Hénon mappings H a,c : C 2 C 2 )) (( x 2 + c ay ) H a,c ( x y = x ( ) ( Note that when a = 0, this reduces to x x 2 + c. When a 0, the map H a,c is invertible. ( ) ( ( )
8 How do we investigate a family of mappings depending on a parameter? We try to draw the parameter space, so that when we click at a specific parameter, a new window comes up showing the corresponding dynamical space.
9 Let us illustrate this for the family x x 2 + c.
10 Let us illustrate this for the family x x 2 + c. In the parameter space, we see the
11 Let us illustrate this for the family x x 2 + c. In the parameter space, we see the Mandelbrot set.
12 Let us illustrate this for the family x x 2 + c. In the parameter space, we see the Mandelbrot set. In the dynamical plane, for each parameter c, we see the corresponding
13 Let us illustrate this for the family x x 2 + c. In the parameter space, we see the Mandelbrot set. In the dynamical plane, for each parameter c, we see the corresponding filled-in Julia set Kc.
14 What makes this exploration work so well is the dichotomy:
15 What makes this exploration work so well is the dichotomy: 0 K c if and only if K c is connected.
16 What makes this exploration work so well is the dichotomy: 0 K c if and only if K c is connected. 0 / K c if and only if K c is a Cantor set.
17 What makes this exploration work so well is the dichotomy: { 0 K c if and only if K c is connected. 0 / Kc if and only if K c is a Cantor set.
18 What makes this exploration work so well is the dichotomy: { 0 K c if and only if K c is connected. 0 / K c if and only if K c is a Cantor set. The set M is the set
19 What makes this exploration work so well is the dichotomy: { 0 K c if and only if K c is connected. 0 / K c if and only if K c is a Cantor set. The set M is the set {c C K c is connected } = {c C 0 K c } ( ) ( )
20 What makes this exploration work so well is the dichotomy: { 0 K c if and only if K c is connected. 0 / K c if and only if K c is a Cantor set. The set M is the set {c C K c is connected } = {c C 0 K c } ( ) ( ) This interesting because of the first characterization, and easy to compute because of the second.
21 Sadly, we do not have the same dichotomy for Hénon mappings: There are Hénon mappings where K is connected and others for which K is a Cantor set and many other things in between. In addition, there is no clear criterion for determining which alternative occurs. Still, the program SaddleDrop does its best.
22 What is being drawn in the dynamical space? If H(p) = p, [DH(p)] has eigenvalues λ, µ with λ > 1, µ < 1, [ ( )] has ei [DH(p)]v = λv, then ( γ(z) = lim n H n p + z ) ( λ n v exists, and the is a unstable parametrization manifold of the of unstable, sa manifold of p, satisfying γ(λz) = H(γ(z)).
23 SaddleDrop draws in the domain C the level curves of the rate of escape function ( ) G + ( x y = lim n 1 2 n log+ ( y) x H n Thus the black is the intersection of the unstable manifold with K.
24 The map γ : C C 2 folds C in some very complicated way into ) ( C 2 (
25 What are we seeing in parameter space Let us look at a movie, Each frame is a picture of the c -plane for fixed a. The parameter a (the Jacobian) describes leaves 0 spiraling
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27 What are we seeing in parameter space This is harder to answer For one thing, there are two setttings: --fastest escaping, which tries to draw the locus where K is connected, and --slowest escaping, which tries to draw the locus where H is a horseshoe mapping, and hence K is a Cantor set on which the dynamics is conjugate to the full shift on two symbols
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29 What are we seeing?
30 What are we seeing? Most points on the boundary of M are the the vertex of a
31 What are we seeing? Most points on the boundary of M are the the vertex of a Cone on the product of a circle with a Cantor set
32 What are we seeing? Most points on the boundary of M are the the vertex of a Cone on the product of a circle with a Cantor set This locus intersects every complex line in a Cantor set
33 What are we seeing? Most points on the boundary of M are the the vertex of a Cone on the product of a circle with a Cantor set This locus intersects every complex line in a Cantor set At points of this locus, there is a tangency of and
34 What are we seeing? Most points on the boundary of M are the the vertex of a Cone on the product of a circle with a Cantor set This locus intersects every complex line in a Cantor set At points of this locus, there is a tangency of J + ( and
35 What are we seeing? Most points on the boundary of M are the the vertex of a Cone on the product of a circle with a Cantor set This locus intersects every complex line in a Cantor set At points of this locus, there is a tangency of J + ( and J 2 +
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38 But there are other points: Those for which there is a periodic cycle with one multiplier of absolute value 1.
39 But there are other points: Those for which there is a periodic cycle with one multiplier of absolute value 1. These form algebraic curves, which will intersect a complex line in parameter space in finitely many points.
40 But there are other points: Those for which there is a periodic cycle with one multiplier of absolute value 1. These form algebraic curves, which will intersect a complex line in parameter space in finitely many points. The little Mandelbrot sets live in only one sheet of the cone over Cantor set cross circle, leaving gaps in the others.
41 Hénon Horseshoes If you pick c outside the Mandelbrot set so that Kc is a Cantor set, then for a sufficiently small, the Hénon map will be a horseshoe. For c real and < -2, and a real and small, it is an ordinary horseshoe.
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55 ( Define Π : K {A, B} by Π(x) = { A B x in A-region x in B-region ) Define Φ : K {A, B} Z by ) ( Φ(x) = (..., Π ( H 1 (x) ), Π(x), Π(H(x)),... ))
56 Kneading Sequences If H is a horseshoe map, then Every bounded point of K has a bi-infinite itinerary of A s and B s
57 Kneading Sequences If H is a horseshoe map, then Every bounded point of K has a bi-infinite itinerary of A s and B s Every itinerary corresponds to exactly one point, so
58 Kneading Sequences If H is a horseshoe map, then Every bounded point of K has a bi-infinite itinerary of A s and B s Every itinerary corresponds to exactly one point, so We have a homeomorphism between K and {A,B} Z
59 Kneading Sequences K H a,c K Φ Φ {A, B} Z σ {A, B} Z
60 Horseshoe Locus The complex horseshoe locus L is the open region of parameter space where the action of the Hénon map on K set is conjugate to the horseshoe map.
61 Horseshoe Locus The complex horseshoe locus L is the open region of parameter space where the action of the Hénon map on K set is conjugate to the horseshoe map. The set
62 Horseshoe Locus The complex horseshoe locus L is the open region of parameter space where the action of the Hénon map on K set is conjugate to the horseshoe map. The set {( c) a ( x y) ( x y) K a,c }, C 2 C 2 ( ) (
63 Horseshoe Locus The complex horseshoe locus L is the open region of parameter space where the action of the Hénon map on K set is conjugate to the horseshoe map. The set {( c) a ( x y) ( x y) K a,c }, C 2 C 2 ( ) ( is a locally trivial bundle of Cantor sets over L.
64 Therefore A loop in the horseshoe locus L gives an automorphism of {A,B} Z. Π 1 (L) Aut(Σ 2, σ) Let us consider the analog for quadratic polynomials
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68 Kneading Sequences K f c K Φ Φ {A, B} N σ {A, B} N
69 Monodromy in C-M Loops in C-M give a monodromy action on Kc π 1 (C M, c 0 ) Aut(Σ +, σ + )
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72 Monodromy Actions π 1 (C M, c 0 ) Aut(Σ +, σ + ) C π 1 (L, (a 0, c 0 )) Aut(Σ, σ)
73 Automorphism of the shift
74 Automorphism of the shift Automorphisms of the one-sided shift are simple: there is only the exchange of 0 and 1
75 Automorphism of the shift Automorphisms of the one-sided shift are simple: there is only the exchange of 0 and 1 Automorphisms of the full shift are quite a different matter:
76 Automorphism of the shift Automorphisms of the one-sided shift are simple: there is only the exchange of 0 and 1 Automorphisms of the full shift are quite a different matter: The group is enormous, it contains all finite groups, and a countable sum of Z s
77 Automorphism of the shift Automorphisms of the one-sided shift are simple: there is only the exchange of 0 and 1 Automorphisms of the full shift are quite a different matter: The group is enormous, it contains all finite groups, and a countable sum of Z s No system of generators is known.
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91 What automorphism are we seeing?
92 What automorphism It is AAB*BABAA are we seeing?
93 What automorphism are we seeing? It is AAB*BABAA This means that any time you see such a sequence, you exchange whatever symbol * is for the opposite symbol.
94 What automorphism are we seeing? It is AAB*BABAA This means that any time you see such a sequence, you exchange whatever symbol * is for the opposite symbol. The formula is obtained from the kneading sequence *BABAA of all points of M beyond the *BABA polynomial.
95 What automorphism are we seeing? It is AAB*BABAA This means that any time you see such a sequence, you exchange whatever symbol * is for the opposite symbol. The formula is obtained from the kneading sequence *BABAA of all points of M beyond the *BABA polynomial. AAB the name of the sheet of cone over Cantor x Circle that the loop surrounds.
96 Chris Lipa has checked that about 30 other loops constructed in a similar way have monodromies given by similar formulas. Questions: Are there loops in L corresponding to all the automorphisms of the shift? Why are they automorphisms? Do such automorphisms, together with the shift itself, generate the full group of automorphisms.
97 Pictures and movies made with PlanarIterations by Ben Hinkle and FractalAsm and SaddleDrop by Karl Papadantonakis, all available at
98 Pictures and movies made with PlanarIterations by Ben Hinkle and FractalAsm and SaddleDrop by Karl Papadantonakis, all available at That s all folks
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