Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Rauzy fractals and Equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Jean-Louis Verger-Gaugry Prague, Journées Numération, Doppler Institute for Mathematical Physics and Applied Mathematics May 28th 2008
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Contents 1 Introduction, example : Bassino’s family of cubic Pisot numbers 2 Concentration and equi-distribution of Galois and beta-conjugates Dichotomy – Szegö’s Theorem In Solomyak’s set Ω Erd˝ os-Turán’s Theorem and Mignotte’s Theorem Factorization of the Parry polynomial 3 Degree of Parry polynomial and Rauzy fractal (central tile) Cyclotomic factors – Riemann Hypothesis – Amoroso Non-cyclotomic factors Non-reciprocal factors An Equidistribution Limit Theorem 4 Rauzy fractal from Galois- and beta-conjugates of a Parry number 5
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Introduction, example : Bassino’s family of cubic Pisot numbers Contents 1 Introduction, example : Bassino’s family of cubic Pisot numbers 2 Concentration and equi-distribution of Galois and beta-conjugates Dichotomy – Szegö’s Theorem In Solomyak’s set Ω Erd˝ os-Turán’s Theorem and Mignotte’s Theorem Factorization of the Parry polynomial 3 Degree of Parry polynomial and Rauzy fractal (central tile) Cyclotomic factors – Riemann Hypothesis – Amoroso Non-cyclotomic factors Non-reciprocal factors An Equidistribution Limit Theorem 4 Rauzy fractal from Galois- and beta-conjugates of a Parry number 5
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Introduction, example : Bassino’s family of cubic Pisot numbers Let k ≥ 2. β (= β k ) is the dominant root of the minimal polynomial P β ( X ) = X 3 − ( k + 2 ) X 2 + 2 kX − k . We have : k < β k < k + 1 and lim k → + ∞ ( β k − k ) = 0. The length of d β k ( 1 ) is 2 k + 2 = d P ; k − 1 � � ( i − 1 ) z i +( k − i + 1 ) z k + i + 1 � + kz k + z k + 1 + kz 2 k + 2 f β k ( z ) = − 1 + kz + i = 2 is minus the reciprocal polynomial of the Parry polynomial n ∗ β ( X ) . k = 30 : the beta-conjugates are the roots of ( φ 2 ( X ) φ 3 ( X ) φ 6 ( X ) φ 10 ( X ) φ 30 ( X ) φ 31 ( X )) × ( φ 10 ( − X ) φ 30 ( − X )) .
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Introduction, example : Bassino’s family of cubic Pisot numbers 1 0.5 � -1 -0.5 0.5 1 � -0.5 -1 F IG .: Galois conjugates ( ⋄ ) and beta-conjugates ( • ) of the cubic Pisot number β = 30 . 0356 . . . , dominant root of X 3 − 32 X 2 + 60 X − 30.
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set Contents 1 Introduction, example : Bassino’s family of cubic Pisot numbers 2 Concentration and equi-distribution of Galois and beta-conjugates Dichotomy – Szegö’s Theorem In Solomyak’s set Ω Erd˝ os-Turán’s Theorem and Mignotte’s Theorem Factorization of the Parry polynomial 3 Degree of Parry polynomial and Rauzy fractal (central tile) Cyclotomic factors – Riemann Hypothesis – Amoroso Non-cyclotomic factors Non-reciprocal factors An Equidistribution Limit Theorem 4 Rauzy fractal from Galois- and beta-conjugates of a Parry number 5
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set β > 1 Perron number if algebraic integer and all its Galois conjugates β ( i ) satisfy : | β ( i ) | < β for all i = 1 , 2 , . . . , d − 1 (degree d ≥ 1, with β ( 0 ) = β ). Let β > 1. Rényi β -expansion of 1 + ∞ � t i β − i , and corresponds to d β ( 1 ) = 0 . t 1 t 2 t 3 . . . 1 = i = 1 t 1 = ⌊ β ⌋ , t 2 = ⌊ β { β }⌋ = ⌊ β T β ( 1 ) ⌋ , t 3 = ⌊ β { β { β }}⌋ = ⌊ β T 2 β ( 1 ) ⌋ , . . . The digits t i belong to A β := { 0 , 1 , 2 , . . . , ⌈ β − 1 ⌉} . Parry number : if d β ( 1 ) is finite or ultimately periodic (i.e. eventually periodic) ; in particular, simple if d β ( 1 ) is finite. Lothaire : a Parry number is a Perron number. Dichotomy : set of Perron numbers P = P P ∪ P a
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set Exploration of this dichotomy by the Erd˝ os-Turán approach and its improvements (Mignotte, Amoroso) applied to + ∞ � t i z i f β ( z ) := for β ∈ P , z ∈ C , i = 0 with t 0 = − 1, where d β ( 1 ) = 0 . t 1 t 2 t 3 . . . , for which f β ( z ) is a rational fraction if and only if β ∈ P P . Beta-conjugates : D. Boyd 1996
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set Dichotomy – Szegö’s Theorem Theorem (Szeg˝ o) A Taylor series � n ≥ 0 a n z n with coefficients in a finite subset S of C is either equal to V ( z ) (i) a rational fraction U ( z ) + z m + 1 1 − z p + 1 where i = 1 b i z i , V ( z ) = � p i = 0 e i z i are polynomials U ( z ) = − 1 + � m with coefficients in S and m ≥ 1 , p ≥ 0 integers, or (ii) it is an analytic function defined on the open unit disk which is not continued beyond the unit circle (which is its natural boundary). Dichotomy of Perron numbers β <—> dichotomy of analytical functions f β ( z ) .
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set In Solomyak’s set Ω ∞ � a j z j | 0 ≤ a j ≤ 1 } B := { f ( z ) = 1 + j = 1 functions analytic in the open unit disk D ( 0 , 1 ) . G := { ξ ∈ D ( 0 , 1 ) | f ( ξ ) = 0 for some f ∈ B} and G − 1 := { ξ − 1 | ξ ∈ G} . External boundary ∂ G − 1 of G − 1 : curve with a cusp at z = 1, a √ � � − 1 + 5 spike on the negative real axis, = , − 1 , and is fractal at 2 an infinite number of points. Ω := G − 1 ∪ D ( 0 , 1 ) .
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set In Solomyak’s set Ω Theorem (Solomyak) The Galois conjugates ( � = β ) and the beta-conjugates of all Parry numbers β belong to Ω , occupy it densely, and P P ∩ Ω = ∅ . ∞ ∞ � � t i z i = ( − 1 + β z ) � β ( 1 ) z j � T j f β ( z ) = − 1 + 1 + , | z | < 1 , i = 1 j = 1 -> the zeros � = β − 1 of f β ( z ) are those of 1 + � ∞ β ( 1 ) z j ; but j = 1 T j β ( 1 ) z j is a Taylor series which belongs to B . 1 + � ∞ j = 1 T j
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set In Solomyak’s set Ω F IG .: Solomyak’s set Ω .
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set In Solomyak’s set Ω -> phenomenon of high concentration and equi-distribution of Galois conjugates ( � = β ) and beta-conjugates of a Parry number β occurs by clustering near the unit circle in Ω . Theorem Let β > 1 be a Parry number. Let ǫ > 0 and µ ǫ the proportion of roots of the Parry polynomial n ∗ β ( X ) of β , with d P = deg ( n ∗ β ( X )) ≥ 1 , which lie in Ω outside the annulus � � D ( 0 , ( 1 − ǫ ) − 1 ) \ D ( 0 , ( 1 − ǫ )) . Then � � 2 β � 2 − 1 Log � n ∗ 2 Log β ( i ) µ ǫ ≤ , ǫ d P � � 2 β � 1 − 1 � � Log � n ∗ 2 Log � n ∗ µ ǫ ≤ ( ii ) β ( 0 ) . � ǫ d P
Rauzy fractal and equi-distribution of Galois- and beta-conjugates of Parry Numbers near the unit circle Concentration and equi-distribution of Galois and beta-conjugates of Parry numbers near the unit circle in Solomyak’s set In Solomyak’s set Ω (i) Let µ 1 d P the number of roots of n ∗ β ( X ) outside D ( 0 , ( 1 − ǫ ) − 1 ) in Ω , except β (since β �∈ Ω ). By Landau’s inequality M ( f ) ≤ � f � 2 for f ( x ) ∈ C [ X ] applied to n ∗ β ( X ) we deduce β ( 1 − ǫ ) − µ 1 d P ≤ M ( n ∗ β ) ≤ � n ∗ β � 2 . Since − Log ( 1 − ǫ ) ≥ ǫ , � � Log � n ∗ − Log β β � 2 µ 1 ≤ 1 . ǫ d P d P
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