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 Introduction 1 Main results 2 Cusick conjecture 3 dim ( M \ L ) < 0 . 888 4 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. | q α − p | ≤ 1 2 , i.e., | α − p 1 q | ≤ 2 q . 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. | q α − p | ≤ 1 2 , i.e., | α − p 1 q | ≤ 2 q . Dirichlet (1841): pigeonhole principle = ⇒ ∀ α ∈ R \ Q , one has � p � q ∈ Q : | α − p q | < 1 # = ∞ q 2 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. | q α − p | ≤ 1 2 , i.e., | α − p 1 q | ≤ 2 q . Dirichlet (1841): pigeonhole principle = ⇒ ∀ α ∈ R \ Q , one has � p � q ∈ Q : | α − p q | < 1 # = ∞ q 2 Definition The Lagrange spectrum L ⊂ R is L := { l ( α ) < ∞ : α ∈ R \ Q } , 1 l ( α ) := lim sup | q ( q α − p ) | 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 ) = ax 2 + bxy + cy 2 a real, indefinite, binary quadratic form with positive discriminant ∆( h ) := b 2 − 4 ac > 0, let � ∆( h ) m ( h ) := sup | h ( p , q ) | ( p , q ) ∈ Z 2 \{ (0 , 0) } C. Matheus L , M and horseshoes
Introduction Main results Cusick conjecture dim ( M \ L ) < 0 . 888 Diophantine approximations (II) Given h ( x , y ) = ax 2 + bxy + cy 2 a real, indefinite, binary quadratic form with positive discriminant ∆( h ) := b 2 − 4 ac > 0, let � ∆( h ) m ( h ) := sup | h ( p , q ) | ( p , q ) ∈ Z 2 \{ (0 , 0) } 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 1 # q | < √ = ∞ , ∀ α ∈ R \ Q , 5 q 2 and � � √ p q ∈ Q : | 1 + 5 − p 1 q | < √ < ∞ , ∀ ε > 0 . # 2 5 + ε ) q 2 ( 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) = �� � � � √ √ √ 221 9 − 4 5 < 8 < < . . . = : n ∈ N z 2 5 n where x n ≤ y n ≤ z n , ( x n , y n , z n ) ∈ N 3 is a Markov triple , i.e., x 2 n + y 2 n + z 2 n = 3 x n y n z n 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) = �� � � � √ √ √ 221 9 − 4 5 < 8 < < . . . = : n ∈ N z 2 5 n where x n ≤ y n ≤ z n , ( x n , y n , z n ) ∈ N 3 is a Markov triple , i.e., x 2 n + y 2 n + z 2 n = 3 x n y n z n Remark All Markov triples are deduced from (1 , 1 , 1) via Vieta’s involutions ( x , y , z ) �→ (3 yz − 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 (1,34,89) (1,13,34) (13,34,1325) (1,5,13) (13,194,7561) (5,13,194) (5,194,2897) (1,1,1) (1,1,2) (1,2,5) (5,433,6466) (5,29,433) (29,433,37666) (2,5,29) (29,169,14701) (2,29,169) (2,169,985) C. Matheus L , M and horseshoes
Introduction Main results Cusick conjecture dim ( M \ L ) < 0 . 888 L and M after Perron (I) Let σ (( a n ) n ∈ Z ) = ( a n +1 ) n ∈ Z be the shift dynamics on Σ = ( N ∗ ) Z , and consider the height function f : Σ → R , f (( a n ) n ∈ Z ) := [ a 0 ; a 1 , . . . ] + [0; a − 1 , . . . ] 1 1 = a 0 + + 1 1 a 1 + a − 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 (( a n ) n ∈ Z ) = ( a n +1 ) n ∈ Z be the shift dynamics on Σ = ( N ∗ ) Z , and consider the height function f : Σ → R , f (( a n ) n ∈ Z ) := [ a 0 ; a 1 , . . . ] + [0; a − 1 , . . . ] 1 1 = a 0 + + 1 1 a 1 + a − 1 + ... ... Perron proved in 1921 that n →∞ f ( σ n ( a )) < ∞ : a ∈ Σ } L = { lim sup and f ( σ n ( a )) < ∞ : a ∈ Σ } M = { sup n ∈ Z 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 − f ( N ∗ ) N 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 √ √ 13 ∈ L , 12 , 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 H x g t ( 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 c F = 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 c F : √ √ L ∩ [ 5 , 3] = M ∩ [ 5 , 3] and L ∩ [ c F , ∞ ) = M ∩ [ c F , ∞ ) = [ c F , ∞ ) . 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 c F : √ √ L ∩ [ 5 , 3] = M ∩ [ 5 , 3] and L ∩ [ c F , ∞ ) = M ∩ [ c F , ∞ ) = [ c F , ∞ ) . 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 c F : √ √ L ∩ [ 5 , 3] = M ∩ [ 5 , 3] and L ∩ [ c F , ∞ ) = M ∩ [ c F , ∞ ) = [ c F , ∞ ) . 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¨ older function of t such that √ ∀ ε > 0 d (3 + 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 Moreira theorem (2016) Markov theorem Hall's ray (1880) (1947) Cusick's conjecture (1975) 3,11... 3,29... 4,5278... √ 5 3 √ 8 √ 12 √ 13 ∈ ∈ M-L M-L 9 √ 3 + 65 Hurwitz theorem Freiman constant 22 (1890) (1975) Freiman (1968) Perron Freiman (1973) (1921) Flahive (1977) 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 J 1 , J 2 , J 3 near 3 . 11 , 3 . 29 and 3 . 7 (resp.) of sizes ∼ 2 × 10 − 10 , 2 × 10 − 7 and 10 − 10 (resp.) such that ∂ J n ⊂ L and int ( J n ) ∩ L = ∅ ; dim ( M ∩ J n ) = dim ( X n ) where X n are explicit Gauss-Cantor sets of dimensions > 0 . 26 , 0 . 353 and 0 . 53 (resp.). C. Matheus L , M and horseshoes
Recommend
More recommend
