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Topological Entropy of Quadratic Polynomials and Sections of the Mandelbrot Set

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Topological Entropy of Quadratic Polynomials and Sections of the Mandelbrot Set Topological entropy of quadratic polynomials and sections of the Mandelbrot set Giulio Tiozzo Harvard University Warwick, April 2012 2. External rays 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 2. External rays 4. Ideas of proof (maybe) 5. Complex version Summary 1. Topological entropy 2. External rays 3. Main theorem 5. Complex version Summary 1. Topological entropy 2. External rays 3. Main theorem 4. Ideas of proof (maybe) Summary 1. Topological entropy 2. External rays 3. Main theorem 4. Ideas of proof (maybe) 5. Complex version Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Topological entropy of real maps Let f : I ! I, continuous. logf#laps of f ng htop(f ; R) := lim n!1 n Entropy measures the randomness of the dynamics. How does entropy change with the parameter c? Topological entropy of real maps logf#laps of f ng htop(f ; R) := lim n!1 n Consider the real quadratic family 2 fc(z) := z + c c 2 [−2; 1=4] Topological entropy of real maps logf#laps of f ng htop(f ; R) := lim n!1 n Consider the real quadratic family 2 fc(z) := z + c c 2 [−2; 1=4] How does entropy change with the parameter c? I is monotone decreasing (Milnor- Thurston ’77). I is continuous (Douady). I 0 ≤ htop(fc; R) ≤ log 2. The function c ! htop(fc; R): I is continuous (Douady). I 0 ≤ htop(fc; R) ≤ log 2. The function c ! htop(fc; R): I is monotone decreasing (Milnor- Thurston ’77). I 0 ≤ htop(fc; R) ≤ log 2. The function c ! htop(fc; R): I is monotone decreasing (Milnor- Thurston ’77). I is continuous (Douady). The function c ! htop(fc; R): I is monotone decreasing (Milnor- Thurston ’77). I is continuous (Douady). I 0 ≤ htop(fc; R) ≤ log 2. The function c ! htop(fc; R): I is monotone decreasing (Milnor- Thurston ’77). I is continuous (Douady). I 0 ≤ htop(fc; R) ≤ log 2. ^ ^ Remark. If we consider fc : C ! C entropy is constant ^ htop(fc; C) = log 2. Mandelbrot set The Mandelbrot set M is the connectedness locus of the quadratic family n M = fc 2 C : fc (0) 9 1g External rays Since C^ n M is simply-connected, it can be uniformized by the exterior of the unit disk ^ ^ ΦM : C n D ! C n M External rays Since C^ n M is simply-connected, it can be uniformized by the exterior of the unit disk ^ ^ ΦM : C n D ! C n M An external ray R(θ) is said to land at x if 2πiθ lim ΦM(ρe ) = x ρ!1 External rays Since C^ n M is simply-connected, it can be uniformized by the exterior of the unit disk ^ ^ ΦM : C n D ! C n M The images of radial arcs in the disk are called external rays. Every angle θ 2 S1 determines an external ray 2πiθ R(θ) := ΦM(fρe : ρ > 1g) External rays Since C^ n M is simply-connected, it can be uniformized by the exterior of the unit disk ^ ^ ΦM : C n D ! C n M The images of radial arcs in the disk are called external rays. Every angle θ 2 S1 determines an external ray 2πiθ R(θ) := ΦM(fρe : ρ > 1g) An external ray R(θ) is said to land at x if 2πiθ lim ΦM(ρe ) = x ρ!1 External rays Since C^ n M is simply-connected, it can be uniformized by the exterior of the unit disk ^ ^ ΦM : C n D ! C n M The images of radial arcs in the disk are called external rays. Every angle θ 2 S1 determines an external ray 2πiθ R(θ) := ΦM(fρe : ρ > 1g) An external ray R(θ) is said to land at x if 2πiθ lim ΦM(ρe ) = x ρ!1 For instance, take A = M\ R the real section of the Mandelbrot set. How common is it for a ray to land on the real axis? Harmonic measure Given a subset A of @M, the harmonic measure µM is the probability that a random ray lands on A: 1 µM(A) := jfθ 2 S : R(θ) lands on Agj How common is it for a ray to land on the real axis? Harmonic measure Given a subset A of @M, the harmonic measure µM is the probability that a random ray lands on A: 1 µM(A) := jfθ 2 S : R(θ) lands on Agj For instance, take A = M\ R the real section of the Mandelbrot set. Harmonic measure Given a subset A of @M, the harmonic measure µM is the probability that a random ray lands on A: 1 µM(A) := jfθ 2 S : R(θ) lands on Agj For instance, take A = M\ R the real section of the Mandelbrot set. How common is it for a ray to land on the real axis? However: The Hausdorff dimension of the set of rays landing on the real axis is1. Real section of the Mandelbrot set Theorem (Zakeri, 2000) The harmonic measure of the real axis is0. The Hausdorff dimension of the set of rays landing on the real axis is1. Real section of the Mandelbrot set Theorem (Zakeri, 2000) The harmonic measure of the real axis is0. However: Real section of the Mandelbrot set Theorem (Zakeri, 2000) The harmonic measure of the real axis is0. However: The Hausdorff dimension of the set of rays landing on the real axis is1. Real section of the Mandelbrot set Theorem (Zakeri, 2000) The harmonic measure of the real axis is0. However: The Hausdorff dimension of the set of rays landing on the real axis is1. Sectioning M Given c 2 [−2; 1=4], we can consider the set of external rays which land on the real axis to the right of c: 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg Sectioning M Given c 2 [−2; 1=4], we can consider the set of external rays which land on the real axis to the right of c: 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg Sectioning M Given c 2 [−2; 1=4], we can consider the set of external rays which land on the real axis to the right of c: 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg Sectioning M Given c 2 [−2; 1=4], we can consider the set of external rays which land on the real axis to the right of c: 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg Sectioning M Given c 2 [−2; 1=4], we can consider the set of external rays which land on the real axis to the right of c: 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg The function c 7! H.dim Pc decreases with c, taking values between 0 and 1. 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg The function c 7! H.dim Pc decreases with c, taking values between 0 and 1. 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg The function c 7! H.dim Pc decreases with c, taking values between 0 and 1. 1 Pc := fθ 2 S : R(θ) lands on @M\ R to the right of cg The function c 7! H.dim Pc decreases with c, taking values between 0 and 1. Main theorem Theorem Let c 2 [−2; 1=4]. Then h (f ; ) top c R = H.dim P log 2 c Main theorem Theorem Let c 2 [−2; 1=4]. Then h (f ; ) top c R = H.dim P log 2 c I It relates dynamical properties of a particular map to the geometry of parameter space near the chosen parameter. I log 2 is the “Lyapunov exponent” of the doubling map (Bowen’s formula). I The proof is purely combinatorial. I It does not depend on MLC. I It can be generalized to (some) non-real veins. Main theorem Theorem Let c 2 [−2; 1=4].
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