Lagrange and Markov spectra & dynamics of horseshoes Carlos - - PowerPoint PPT Presentation

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Lagrange and Markov spectra & dynamics of horseshoes

Carlos Matheus

CNRS – ´ Ecole Polytechnique

November 5, 2019

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Table of contents

1

Introduction

2

Main results

3

Cusick conjecture

4

dim(M \ L) < 0.888

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Diophantine approximations (I)

Given α ∈ R, q ∈ N∗, ∃ p ∈ Z s.t. |qα − p| ≤ 1

2, i.e., |α − p q| ≤ 1 2q.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Diophantine approximations (I)

Given α ∈ R, q ∈ N∗, ∃ p ∈ Z s.t. |qα − p| ≤ 1

2, i.e., |α − p q| ≤ 1 2q.

Dirichlet (1841): pigeonhole principle = ⇒ ∀ α ∈ R \ Q, one has # p q ∈ Q : |α − p q | < 1 q2

• = ∞
• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Diophantine approximations (I)

Given α ∈ R, q ∈ N∗, ∃ p ∈ Z s.t. |qα − p| ≤ 1

2, i.e., |α − p q| ≤ 1 2q.

Dirichlet (1841): pigeonhole principle = ⇒ ∀ α ∈ R \ Q, one has # p q ∈ Q : |α − p q | < 1 q2

• = ∞

Definition The Lagrange spectrum L ⊂ R is L := {l(α) < ∞ : α ∈ R \ Q}, l(α) := lim sup

p,q→∞

1 |q(qα − p)|

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Diophantine approximations (II)

Given h(x, y) = ax2 + bxy + cy2 a real, indefinite, binary quadratic form with positive discriminant ∆(h) := b2 − 4ac > 0, let m(h) := sup

(p,q)∈Z2\{(0,0)}

• ∆(h)

|h(p, q)|

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Diophantine approximations (II)

Given h(x, y) = ax2 + bxy + cy2 a real, indefinite, binary quadratic form with positive discriminant ∆(h) := b2 − 4ac > 0, let m(h) := sup

(p,q)∈Z2\{(0,0)}

• ∆(h)

|h(p, q)| Definition The Markov spectrum M ⊂ R is M := {m(h) < ∞ : h as above }.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Beginning of L and M (I)

Hurwitz (1890): √ 5 = min L because # p q ∈ Q : |α − p q | < 1 √ 5q2

• = ∞,

∀ α ∈ R \ Q, and #

• p

q ∈ Q : |1 + √ 5 2 − p q | < 1 ( √ 5 + ε)q2

• < ∞,

∀ ε > 0.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Beginning of L and M (II)

Markov (1880) : L ∩ [ √ 5, 3) = M ∩ [ √ 5, 3) =

• √

5 < √ 8 < √ 221 5 < . . .

• =
• 9 − 4

z2

n

: n ∈ N

• where xn ≤ yn ≤ zn, (xn, yn, zn) ∈ N3 is a Markov triple, i.e.,

x2

n + y2 n + z2 n = 3xnynzn

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Beginning of L and M (II)

Markov (1880) : L ∩ [ √ 5, 3) = M ∩ [ √ 5, 3) =

• √

5 < √ 8 < √ 221 5 < . . .

• =
• 9 − 4

z2

n

: n ∈ N

• where xn ≤ yn ≤ zn, (xn, yn, zn) ∈ N3 is a Markov triple, i.e.,

x2

n + y2 n + z2 n = 3xnynzn

Remark All Markov triples are deduced from (1, 1, 1) via Vieta’s involutions (x, y, z) → (3yz − x, y, z), etc. Keywords: Markov tree, Markov uniqueness conjecture [Zagier], [Bombieri] ..., Markov expanders [Bourgain-Gamburd-Sarnak], ...

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Markov’s tree

(1,5,13) (2,5,29) (1,13,34) (5,13,194) (5,29,433) (2,29,169) (1,34,89) (13,34,1325) (13,194,7561) (5,194,2897) (5,433,6466) (29,433,37666) (29,169,14701) (2,169,985) (1,1,1) (1,1,2) (1,2,5)

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

L and M after Perron (I)

Let σ((an)n∈Z) = (an+1)n∈Z be the shift dynamics on Σ = (N∗)Z, and consider the height function f : Σ → R, f ((an)n∈Z) := [a0; a1, . . . ] + [0; a−1, . . . ] = a0 + 1 a1 +

1

... + 1 a−1 +

1

...

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

L and M after Perron (I)

Let σ((an)n∈Z) = (an+1)n∈Z be the shift dynamics on Σ = (N∗)Z, and consider the height function f : Σ → R, f ((an)n∈Z) := [a0; a1, . . . ] + [0; a−1, . . . ] = a0 + 1 a1 +

1

... + 1 a−1 +

1

... Perron proved in 1921 that L = {lim sup

n→∞ f (σn(a)) < ∞ : a ∈ Σ}

and M = {sup

n∈Z

f (σn(a)) < ∞ : a ∈ Σ}

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Perron’s description of L and M

(N∗)Z− (N∗)N f

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

L and M after Perron (II)

This dynamical characterisation of L and M gives access to several statements: Perron also showed in 1921 that ( √ 12, √ 13) ∩ M = ∅ and √ 12, √ 13 ∈ L, it is not hard to use this description of L and M to prove that L ⊂ M are closed subsets of the real line, etc.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

L, M and the modular surface

The relation between continued fractions and geodesics on the modular surface H/SL(2, Z) says that L and M correspond to heights of excursions of geodesics into the cusp of H/SL(2, Z). Movie by Pierre Arnoux and Edmund Harriss.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Impressionistic picture of the modular surface

gt(x) H x

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Ending of L and M

The works of Hall (1947), ..., Freiman (1975) give that the largest half-line of the form [c, ∞) contained in L ⊂ M is

• 2221564096 + 283748

√ 462 491993569 , ∞

• This half-line is called Hall’s ray in the literature and its left

endpoint is called Freiman’s constant cF = 4.5278 . . . .

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Intermediate portion of L and M (I)

We saw that L and M coincide before 3 and after cF: L ∩ [ √ 5, 3] = M ∩ [ √ 5, 3] and L ∩ [cF, ∞) = M ∩ [cF, ∞) = [cF, ∞).

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Intermediate portion of L and M (I)

We saw that L and M coincide before 3 and after cF: L ∩ [ √ 5, 3] = M ∩ [ √ 5, 3] and L ∩ [cF, ∞) = M ∩ [cF, ∞) = [cF, ∞). Nevertheless, Freiman (1968, 1973) and Flahive (1977) proved that M \ L contains infinite countable subsets near 3.11 and 3.29

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Intermediate portion of L and M (I)

We saw that L and M coincide before 3 and after cF: L ∩ [ √ 5, 3] = M ∩ [ √ 5, 3] and L ∩ [cF, ∞) = M ∩ [cF, ∞) = [cF, ∞). Nevertheless, Freiman (1968, 1973) and Flahive (1977) proved that M \ L contains infinite countable subsets near 3.11 and 3.29 On the other hand, Cusick conjectured in 1975 that L and M coincide after √ 12.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Intermediate portion of L and M (II)

More recently, Moreira (2016) showed that dim(L ∩ (−∞, t)) = dim(M ∩ (−∞, t)) for all t ∈ R. Hence, M \ L doesn’t create “jumps in dimension” between L and M. Moreira also proved that d(t) := dim(L ∩ (−∞, t)) is a continuous non-H¨

• lder function of t such that

d(3 + ε) > 0 ∀ε > 0 and d( √ 12) = 1.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Global view of the Lagrange and Markov spectra

Markov theorem Moreira theorem Hurwitz theorem Freiman constant

(1975)

Perron

(1921)

Freiman Cusick's conjecture

(1890) (1880) (1968)

Freiman (1973) Flahive (1977)

(2016) (1975)

Hall's ray

(1947) 3 4,5278... 3,11...

∈ M-L 3,29... ∈ M-L √12 √13 √8 22 9√3 + 65 √5

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Main results (I)

Theorem (M.–Moreira) There are 3 intervals J1, J2, J3 near 3.11, 3.29 and 3.7 (resp.) of sizes ∼ 2 × 10−10, 2 × 10−7 and 10−10 (resp.) such that ∂Jn ⊂ L and int(Jn) ∩ L = ∅; dim(M ∩ Jn) = dim(Xn) where Xn are explicit Gauss-Cantor sets of dimensions > 0.26, 0.353 and 0.53 (resp.).

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Main results (I)

Theorem (M.–Moreira) There are 3 intervals J1, J2, J3 near 3.11, 3.29 and 3.7 (resp.) of sizes ∼ 2 × 10−10, 2 × 10−7 and 10−10 (resp.) such that ∂Jn ⊂ L and int(Jn) ∩ L = ∅; dim(M ∩ Jn) = dim(Xn) where Xn are explicit Gauss-Cantor sets of dimensions > 0.26, 0.353 and 0.53 (resp.). In particular, Cusick’s conjecture is not true: for instance, 7940451225305 − √ 3 2326589591051 + √ 330629 − 483 310 = 3.70969985 · · · ∈ M \L is the largest element of (M \ L) ∩ J3. Moreover, 0.53 < dim(M \ L) < 0.888.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Main results (II)

Interestingly enough, the distance between (M \ L) ∩ Jn and ∂Jn is positive ∀ 1 ≤ n ≤ 3. Thus, (M \ L) ∩ Jn are closed subsets of M. This scenario led T. Bousch to ask if M \ L could be a closed subset of M. In this direction, we have the following partial result: Theorem (Lima–M.–Moreira–Vieira) There is a decreasing sequence mn → 3 such that mi − 3 ∈ {0055..., 00016..., 0000048..., 00000014...} and mi ∈ M \ L for 1 ≤ i ≤ 4; if mn is “locally unique”, then mn ∈ M \ L.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Main results (III)

Theorem (Delecroix–M.–Moreira) There is an algorithm providing finite sets 1/Q-close (in Hausdorff topology) to L and M after time O(Q2.367). Remark The naive idea of using periodic and eventually periodic sequences in Perron’s characterisation of L and M leads to a poor algorithm: morally, one has to compute with 4Q4 words of lengths ≤ Q4 to get 1/Q-dense subsets of L and M.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Rigorous drawing of L

Our algorithm with Delecroix and Moreira allows to produce rigorous approximate pictures of L: for instance, the picture below displays a finite collection of (blue) intervals with Hausdorff distance ≤

1 150000 from L2 = L ∩ [

√ 5, √ 12].

2.9 3.0 3.1 3.2 3.3 3.4

Lagrange spectrum L2 at precision Q2 = 150000

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Common theme behind main results: horseshoes

After Perron, L ∩ [ √ 5, √ 21] and M ∩ [ √ 5, √ 21] are related to the heights of the orbits of the restriction of the natural extension G of the Gauss map to the horseshoe Λ4 = {([a0; a1, . . . ], [0; a−1, . . . ]) ∈ R2 : (an)n∈Z ∈ {1, 2, 3, 4}Z} with respect to the function f (z, w) = z + w.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Common theme behind main results: horseshoes

After Perron, L ∩ [ √ 5, √ 21] and M ∩ [ √ 5, √ 21] are related to the heights of the orbits of the restriction of the natural extension G of the Gauss map to the horseshoe Λ4 = {([a0; a1, . . . ], [0; a−1, . . . ]) ∈ R2 : (an)n∈Z ∈ {1, 2, 3, 4}Z} with respect to the function f (z, w) = z + w. As it turns out, our main results above are inspired from dynamical considerations about Λ4. In order to illustrate this point, let us now sketch the proof of 0.53 < dim(M \ L) < 0.888

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Some notations

Given x = (xm)m∈Z ∈ (N∗)Z and n ∈ Z, let λn(x) = f (σn(x)) = [xn; xn+1, . . . ] + [0; xn−1, . . . ]

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Some notations

Given x = (xm)m∈Z ∈ (N∗)Z and n ∈ Z, let λn(x) = f (σn(x)) = [xn; xn+1, . . . ] + [0; xn−1, . . . ] In this context, the Markov value of x is m(x) = sup

n∈Z

λn(x) and the Lagrange value of x is ℓ(x) = lim sup

n→∞ λn(x).

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Some notations

Given x = (xm)m∈Z ∈ (N∗)Z and n ∈ Z, let λn(x) = f (σn(x)) = [xn; xn+1, . . . ] + [0; xn−1, . . . ] In this context, the Markov value of x is m(x) = sup

n∈Z

λn(x) and the Lagrange value of x is ℓ(x) = lim sup

n→∞ λn(x).

Also, given a finite string z1 . . . zk, we denote by z1 . . . zk, resp. z1 . . . zk, the half-infinite sequence obtained by indefinite periodic repetition of z1 . . . zk to the left, resp. right.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Non-semi-symmetric words

Our solution of Cusick’s conjecture starts by the following general method to find potential elements in M \ L.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Non-semi-symmetric words

Our solution of Cusick’s conjecture starts by the following general method to find potential elements in M \ L. Consider α = 3322212. It is not semi-symmetric in Flahive’s sense, i.e., it is not a concatenation of two palindromes.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Non-semi-symmetric words

Our solution of Cusick’s conjecture starts by the following general method to find potential elements in M \ L. Consider α = 3322212. It is not semi-symmetric in Flahive’s sense, i.e., it is not a concatenation of two palindromes. In this situation, we can hope to get different answers to the problems of gluing 21 and α, and α and 21.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Non-semi-symmetric words

Our solution of Cusick’s conjecture starts by the following general method to find potential elements in M \ L. Consider α = 3322212. It is not semi-symmetric in Flahive’s sense, i.e., it is not a concatenation of two palindromes. In this situation, we can hope to get different answers to the problems of gluing 21 and α, and α and 21. As it turns out, µ = m(α21) = 3.70969985975 . . . , and the smallest Markov value amongst 21wα with w ∈

n∈N

{1, 2}n is ν = m(212212α) = 3.70969985982 · · · > µ

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

M \ L and geometry of stable manifolds

√ 12 ℓ µ ν G n0(z) qα G n0+m0(z) G n1(z) Λα p21 pα W u

loc(Λα)

• f
• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Local uniqueness and self-replication mechanism

This lack of symmetry between the gluing problems above helps in getting the following local uniqueness statement: Lemma (i) If m(x) < 3.70969985975033 and λ0(x) > 3.70969985968, then x = . . . x−1x∗

0x1 · · · = α33∗222123322212 . . . up to transposition.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Local uniqueness and self-replication mechanism

This lack of symmetry between the gluing problems above helps in getting the following local uniqueness statement: Lemma (i) If m(x) < 3.70969985975033 and λ0(x) > 3.70969985968, then x = . . . x−1x∗

0x1 · · · = α33∗222123322212 . . . up to transposition.

Corollary (i) The Lagrange spectrum does not intersect the interval J = (3.70969985968, 3.70969985975033) Proof. λ ∈ L ∩ J & Lemma (i) imply λ = m(αα) < 3.70969985968.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Gauss–Cantor sets in M ∩ J

A straightforward calculation reveals that: Lemma (ii) M ∩ J contains {m(α2121212θ) : θ ∈ {1, 2}N}.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Gauss–Cantor sets in M ∩ J

A straightforward calculation reveals that: Lemma (ii) M ∩ J contains {m(α2121212θ) : θ ∈ {1, 2}N}. Consequently: Corollary (ii) M ∩ J contains a Cantor set bi-Lipschitz homeomorphic to C(2) := {[0; θ] : θ ∈ {1, 2}N}

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

End of proof of dim(M \ L) > 0.53

By combining Corollaries (i) and (ii), we derive that dim((M \ L) ∩ (3.709, 3.71)) ≥ dim(C(2)) > 0.53 thanks to the work of Good, Hensley, ..., Jenkinson–Pollicott. (Keywords: Thermodynamical formalism, Bowen’s formula, ...)

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of M ∩ (3, √ 10)

Perron showed in 1921 that m(θ) ≤ √ 12 ⇐ ⇒ θ ∈ {1, 2}Z.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of M ∩ (3, √ 10)

Perron showed in 1921 that m(θ) ≤ √ 12 ⇐ ⇒ θ ∈ {1, 2}Z. Hall proved in 1971 that dim({[0; ρ] : ρ ∈ {1, 2}N not containing 121}) ≤ 0.465

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of M ∩ (3, √ 10)

Perron showed in 1921 that m(θ) ≤ √ 12 ⇐ ⇒ θ ∈ {1, 2}Z. Hall proved in 1971 that dim({[0; ρ] : ρ ∈ {1, 2}N not containing 121}) ≤ 0.465 Since m(θ) < √ 10 ⇐ ⇒ θ ∈ {1, 2}Z doesn’t contain 121, it follows that dim(M ∩ (3, √ 10)) ≤ 0.93

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of M ∩ (3, √ 10)

Perron showed in 1921 that m(θ) ≤ √ 12 ⇐ ⇒ θ ∈ {1, 2}Z. Hall proved in 1971 that dim({[0; ρ] : ρ ∈ {1, 2}N not containing 121}) ≤ 0.465 Since m(θ) < √ 10 ⇐ ⇒ θ ∈ {1, 2}Z doesn’t contain 121, it follows that dim(M ∩ (3, √ 10)) ≤ 0.93 It follows that dim((M \ L) ∩ (3, √ 12)) ≤ max{0.93, dim((M \ L) ∩ ( √ 10, √ 12))}

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (I)

The key idea is: m(θ) ∈ (M \ L) ∩ ( √ 10, √ 12) = ⇒ severe restrictions on the past or future dynamics of θ ∈ {1, 2}Z.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (I)

The key idea is: m(θ) ∈ (M \ L) ∩ ( √ 10, √ 12) = ⇒ severe restrictions on the past or future dynamics of θ ∈ {1, 2}Z. Very roughly speaking, a quantitative version of the shadowing lemma says that, up to transposition, the future dynamics of θ with m(θ) ∈ (M \ L) ∩ ( √ 10, √ 12) lives in the gaps of B = {11, 22}Z ⊂ {x ∈ {1, 2}Z : x doesn’t contain 121}

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

M \ L and symmetric blocks

µ Λ11,22 Λ1,2 √ 10 √ 12 z z+∞ z−∞

• f
• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (II)

Concretely, m(θ) ∈ (M \ L) ∩ ( √ 10, √ 12) implies that, up to transposition, θ = (θm)m∈Z satisfies either θ1 . . . θn has an unique continuation,

• r its continuations are θ1 . . . θn112 and θ1 . . . θn221,

for all large n ∈ N.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (II)

Concretely, m(θ) ∈ (M \ L) ∩ ( √ 10, √ 12) implies that, up to transposition, θ = (θm)m∈Z satisfies either θ1 . . . θn has an unique continuation,

• r its continuations are θ1 . . . θn112 and θ1 . . . θn221,

for all large n ∈ N. This non-trivial control of the bifurcation tree of θ implies that dim((M \ L) ∩ [ √ 10, √ 12]) ≤ dim(C(2)) + s0 where |I(a1 . . . an112)|s0 + |I(a1 . . . an221)|s0 ≤ |I(a1 . . . an)|s0 for all (a1 . . . an) ∈ {1, 2}n and I(a1 . . . ak) = {[0; a1 . . . akz] : z > 1}.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (III)

Since |I(a1 . . . ak)| =

1 qk(qk+qk−1) where pm qm = [0; a1, . . . , am] and

qm = amqm−1 + qm−2, we have |I(a1 . . . an112)|s0 + |I(a1 . . . an221)|s0 |I(a1 . . . an)|s0 =

• r + 1

(3r + 5)(4r + 7) s0 +

• r + 1

(3r + 7)(5r + 12) s0 with 0 < r = qn−1/qn < 1.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (IV)

Because

r+1 (3r+5)(4r+7) ≤ 1 35, r+1 (3r+7)(5r+12) ≤ 1 81.98 for 0 < r < 1, and

1

35

0.174813 +

• 1

81.98

0.174813 < 1, we obtain that dim((M \ L) ∩ [ √ 10, √ 12]) ≤ dim(C(2)) + 0.174813

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Control of (M \ L) ∩ ( √ 10, √ 12) (IV)

Because

r+1 (3r+5)(4r+7) ≤ 1 35, r+1 (3r+7)(5r+12) ≤ 1 81.98 for 0 < r < 1, and

1

35

0.174813 +

• 1

81.98

0.174813 < 1, we obtain that dim((M \ L) ∩ [ √ 10, √ 12]) ≤ dim(C(2)) + 0.174813 Since dim(C(2)) < 0.531291 by Jenkinson–Pollicott work, we conclude that dim((M \ L) ∩ [ √ 10, √ 12]) < 0.706104

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

End of proof of the 2nd main result

By pursuing similar arguments, we show that dim((M \ L) ∩ I) < 0.888 for I = (3, 3.06), (3.06, √ 12), ( √ 12, 3.84), (3.84, 3.92), (3.92, 4.01), (4.01, √ 20), ( √ 20, √ 21). This completes the proof of dim(M \ L) < 0.888.

• C. Matheus

L, M and horseshoes

Introduction Main results Cusick conjecture dim(M \ L) < 0.888

Final words

En guise de conclusion, je voudrais finir l’expos´ e par une photo d’un coin de Rio tr` es appr´ eci´ e par Christian...

• C. Matheus

L, M and horseshoes