The core entropy of polynomials of higher degree

Transcription

1 The core entropy of polynomials of higher degree Giulio Tiozzo University of Toronto In memory of Tan Lei Angers, October 23, 2017

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3 First March 4, 2012 Hi Mr. Giulio Tiozzo, My name is Tan Lei. I am a chinese mathematician working in France in the field of holomorphic dynamics. Curt McMullen suggested me to contact you for the following questions that you might help. It seems that one can think of the core entropy as a function on the Mandelbrot set itself. And Milnor had a student who proved entropy is monotone on M. Do you have a copy of this thesis? How to define the core entropy when the Hubbard tree is topologically infinite? Or worse when the critical orbit is dense in J? Is the monotonicity proved using puzzles? Is there a continuity result of the core entropy as a function of the external angle? Many thanks in advance for your help. Sincerely yours, Tan Lei

4 In total: 569 s

5 In total: 569 s It s the fourth visit of Giulio to Angers and he seems still enjoy it. (December 20, 2014)

6 In total: 569 s It s the fourth visit of Giulio to Angers and he seems still enjoy it. (December 20, 2014) Angers is a small city, 1h30m TGV distance away from Paris, and has a 104m long tapestry on Apocalypse that seems to have impressed most of my Western visitors

7 In total: 569 s It s the fourth visit of Giulio to Angers and he seems still enjoy it. (December 20, 2014) Angers is a small city, 1h30m TGV distance away from Paris, and has a 104m long tapestry on Apocalypse that seems to have impressed most of my Western visitors

8 Topological entropy of real interval maps

9 Topological entropy of real interval maps A lap of f is a maximal interval on which f is monotone.

10 Topological entropy of real interval maps A lap of f is a maximal interval on which f is monotone. The topological entropy of f also equals

11 Topological entropy of real interval maps A lap of f is a maximal interval on which f is monotone. The topological entropy of f also equals h top (f, R) = lim n log #{laps(f n )} n

12 Topological entropy of real interval maps A lap of f is a maximal interval on which f is monotone. The topological entropy of f also equals h top (f, R) = lim n log #{laps(f n )} n

13 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

14 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

15 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

16 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

17 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

18 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

19 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

20 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n

21 Topological entropy of real maps Let f : I I, continuous. h top (f, R) := lim n log #{laps(f n )} n Agrees with general definition for maps on compact spaces using open covers (Misiurewicz-Szlenk)

22 Topological entropy of real maps h top (f, R) := lim n log #{laps(f n )} n Consider the real quadratic family f c (z) := z 2 + c c [ 2, 1/4]

23 Topological entropy of real maps h top (f, R) := lim n log #{laps(f n )} n Consider the real quadratic family f c (z) := z 2 + c c [ 2, 1/4] How does entropy change with the parameter c?

24 The function c h top (f c, R):

25 The function c h top (f c, R): is continuous

26 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard).

27 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard). 0 h top (f c, R) log 2.

28 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard). 0 h top (f c, R) log 2.

29 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard). 0 h top (f c, R) log 2.

30 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard). 0 h top (f c, R) log 2. Question : Can we extend this theory to complex polynomials?

31 The function c h top (f c, R): is continuous and monotone (Milnor-Thurston 1977, Douady-Hubbard). 0 h top (f c, R) log 2. Remark. If we consider f : Ĉ Ĉ entropy is constant h top (f, Ĉ) = log d.

32 The complex case: Hubbard trees The Hubbard tree T of a postcritically finite polynomial is a forward invariant, connected subset of the filled Julia set which contains the critical orbit.

33 The complex case: Hubbard trees The Hubbard tree T of a postcritically finite polynomial is a forward invariant, connected subset of the filled Julia set which contains the critical orbit.

34 Complex Hubbard trees The Hubbard tree T of a postcritically finite polynomial f is a forward invariant, connected subset of the filled Julia set which contains the critical orbit.

35 Complex Hubbard trees The Hubbard tree T of a postcritically finite polynomial f is a forward invariant, connected subset of the filled Julia set which contains the critical orbit. The map f acts on it.

36 The core entropy Let f be a postcritically finite polynomial.

37 The core entropy Let f be a postcritically finite polynomial. Let T be its Hubbard tree.

38 The core entropy Let f be a postcritically finite polynomial. Let T be its Hubbard tree. Then its core entropy is defined as

39 The core entropy Let f be a postcritically finite polynomial. Let T be its Hubbard tree. Then its core entropy is defined as h(f ) := h(f T )

40 The core entropy Let f be a postcritically finite polynomial. Let T be its Hubbard tree. Then its core entropy is defined as h(f ) := h(f T ) Question: How does h(f ) vary with the polynomial f?

41 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s }

42 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that:

43 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D;

44 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ;

45 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ; 3. s k=1 ( #(lk D) 1 ) = d 1.

46 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ; 3. s k=1 ( #(lk D) 1 ) = d 1. A critical portrait m is said to be a primitive major if moreover the elements of m are pairwise disjoint.

47 The space PM(d) of primitive majors PM(d) = {primitive majors of degree d}

48 The space PM(d) of primitive majors has a canonical metric. PM(d) = {primitive majors of degree d}

49 The space PM(d) of primitive majors PM(d) = {primitive majors of degree d} has a canonical metric. The quotient X m := D/m is a graph (tree of circles)

50 The space PM(d) of primitive majors PM(d) = {primitive majors of degree d} has a canonical metric. The quotient X m := D/m is a graph (tree of circles) Let π m : D D/m the projection map.

51 The space PM(d) of primitive majors PM(d) = {primitive majors of degree d} has a canonical metric. The quotient X m := D/m is a graph (tree of circles) Let π m : D D/m the projection map. Define the distance between primitive majors as d(m 1, m 2 ) := sup d(π m1 (x), π m1 (y)) d(π m2 (x), π m2 (y)) x,y

52 Critical markings Let f be a postcritically finite polynomial.

53 Critical markings Let f be a postcritically finite polynomial. For each critical point c, we define the critical leaf Θ(c) as follows.

54 Critical markings Let f be a postcritically finite polynomial. For each critical point c, we define the critical leaf Θ(c) as follows. 1. If c is in the Julia set, then pick one ray θ of minimal period which lands at f (c), and take Θ(c) to be the preimage of θ.

55 Critical markings Let f be a postcritically finite polynomial. For each critical point c, we define the critical leaf Θ(c) as follows. 1. If c is in the Julia set, then pick one ray θ of minimal period which lands at f (c), and take Θ(c) to be the preimage of θ. 2. If c is in the Fatou component U, then pick one ray θ which lands on the boundary of f (U), and take Θ(c) to be the preimage of θ.

56 Critical markings Let f be a postcritically finite polynomial. For each critical point c, we define the critical leaf Θ(c) as follows. 1. If c is in the Julia set, then pick one ray θ of minimal period which lands at f (c), and take Θ(c) to be the preimage of θ. 2. If c is in the Fatou component U, then pick one ray θ which lands on the boundary of f (U), and take Θ(c) to be the preimage of θ. Then Θ := {Θ(c 1 ),..., Θ(c k )} is a critical marking (Poirier).

57 The space PM(d) of primitive majors For d = 2,

58 The space PM(d) of primitive majors For d = 2, l is a diameter.

59 The space PM(d) of primitive majors For d = 2, l is a diameter. PM(2) = D

60 Core entropy for quadratic polynomials

61 Core entropy for quadratic polynomials

62 Core entropy for quadratic polynomials Question Can you see the Mandelbrot set in this picture?

63 The entropy as a function of external angle Monotonicity still holds along veins:

64 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite,

65 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose,

66 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose, Tan Lei,

67 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose, Tan Lei, Zeng The core entropy is also proportional to the dimension of the set of biaccessible angles (Zakeri, Smirnov, Zdunik, Bruin-Schleicher...)

68 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose, Tan Lei, Zeng The core entropy is also proportional to the dimension of the set of biaccessible angles (Zakeri, Smirnov, Zdunik, Bruin-Schleicher...) θ is biaccessible if η θ s.t. R(θ) and R(η) land at the same point. B c := {θ R/Z : θ is biaccessible }

69 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose, Tan Lei, Zeng The core entropy is also proportional to the dimension of the set of biaccessible angles (Zakeri, Smirnov, Zdunik, Bruin-Schleicher...) θ is biaccessible if η θ s.t. R(θ) and R(η) land at the same point. B c := {θ R/Z : θ is biaccessible } H. dim B c = h(f c) log d

70 The entropy as a function of external angle Monotonicity still holds along veins: Li Tao for postcritically finite, Penrose, Tan Lei, Zeng The core entropy is also proportional to the dimension of the set of biaccessible angles (Zakeri, Smirnov, Zdunik, Bruin-Schleicher...) θ is biaccessible if η θ s.t. R(θ) and R(η) land at the same point. B c := {θ R/Z : θ is biaccessible } H. dim B c = h(f c) log d Core entropy also proportional to Hausdorff dimension of angles landing on the corresponding vein (T., Jung)

71 Tan Lei s proof of monotonicity (Feb 16, 2013) Dear Giulio, I think the following strategy might prove trivially a generalization of Tao Li s results, even in higher degree. I ll concentrate on the quadratic case.

72 Tan Lei s proof of monotonicity (Feb 16, 2013) Dear Giulio, I think the following strategy might prove trivially a generalization of Tao Li s results, even in higher degree. I ll concentrate on the quadratic case. Any pair of distinct angles θ ± defines four partitions of the circle: L(θ ± ) is the circle minus the four points θ ± /2, and θ ± /2 + 1/2 and Full(θ ± ) is S 1 minus the two intervals [ θ 2, θ+ 2 ] and [ θ , θ ].

73 Tan Lei s proof of monotonicity (Feb 16, 2013) Dear Giulio, I think the following strategy might prove trivially a generalization of Tao Li s results, even in higher degree. I ll concentrate on the quadratic case. Any pair of distinct angles θ ± defines four partitions of the circle: L(θ ± ) is the circle minus the four points θ ± /2, and θ ± /2 + 1/2 and Full(θ ± ) is S 1 minus the two intervals [ θ 2, θ+ 2 ] and [ θ , θ ]. Now, rather than, as Douady and Tao Li, looking at angles landing as the Hubbard tree, we look at pairs of angles landing together and pairs of angles landing at the tree.

74 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± )

75 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± ) Then H(θ ± ) F (θ ± ) B(θ ± ) where B(θ ± ) is the set of pairs of biaccessible angles and H(θ ± ) is the set of angles of rays landing on the tree.

76 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± ) Then H(θ ± ) F (θ ± ) B(θ ± ) where B(θ ± ) is the set of pairs of biaccessible angles and H(θ ± ) is the set of angles of rays landing on the tree. Once all these are set up cleanly, the result becomes trivial: If you take c further than c, than Full(θ ± ) contains Full(θ ± )

77 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± ) Then H(θ ± ) F (θ ± ) B(θ ± ) where B(θ ± ) is the set of pairs of biaccessible angles and H(θ ± ) is the set of angles of rays landing on the tree. Once all these are set up cleanly, the result becomes trivial: If you take c further than c, than Full(θ ± ) contains Full(θ ± ) and trivially F(θ ± ) F(θ ± ).

78 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± ) Then H(θ ± ) F (θ ± ) B(θ ± ) where B(θ ± ) is the set of pairs of biaccessible angles and H(θ ± ) is the set of angles of rays landing on the tree. Once all these are set up cleanly, the result becomes trivial: If you take c further than c, than Full(θ ± ) contains Full(θ ± ) and trivially So the entropy increases. F(θ ± ) F(θ ± ).

79 Tan Lei s proof of monotonicity So let F (θ ± ) = the set of pairs (η, ζ) having the same itinerary with respect to components of Full(θ ± ) Then H(θ ± ) F (θ ± ) B(θ ± ) where B(θ ± ) is the set of pairs of biaccessible angles and H(θ ± ) is the set of angles of rays landing on the tree. Once all these are set up cleanly, the result becomes trivial: If you take c further than c, than Full(θ ± ) contains Full(θ ± ) and trivially So the entropy increases. F(θ ± ) F(θ ± ). With pictures the idea would be a lot easer to explain. All the best, Tan Lei

80 Continuity in the quadratic case Question (Thurston, Hubbard): Is h(θ) a continuous function of θ?

81 Continuity in the quadratic case Question (Thurston, Hubbard): Is h(θ) a continuous function of θ? Theorem (T., Dudko-Schleicher) The core entropy function h(θ) extends to a continuous function from R/Z to R.

82 The core entropy for cubic polynomials

83 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s }

84 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that:

85 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D;

86 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ;

87 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ; 3. s k=1 ( #(lk D) 1 ) = d 1.

88 Primitive majors A critical portrait of degree d is defined as a collection m = {l 1,..., l s } of leaves and ideal polygons in D such that: 1. any two distinct elements l k and l l either are disjoint or intersect at one point on D; 2. the vertices of each l k are identified under z z d ; 3. s k=1 ( #(lk D) 1 ) = d 1. A critical portrait m is said to be a primitive major if moreover the elements of m are pairwise disjoint.

89 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d.

90 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d. Define the potential function of f at c as G f (c) := lim n 1 d n log f n (c)

91 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d. Define the potential function of f at c as 1 G f (c) := lim n d n log f n (c) which measure the rate of escape of the critical point.

92 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d. Define the potential function of f at c as 1 G f (c) := lim n d n log f n (c) which measure the rate of escape of the critical point. For r > 0, define the equipotential locus

93 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d. Define the potential function of f at c as 1 G f (c) := lim n d n log f n (c) which measure the rate of escape of the critical point. For r > 0, define the equipotential locus Y d (r) := {f P d : G f (c) = r for all c Crit(f )}

94 Primitive majors and polynomials Let P d be the space of monic, centered polynomials of degree d. Define the potential function of f at c as 1 G f (c) := lim n d n log f n (c) which measure the rate of escape of the critical point. For r > 0, define the equipotential locus Y d (r) := {f P d : G f (c) = r for all c Crit(f )} Theorem (Thurston) For each r > 0, we have a homeomorphism Y d (r) = PM(d)

95 The space PM(3) of cubic primitive majors For d = 3,

96 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3)

97 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3) non-intersecting: a + 1/3 b a + 2/3

98 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3) non-intersecting: a + 1/3 b a + 2/3 symmetry: a + 1/3 b a + 1/2

99 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3) non-intersecting: a + 1/3 b a + 2/3 symmetry: a + 1/3 b a + 1/2 PM(3) = { (a, b) S 1 S 1 : a b a + 1 } / 2

100 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3) non-intersecting: a + 1/3 b a + 2/3 symmetry: a + 1/3 b a + 1/2 PM(3) = { (a, b) S 1 S 1 : a b a + 1 } / 2 where: (a, a + 1/3) (a + 1/3, a + 2/3) (wraps 3 times around)

101 The space PM(3) of cubic primitive majors For d = 3, generically two leaves: (a, a + 1/3), (b, b + 1/3) non-intersecting: a + 1/3 b a + 2/3 symmetry: a + 1/3 b a + 1/2 PM(3) = { (a, b) S 1 S 1 : a b a + 1 } / 2 where: (a, a + 1/3) (a + 1/3, a + 2/3) (wraps 3 times around) (a, a + 1/2) (a + 1/2, a) (wraps 2 times around)

102 The core entropy for cubic polynomials

103 The core entropy for cubic polynomials

104 The core entropy for cubic polynomials

105 The unicritical slice f (z) = z 3 + c

106 The symmetric slice f (z) = z 3 + cz

107 Main theorem, combinatorial version Theorem (T. - Yan Gao) Fix d 2. Then the core entropy extends to a continuous function on the space PM(d) of primitive majors.

108 Main theorem, combinatorial version Theorem (T. - Yan Gao) Fix d 2. Then the core entropy extends to a continuous function on the space PM(d) of primitive majors.

109 Main theorem, analytic version Define P d as the space of monic, centered polynomials of degree d.

110 Main theorem, analytic version Define P d as the space of monic, centered polynomials of degree d. One says f n f if the coefficients of f n converge to the coefficients of f.

111 Main theorem, analytic version Define P d as the space of monic, centered polynomials of degree d. One says f n f if the coefficients of f n converge to the coefficients of f. Theorem (T. - Yan Gao) Let d 2. Then the core entropy is a continuous function on the space of monic, centered, postcritically finite polynomials of degree d.

112 Main theorem, analytic version Define P d as the space of monic, centered polynomials of degree d. One says f n f if the coefficients of f n converge to the coefficients of f. Theorem (T. - Yan Gao) Let d 2. Then the core entropy is a continuous function on the space of monic, centered, postcritically finite polynomials of degree d. I hope you two will make a great paper together! (January 28, 2015)

113 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue

114 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree,

115 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter!

116 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter! Idea 2: (Thurston): look at set of pairs of postcritical points

117 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter! Idea 2: (Thurston): look at set of pairs of postcritical points, which correspond to arcs between postcritical points.

118 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter! Idea 2: (Thurston): look at set of pairs of postcritical points, which correspond to arcs between postcritical points. Denote i α := f i (c α ) the i th iterate of the critical point c α, let the postcritical set, P := {i α : i 1, 1 α s}

119 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter! Idea 2: (Thurston): look at set of pairs of postcritical points, which correspond to arcs between postcritical points. Denote i α := f i (c α ) the i th iterate of the critical point c α, let the postcritical set, and P := {i α : i 1, 1 α s} A := P P the set of pairs of postcritical points

120 Computing the entropy Idea 1: look at Markov partition, write matrix and take leading eigenvalue This works, but you need to know the topology of the tree, and that varies wildly with the parameter! Idea 2: (Thurston): look at set of pairs of postcritical points, which correspond to arcs between postcritical points. Denote i α := f i (c α ) the i th iterate of the critical point c α, let the postcritical set, and P := {i α : i 1, 1 α s} A := P P the set of pairs of postcritical points (= arcs between them)

121 Computing the entropy: non-separated pair A pair (i α, j β ) is non-separated if the corresponding arc does not contain critical points.

122 Computing the entropy: non-separated pair A pair (i α, j β ) is non-separated if the corresponding arc does not contain critical points.

123 Computing the entropy: non-separated pair A pair (i α, j β ) is non-separated if the corresponding arc does not contain critical points.

124 Computing the entropy: non-separated pair A pair (i α, j β ) is non-separated if the corresponding arc does not contain critical points. (1, 2) (2, 3)

125 Computing the entropy: separated pair A pair (i α, j β ) is separated if the corresponding arc contains critical points c γ1,..., c γk.

126 Computing the entropy: separated pair A pair (i α, j β ) is separated if the corresponding arc contains critical points c γ1,..., c γk. We record the critical points we cross in the separation vector (γ 1,..., γ k )

127 Computing the entropy: separated pair A pair (i α, j β ) is separated if the corresponding arc contains critical points c γ1,..., c γk. We record the critical points we cross in the separation vector (γ 1,..., γ k )

128 Computing the entropy: separated pair A pair (i α, j β ) is separated if the corresponding arc contains critical points c γ1,..., c γk. We record the critical points we cross in the separation vector (γ 1,..., γ k )

129 Computing the entropy: separated pair A pair (i α, j β ) is separated if the corresponding arc contains critical points c γ1,..., c γk. We record the critical points we cross in the separation vector (γ 1,..., γ k ) (1, 3) (1, 2) + (1, 4)

130 The algorithm Let r the cardinality of the set of pairs of postcritical points, and consider A : R r R r given by

131 The algorithm Let r the cardinality of the set of pairs of postcritical points, and consider A : R r R r given by If (i α, j β ) is non-separated, then A(e iα,j β ) = e (i+1)α,(j+1) β

132 The algorithm Let r the cardinality of the set of pairs of postcritical points, and consider A : R r R r given by If (i α, j β ) is non-separated, then A(e iα,j β ) = e (i+1)α,(j+1) β If (i α, j β ) is separated by (γ 1,..., γ k ), then A(e iα,j β ) = e (i+1)α,1 γ1 + e 1γ1,1 γ2 + + e 1γk,(j+1) β

133 The algorithm Let r the cardinality of the set of pairs of postcritical points, and consider A : R r R r given by If (i α, j β ) is non-separated, then A(e iα,j β ) = e (i+1)α,(j+1) β If (i α, j β ) is separated by (γ 1,..., γ k ), then A(e iα,j β ) = e (i+1)α,1 γ1 + e 1γ1,1 γ2 + + e 1γk,(j+1) β Theorem (Thurston; Tan Lei; Gao Yan) The core entropy of f is given by h(f ) = log λ where λ is the leading eigenvalue of A.

134 The algorithm

135 Computing entropy: the clique polynomial Let Γ be a finite, directed graph.

136 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j}

137 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta)

138 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t).

139 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial

140 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial P(t) = ( 1) C(γ) t l(γ) γ simple multicycle

141 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial P(t) = ( 1) C(γ) t l(γ) where: γ simple multicycle

142 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial P(t) = ( 1) C(γ) t l(γ) where: γ simple multicycle a simple multicycle is a disjoint union of (vertex)-disjoint cycles

143 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial P(t) = ( 1) C(γ) t l(γ) where: γ simple multicycle a simple multicycle is a disjoint union of (vertex)-disjoint cycles C(γ) is the number of connected components of γ

144 Computing entropy: the clique polynomial Let Γ be a finite, directed graph. Its adjacency matrix is A such that A ij := #{i j} We can consider its spectral determinant P(t) := det(i ta) Note that λ 1 is the smallest root of P(t). Note that P(t) can be obtained as the clique polynomial P(t) = ( 1) C(γ) t l(γ) where: γ simple multicycle a simple multicycle is a disjoint union of (vertex)-disjoint cycles C(γ) is the number of connected components of γ l(γ) its length.

145 The clique polynomial: example P(t) = ( 1) C(γ) t l(γ) γ simple multicycle

146 The clique polynomial: example P(t) = ( 1) C(γ) t l(γ) γ simple multicycle

147 The clique polynomial: example P(t) = γ simple multicycle ( 1) C(γ) t l(γ) two 2-cycles

148 The clique polynomial: example P(t) = γ simple multicycle ( 1) C(γ) t l(γ) two 2-cycles one 3-cycle

149 The clique polynomial: example P(t) = γ simple multicycle ( 1) C(γ) t l(γ) two 2-cycles one 3-cycle one pair of disjoint cycles (2 + 3)

150 The clique polynomial: example P(t) = γ simple multicycle ( 1) C(γ) t l(γ) two 2-cycles one 3-cycle one pair of disjoint cycles (2 + 3) P(t) = 1 2t 2 t 3 + t 5

151 Computing entropy: the infinite clique polynomial Let Γ be a countable, directed graph.

152 Computing entropy: the infinite clique polynomial Let Γ be a countable, directed graph. Let us suppose that:

153 Computing entropy: the infinite clique polynomial Let Γ be a countable, directed graph. Let us suppose that: Γ has bounded outgoing degree;

154 Computing entropy: the infinite clique polynomial Let Γ be a countable, directed graph. Let us suppose that: Γ has bounded outgoing degree; Γ has bounded cycles: for every n there exists finitely many simple cycles of length n

155 Computing entropy: the infinite clique polynomial Let Γ be a countable, directed graph. Let us suppose that: Γ has bounded outgoing degree; Γ has bounded cycles: for every n there exists finitely many simple cycles of length n Then we define the growth rate of Γ as : r(γ) := lim sup n C(Γ, n) where C(Γ, n) is the number of closed paths of length n.

156 Computing entropy: the infinite clique polynomial Let Γ with bounded outgoing degree and bounded cycles.

157 Computing entropy: the infinite clique polynomial Let Γ with bounded outgoing degree and bounded cycles.then one can define as a formal power series P(t) = γ simple multicycle ( 1) C(γ) t l(γ)

158 Computing entropy: the infinite clique polynomial Let Γ with bounded outgoing degree and bounded cycles.then one can define as a formal power series P(t) = γ simple multicycle ( 1) C(γ) t l(γ) Let now σ := lim sup n S(n) where S(n) is the number of simple multi-cycles of length n.

159 Computing entropy: the infinite clique polynomial Let Γ with bounded outgoing degree and bounded cycles.then one can define as a formal power series P(t) = γ simple multicycle ( 1) C(γ) t l(γ) Let now σ := lim sup n S(n) where S(n) is the number of simple multi-cycles of length n. Theorem Let σ 1. Then P(t) defines a holomorphic function in the unit disk, and its root of minimum modulus is r 1.

160 Wedges (4, 5) (3, 4) (3, 5) (2, 3) (2, 4) (2, 5) (1, 2) (1, 3) (1, 4) (1, 5)

161 Labeled wedges Label all pairs as either separated or non-separated

162 Labeled wedges Label all pairs as either separated or non-separated (3, 4) (2,3) (2, 4) (1, 2) (1,3) (1, 4) (The boxed pairs are the separated ones.)

163 From wedges to graphs Define a graph associated to the wedge as follows:

164 From wedges to graphs Define a graph associated to the wedge as follows: If (i, j) is non-separated, then (i, j) (i + 1, j + 1)

165 From wedges to graphs Define a graph associated to the wedge as follows: If (i, j) is non-separated, then (i, j) (i + 1, j + 1) If (i, j) is separated, then (i, j) (1, i + 1) and (i, j) (1, j + 1).

166 From wedges to graphs Define a graph associated to the wedge as follows: If (i, j) is non-separated, then (i, j) (i + 1, j + 1) If (i, j) is separated, then (i, j) (1, i + 1) and (i, j) (1, j + 1). (3, 4) (2,3) (2, 4) (1, 2) (1,3) (1, 4)

167 From wedges to graphs Define a graph associated to the wedge as follows: If (i, j) is non-separated, then (i, j) (i + 1, j + 1) If (i, j) is separated, then (i, j) (1, i + 1) and (i, j) (1, j + 1). (3, 4) (2,3) (2, 4) (1, 2) (1,3) (1, 4) This sounds like climbing a mountain; you go up step by step, but you chute all the way to the bottom, and in two broken pieces (August 25, 2014)

168 Continuity: sketch of proof Suppose θ n θ (primitive majors)

169 Continuity: sketch of proof Suppose θ n θ (primitive majors) Then W θn W θ (wedges)

170 Continuity: sketch of proof Suppose θ n θ (primitive majors) Then W θn W θ (wedges) so P θn (t) P θ (t) (spectral determinants)

171 Continuity: sketch of proof Suppose θ n θ (primitive majors) Then W θn W θ (wedges) so P θn (t) P θ (t) (spectral determinants) and r(θ n ) r(θ) (growth rates)

172 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set:

173 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set: use h to define vein structure in higher degree?

174 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set: use h to define vein structure in higher degree? 3. The local Hölder exponent of h at m equals h(m) log d

175 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set: use h to define vein structure in higher degree? 3. The local Hölder exponent of h at m equals h(m) log d (Fels) Also true for invariant sets of the doubling map on the circle, (Carminati-T., Bandtlow-Rugh)

176 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set: use h to define vein structure in higher degree? 3. The local Hölder exponent of h at m equals h(m) log d (Fels) Also true for invariant sets of the doubling map on the circle, (Carminati-T., Bandtlow-Rugh) 4. Derivative of core entropy yields measure on lamination for M.set

177 Further directions / questions 1. Conjecture: In each stratum the maximum of the core entropy equals max h(m) = log(depth(π) + 1) m Π where the Depth of a stratum is the maximum length of a chain of nested leaves in the primitive major. 2. The level sets of the function h(θ) determines lamination for the Mandelbrot set: use h to define vein structure in higher degree? 3. The local Hölder exponent of h at m equals h(m) log d (Fels) Also true for invariant sets of the doubling map on the circle, (Carminati-T., Bandtlow-Rugh) 4. Derivative of core entropy yields measure on lamination for M.set

178 Merci!