giulio tiozzo harvard university june 21, 2013 · an entropic tour of the mandelbrot set giulio...
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An entropic tour of the Mandelbrot set
Giulio TiozzoHarvard University
June 21, 2013
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Summary
1. Topological entropy
2. External rays3. Main theorem, real version4. Complex version5. Sketch of proof (maybe)6. Remarks and conjectures
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Summary
1. Topological entropy2. External rays
3. Main theorem, real version4. Complex version5. Sketch of proof (maybe)6. Remarks and conjectures
![Page 4: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/4.jpg)
Summary
1. Topological entropy2. External rays3. Main theorem, real version
4. Complex version5. Sketch of proof (maybe)6. Remarks and conjectures
![Page 5: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/5.jpg)
Summary
1. Topological entropy2. External rays3. Main theorem, real version4. Complex version
5. Sketch of proof (maybe)6. Remarks and conjectures
![Page 6: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/6.jpg)
Summary
1. Topological entropy2. External rays3. Main theorem, real version4. Complex version5. Sketch of proof (maybe)
6. Remarks and conjectures
![Page 7: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/7.jpg)
Summary
1. Topological entropy2. External rays3. Main theorem, real version4. Complex version5. Sketch of proof (maybe)6. Remarks and conjectures
![Page 8: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/8.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
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Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 10: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/10.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
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Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 12: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/12.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 13: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/13.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 14: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/14.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 15: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/15.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 16: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/16.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
![Page 17: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/17.jpg)
Topological entropy of real maps
Let f : I → I, continuous.
htop(f ,R) := limn→∞
log #{laps(f n)}n
Entropy measures the randomness of the dynamics.
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Example: the airplane map
A 7→ A ∪ BB 7→ A
⇒(
1 11 0
)⇒ λ =
√5+12 = ehtop(fc ,R)
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Example: the airplane map
A 7→ A ∪ BB 7→ A
⇒(
1 11 0
)⇒ λ =
√5+12 = ehtop(fc ,R)
![Page 20: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/20.jpg)
Example: the airplane map
A 7→ A ∪ BB 7→ A
⇒(
1 11 0
)
⇒ λ =√
5+12 = ehtop(fc ,R)
![Page 21: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/21.jpg)
Example: the airplane map
A 7→ A ∪ BB 7→ A
⇒(
1 11 0
)⇒ λ =
√5+12
= ehtop(fc ,R)
![Page 22: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/22.jpg)
Example: the airplane map
A 7→ A ∪ BB 7→ A
⇒(
1 11 0
)⇒ λ =
√5+12 = ehtop(fc ,R)
![Page 23: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/23.jpg)
Topological entropy of real maps
htop(f ,R) := limn→∞
log #{laps(f n)}n
Consider the real quadratic family
fc(z) := z2 + c c ∈ [−2,1/4]
How does entropy change with the parameter c?
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Topological entropy of real maps
htop(f ,R) := limn→∞
log #{laps(f n)}n
Consider the real quadratic family
fc(z) := z2 + c c ∈ [−2,1/4]
How does entropy change with the parameter c?
![Page 25: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/25.jpg)
The function c → htop(fc ,R):
I is continuous and monotone (Milnor-Thurston, 1977).I 0 ≤ htop(fc ,R) ≤ log 2.
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The function c → htop(fc ,R):
I is continuous
and monotone (Milnor-Thurston, 1977).I 0 ≤ htop(fc ,R) ≤ log 2.
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The function c → htop(fc ,R):
I is continuous and monotone (Milnor-Thurston, 1977).
I 0 ≤ htop(fc ,R) ≤ log 2.
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The function c → htop(fc ,R):
I is continuous and monotone (Milnor-Thurston, 1977).I 0 ≤ htop(fc ,R) ≤ log 2.
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The function c → htop(fc ,R):
I is continuous and monotone (Milnor-Thurston, 1977).I 0 ≤ htop(fc ,R) ≤ log 2.
Remark. If we consider fc : C→ C entropy is constanthtop(fc , C) = log 2.
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Mandelbrot set
The Mandelbrot setM is the connectedness locus of thequadratic family
M = {c ∈ C : f nc (0) 9∞}
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External rays
Since C \M is simply-connected, it can be uniformized by theexterior of the unit disk
ΦM : C \ D→ C \M
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External rays
Since C \M is simply-connected, it can be uniformized by theexterior of the unit disk
ΦM : C \ D→ C \M
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External raysSince C \M is simply-connected, it can be uniformized by theexterior of the unit disk
ΦM : C \ D→ C \MThe images of radial arcs in the disk are called external rays.Every angle θ ∈ S1 determines an external ray
R(θ) := ΦM({ρe2πiθ : ρ > 1})
An external ray R(θ) is said to land at x if
limρ→1
ΦM(ρe2πiθ) = x
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External raysSince C \M is simply-connected, it can be uniformized by theexterior of the unit disk
ΦM : C \ D→ C \MThe images of radial arcs in the disk are called external rays.Every angle θ ∈ S1 determines an external ray
R(θ) := ΦM({ρe2πiθ : ρ > 1})An external ray R(θ) is said to land at x if
limρ→1
ΦM(ρe2πiθ) = x
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External raysSince C \M is simply-connected, it can be uniformized by theexterior of the unit disk
ΦM : C \ D→ C \MThe images of radial arcs in the disk are called external rays.Every angle θ ∈ S1 determines an external ray
R(θ) := ΦM({ρe2πiθ : ρ > 1})An external ray R(θ) is said to land at x if
limρ→1
ΦM(ρe2πiθ) = x
![Page 36: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/36.jpg)
Rays landing on the real slice of the Mandelbrot set
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Harmonic measureGiven a subset A of ∂M, the harmonic measure νM is theprobability that a random ray lands on A:
νM(A) := Leb({θ ∈ S1 : R(θ) lands on A})
For instance, take A =M∩R the real section of the Mandelbrotset. How common is it for a ray to land on the real axis?
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Harmonic measureGiven a subset A of ∂M, the harmonic measure νM is theprobability that a random ray lands on A:
νM(A) := Leb({θ ∈ S1 : R(θ) lands on A})
For instance, take A =M∩R the real section of the Mandelbrotset.
How common is it for a ray to land on the real axis?
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Harmonic measureGiven a subset A of ∂M, the harmonic measure νM is theprobability that a random ray lands on A:
νM(A) := Leb({θ ∈ S1 : R(θ) lands on A})
For instance, take A =M∩R the real section of the Mandelbrotset. How common is it for a ray to land on the real axis?
![Page 40: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/40.jpg)
Real section of the Mandelbrot setTheorem (Zakeri, 2000)The harmonic measure of the real axis is 0.
However,the Hausdorff dimension of the set of rays landing on the realaxis is 1.
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Real section of the Mandelbrot setTheorem (Zakeri, 2000)The harmonic measure of the real axis is 0. However,
the Hausdorff dimension of the set of rays landing on the realaxis is 1.
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Real section of the Mandelbrot setTheorem (Zakeri, 2000)The harmonic measure of the real axis is 0. However,the Hausdorff dimension of the set of rays landing on the realaxis is 1.
![Page 43: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/43.jpg)
Real section of the Mandelbrot setTheorem (Zakeri, 2000)The harmonic measure of the real axis is 0. However,the Hausdorff dimension of the set of rays landing on the realaxis is 1.
![Page 44: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/44.jpg)
SectioningMGiven c ∈ [−2,1/4], we can consider the set of external rayswhich land on the real axis to the right of c:
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}
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SectioningMGiven c ∈ [−2,1/4], we can consider the set of external rayswhich land on the real axis to the right of c:
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}
![Page 46: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/46.jpg)
SectioningMGiven c ∈ [−2,1/4], we can consider the set of external rayswhich land on the real axis to the right of c:
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}
![Page 47: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/47.jpg)
SectioningMGiven c ∈ [−2,1/4], we can consider the set of external rayswhich land on the real axis to the right of c:
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}
![Page 48: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/48.jpg)
SectioningMGiven c ∈ [−2,1/4], we can consider the set of external rayswhich land on the real axis to the right of c:
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}
![Page 49: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/49.jpg)
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}The function
c 7→ H.dim Pc
decreases with c, taking values between 0 and 1.
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Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}The function
c 7→ H.dim Pc
decreases with c, taking values between 0 and 1.
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Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}The function
c 7→ H.dim Pc
decreases with c, taking values between 0 and 1.
![Page 52: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/52.jpg)
Pc := {θ ∈ S1 : R(θ) lands on ∂M∩ [c,1/4]}The function
c 7→ H.dim Pc
decreases with c, taking values between 0 and 1.
![Page 53: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/53.jpg)
Main theoremTheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
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Main theoremTheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
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Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.I It does not depend on MLC.I It can be generalized to (some) non-real veins.
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Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.I It does not depend on MLC.I It can be generalized to (some) non-real veins.
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Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.I It does not depend on MLC.I It can be generalized to (some) non-real veins.
![Page 58: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/58.jpg)
Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.
I It does not depend on MLC.I It can be generalized to (some) non-real veins.
![Page 59: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/59.jpg)
Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.I It does not depend on MLC.
I It can be generalized to (some) non-real veins.
![Page 60: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/60.jpg)
Main theorem
TheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Pc
I It relates dynamical properties of a particular map to thegeometry of parameter space near the chosen parameter.
I Entropy formula: relates dimension, entropy and Lyapunovexponent (Manning, Bowen, Ledrappier, Young, ...).
I The proof is purely combinatorial.I It does not depend on MLC.I It can be generalized to (some) non-real veins.
![Page 61: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/61.jpg)
In the dynamical planeDouady’s principle : “sow in dynamical plane and reap inparameter space”.
Each fc has a Julia set Jc .
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In the dynamical planeDouady’s principle : “sow in dynamical plane and reap inparameter space”.Each fc has a Julia set Jc .
![Page 63: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/63.jpg)
In the dynamical plane
Douady’s principle : “sow in dynamical plane and reap inparameter space”Each fc has a Julia set Jc .Let
Sc := {θ ∈ S1 : R(θ) lands on Jc ∩ R}
the set of rays landing on the real section (spine) of the Juliaset.
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Main theoremTheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Sc = H.dim Pc
![Page 65: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/65.jpg)
Main theoremTheoremLet c ∈ [−2,1/4]. Then
htop(fc ,R)
log 2= H.dim Sc = H.dim Pc
![Page 66: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/66.jpg)
It also equals:I The entropy of the induced action on the Hubbard tree Tc
(minimal forward-invariant set containing the critical orbit),divided by log 2.
I The dimension of the set of biaccessible angles (Zakeri,Smirnov, Zdunik, Bruin-Schleicher ...)
CorollaryThe set of biaccessible angles for the Feigenbaum parameter(limit of period doubling cascades) cFeig has Hausdorffdimension 0.
![Page 67: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/67.jpg)
It also equals:I The entropy of the induced action on the Hubbard tree Tc
(minimal forward-invariant set containing the critical orbit),divided by log 2.
I The dimension of the set of biaccessible angles (Zakeri,Smirnov, Zdunik, Bruin-Schleicher ...)
CorollaryThe set of biaccessible angles for the Feigenbaum parameter(limit of period doubling cascades) cFeig has Hausdorffdimension 0.
![Page 68: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/68.jpg)
It also equals:I The entropy of the induced action on the Hubbard tree Tc
(minimal forward-invariant set containing the critical orbit),divided by log 2.
I The dimension of the set of biaccessible angles (Zakeri,Smirnov, Zdunik, Bruin-Schleicher ...)
CorollaryThe set of biaccessible angles for the Feigenbaum parameter(limit of period doubling cascades) cFeig has Hausdorffdimension 0.
![Page 69: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/69.jpg)
The complex case: Hubbard treesThe Hubbard tree Tc of a quadratic polynomial is a forwardinvariant subset of the filled Julia set which contains the criticalorbit.
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The complex case: Hubbard treesThe Hubbard tree Tc of a quadratic polynomial is a forwardinvariant subset of the filled Julia set which contains the criticalorbit.
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Complex Hubbard treesThe Hubbard tree Tc of a quadratic polynomial is a forwardinvariant subset of the filled Julia set which contains the criticalorbit.
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Complex Hubbard treesThe Hubbard tree Tc of a quadratic polynomial is a forwardinvariant subset of the filled Julia set which contains the criticalorbit. The map fc acts on it.
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Topologically finite parameters
DefinitionA parameter c is topologically finite if its Hubbard tree ishomeomorphic to a finite tree.
Postcritically finite⇒ Topologically finite (but many more!)
PropositionIf c is biaccessible in parameter space (there are two distinctrays landing on c), then fc is topologically finite (i.e. on all veinsofM).
![Page 74: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/74.jpg)
Topologically finite parameters
DefinitionA parameter c is topologically finite if its Hubbard tree ishomeomorphic to a finite tree.Postcritically finite⇒ Topologically finite
(but many more!)
PropositionIf c is biaccessible in parameter space (there are two distinctrays landing on c), then fc is topologically finite (i.e. on all veinsofM).
![Page 75: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/75.jpg)
Topologically finite parameters
DefinitionA parameter c is topologically finite if its Hubbard tree ishomeomorphic to a finite tree.Postcritically finite⇒ Topologically finite (but many more!)
PropositionIf c is biaccessible in parameter space (there are two distinctrays landing on c), then fc is topologically finite (i.e. on all veinsofM).
![Page 76: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/76.jpg)
Topologically finite parameters
DefinitionA parameter c is topologically finite if its Hubbard tree ishomeomorphic to a finite tree.Postcritically finite⇒ Topologically finite (but many more!)
PropositionIf c is biaccessible in parameter space (there are two distinctrays landing on c), then fc is topologically finite
(i.e. on all veinsofM).
![Page 77: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/77.jpg)
Topologically finite parameters
DefinitionA parameter c is topologically finite if its Hubbard tree ishomeomorphic to a finite tree.Postcritically finite⇒ Topologically finite (but many more!)
PropositionIf c is biaccessible in parameter space (there are two distinctrays landing on c), then fc is topologically finite (i.e. on all veinsofM).
![Page 78: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/78.jpg)
Entropy of topologically finite parameters
Let c be topologically finite, with Hubbard tree Tc .
Let
Hc := {θ ∈ S1 : Rc(θ) lands on Tc}
TheoremLet fc be topologically finite. Then
h(fc ,Tc)
log 2= H.dim Hc
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Entropy of topologically finite parameters
Let c be topologically finite, with Hubbard tree Tc . Let
Hc := {θ ∈ S1 : Rc(θ) lands on Tc}
TheoremLet fc be topologically finite. Then
h(fc ,Tc)
log 2= H.dim Hc
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Entropy of topologically finite parameters
Let c be topologically finite, with Hubbard tree Tc . Let
Hc := {θ ∈ S1 : Rc(θ) lands on Tc}
TheoremLet fc be topologically finite. Then
h(fc ,Tc)
log 2= H.dim Hc
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Entropy of Hubbard trees as a function of externalangle (W. Thurston)
Can you see the Mandelbrot set in this picture?
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Entropy of Hubbard trees as a function of externalangle (W. Thurston)
Can you see the Mandelbrot set in this picture?
![Page 83: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/83.jpg)
The complex caseA vein is an embedded arc in the Mandelbrot set.
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The complex case
A vein is an embedded arc in the Mandelbrot set.
Given a parameter c along a vein, we can look at the set Pc ofparameter rays which land on the vein “below” c.
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VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].
For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 86: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/86.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 87: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/87.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;
I the critical point maps to the β fixed point after exactly qiterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 88: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/88.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 89: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/89.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 90: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/90.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;
I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 91: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/91.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 92: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/92.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 93: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/93.jpg)
VeinsExistence of veins converging to dyadic angles[Branner-Douady, Kahn, Riedl].For each p/q ∈ Q ∩ (0,1), there exists a unique parameter cp/qfor which:
I the rotation number around the α fixed point is p/q;I the critical point maps to the β fixed point after exactly q
iterates: f q(0) = β.
DefinitionThe principal vein vp/q in the p/q limb is the vein joining cp/q tothe center of the main cardioid.
I v1/2 = real axis;I v1/3 = Branner-Douady vein
The Hubbard tree for all parameters along vp/q is a q-prongedstar.
Pc := {θ ∈ S1 : RM(θ) lands on [0, c]}
![Page 94: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/94.jpg)
Complex versionTheoremLet vp/q be the principal vein in the p/q-limb of the Mandelbrotset, and let c ∈ vp/q. Then
htop(fc ,Tc)
log 2= H.dim Hc = H.dim Pc
![Page 95: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/95.jpg)
Sketch of proof1. The rays landing on parameter space land also on the real
section of the Julia set:
Pc ⊆ Sc
2. In order to prove the reverse inequality, one would like toembed the rays landing on the Hubbard tree in parameterspace. This cannot be done in the renormalized copies.
3. However:
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
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Sketch of proof1. The rays landing on parameter space land also on the real
section of the Julia set:
Pc ⊆ Sc
2. In order to prove the reverse inequality, one would like toembed the rays landing on the Hubbard tree in parameterspace. This cannot be done in the renormalized copies.
3. However:
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
![Page 97: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/97.jpg)
Sketch of proof1. The rays landing on parameter space land also on the real
section of the Julia set:
Pc ⊆ Sc
2. In order to prove the reverse inequality, one would like toembed the rays landing on the Hubbard tree in parameterspace. This cannot be done in the renormalized copies.
3. However:
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
![Page 98: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/98.jpg)
Sketch of proof (continues)
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
4. By continuity,H.dim Hc = H.dim Pc
for non-renormalizable parameters.5. By renormalization, the same holds for all parameters
which are not infinitely renormalizable.6. By density of such parameters (in the space of angles), the
result holds.
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Sketch of proof (continues)
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
4. By continuity,H.dim Hc = H.dim Pc
for non-renormalizable parameters.
5. By renormalization, the same holds for all parameterswhich are not infinitely renormalizable.
6. By density of such parameters (in the space of angles), theresult holds.
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Sketch of proof (continues)
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
4. By continuity,H.dim Hc = H.dim Pc
for non-renormalizable parameters.5. By renormalization, the same holds for all parameters
which are not infinitely renormalizable.
6. By density of such parameters (in the space of angles), theresult holds.
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Sketch of proof (continues)
PropositionIf c is a non-renormalizable, real parameter, and c′ > c anotherreal parameter, there exists a non-constant, piecewise linearmap F : R/Z→ R/Z such that
F (Hc′) ⊆ Pc
4. By continuity,H.dim Hc = H.dim Pc
for non-renormalizable parameters.5. By renormalization, the same holds for all parameters
which are not infinitely renormalizable.6. By density of such parameters (in the space of angles), the
result holds.
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Pseudocenters
DefinitionThe (dyadic) pseudocenter of a real interval [a,b] with|a− b| < 1 is the unique dyadic rational number with shortestbinary expansion.
E.g., the pseudocenter of the interval [1315 ,
1415 ] is 7
8 = 0.111,since 13
15 = 0.1101 and 1415 = 0.1110.
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Pseudocenters
DefinitionThe (dyadic) pseudocenter of a real interval [a,b] with|a− b| < 1 is the unique dyadic rational number with shortestbinary expansion.
E.g., the pseudocenter of the interval [1315 ,
1415 ] is 7
8 = 0.111,since 13
15 = 0.1101 and 1415 = 0.1110.
![Page 104: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/104.jpg)
Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 .
Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1. Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ). All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
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Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 . Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1.
Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ). All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
![Page 106: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/106.jpg)
Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 . Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1. Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ). All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
![Page 107: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/107.jpg)
Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 . Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1. Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ). All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
![Page 108: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/108.jpg)
Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 . Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1. Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ).
All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
![Page 109: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/109.jpg)
Bonus level: a bisection algorithm
TheoremLet c1 < c2 be two real parameters on the boundary ofM, withexternal angles 0 ≤ θ2 < θ1 ≤ 1
2 . Let θ∗ be the dyadicpseudocenter of the interval (θ2, θ1), and let
θ∗ = 0.s1s2 . . . sn−1sn
be its binary expansion, with sn = 1. Then the hyperbolicwindow of least period in the interval (θ2, θ1) is the interval ofexternal angles (α2, α1) with
α2 := 0.s1s2 . . . sn−1
α1 := 0.s1s2 . . . sn−1s1s2 . . . sn−1
(where si := 1− si ). All hyperbolic windows are obtained byiteration of this algorithm, starting with θ2 = 0, θ1 = 1/2.
![Page 110: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/110.jpg)
Bonus level: a bisection algorithm
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Pseudocenters and maxima of entropy
DefinitionThe pseudocenter of a real interval [a,b] with |a− b| < 1 is theunique dyadic rational number with shortest binary expansion.
ConjectureLet θ1 < θ2 be two external angles whose rays RM(θ1), RM(θ2)land on the same parameter. Then the maximum of entropy onthe interval [θ1, θ2] is attained at its pseudocenter θ∗:
maxθ∈[θ1,θ2]
h(θ) = h(θ∗)
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Pseudocenters and maxima of entropy
DefinitionThe pseudocenter of a real interval [a,b] with |a− b| < 1 is theunique dyadic rational number with shortest binary expansion.
ConjectureLet θ1 < θ2 be two external angles whose rays RM(θ1), RM(θ2)land on the same parameter.
Then the maximum of entropy onthe interval [θ1, θ2] is attained at its pseudocenter θ∗:
maxθ∈[θ1,θ2]
h(θ) = h(θ∗)
![Page 113: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/113.jpg)
Pseudocenters and maxima of entropy
DefinitionThe pseudocenter of a real interval [a,b] with |a− b| < 1 is theunique dyadic rational number with shortest binary expansion.
ConjectureLet θ1 < θ2 be two external angles whose rays RM(θ1), RM(θ2)land on the same parameter. Then the maximum of entropy onthe interval [θ1, θ2] is attained at its pseudocenter θ∗:
maxθ∈[θ1,θ2]
h(θ) = h(θ∗)
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Thurston’s entropy plot
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Thurston’s quadratic minor lamination
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A transverse measure on QML
Let `1 < `2 two leaves, and τ a transverse arc connecting them.
Then we defineµ(τ) := h(Tc2)− h(Tc1)
“Combinatorial bifurcation measure”?
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A transverse measure on QML
Let `1 < `2 two leaves, and τ a transverse arc connecting them.Then we define
µ(τ) := h(Tc2)− h(Tc1)
“Combinatorial bifurcation measure”?
![Page 118: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/118.jpg)
A transverse measure on QML
Let `1 < `2 two leaves, and τ a transverse arc connecting them.Then we define
µ(τ) := h(Tc2)− h(Tc1)
“Combinatorial bifurcation measure”?
![Page 119: Giulio Tiozzo Harvard University June 21, 2013 · An entropic tour of the Mandelbrot set Giulio Tiozzo Harvard University June 21, 2013](https://reader033.vdocuments.mx/reader033/viewer/2022042407/5f214942d7fba54739269761/html5/thumbnails/119.jpg)
The end
Thank you!
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Coda: from Farey to the tent map, via ?
Minkowski’s question-mark function conjugates the Farey mapwith the tent map
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
0.2 0.4 0.6 0.8 1.0
0.2
0.4
0.6
0.8
1.0
Continued fractions ⇔ Binary expansions
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The dictionary
Continued fractions ⇔ Binary expansions
E ←?→ R
α− continued fractions unimodal maps
numbers of generalized external raysbounded type on Julia sets
cutting sequences for univoque numbersgeodesics on torus(Cassaigne, 1999)
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A unified approach
The dictionary yields a unified proof of the following results:
1. The real part of the boundary of the Mandelbrot set hasHausdorff dimension 1
H.dim(∂M∩ R) = 1
(Zakeri, 2000)2. The set of matching intervals for α-continued fractions has
zero measure and full Hausdorff dimension(Nakada-Natsui conjecture, Carminati-T. 2010)
3. The set of univoque numbers has zero measure and fullHausdorff dimension (Erdos-Horvath-Joo, Daroczy-Katai,Komornik-Loreti)
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Entropy of α-continued fractions vs real hyperboliccomponents ofM