What is Pisot conjecture?

Transcription

1 What is Pisot conjecture? Shigeki Akiyama (Niigata University, Japan) 11 June 2010 at Leiden Typeset by FoilTEX

2 Let (X, B, µ) be a probability space and T : X X be a measure preserving transformation. Then (X, B, µ, T ) forms a measure theoretical dynamical system. By Poincaré s recurrence theorem, for a set Y B with µ(y ) > 0, almost all T -orbit from Y is recurrent. The first return map on Y is defined by: ˆT = T m(x) (x) where m(x) = min{m Z >0 T m (x) Y }. This gives the induced system: (Y, B Y, 1 µ(y ) µ, ˆT ). Typeset by FoilTEX 1

3 From now on let X R d. The system (X, B, µ, T ) is 1 self-inducing if there is a Y such that (Y, B Y, µ(y ) µ, ˆT ) is isomorphic to the original dynamics by the affine isomorphism map ϕ: ϕ X Y T X ϕ ˆT Y Typeset by FoilTEX 2

4 A Pisot number is an algebraic integer > 1 such that all conjugates other than itself has modulus strictly less than 1. A well known property: d(β n, Z) 0 as n. if β is a Pisot number, then A partial converse is shown by Hardy: Let β > 1 be an algebraic number and x 0 is a real number. If d(xβ n, Z) 0 then β is a Pisot number. Typeset by FoilTEX 3

5 Motivation: The self-inducing structure corresponds to pure periodic expansion in arithmetic algorithms. The scaling constant (the maximal eigenvalue of the matrix of ϕ 1 ) often becomes a Pisot number, moreover a Pisot unit. Example 1: Continued fraction is derived from successive induction of irrational rotation. The scaling constant for the self-inducing structure are quadratic Pisot units. This fact is used to effectively find the fundamental unit of real quadratic fields. Typeset by FoilTEX 4

6 Example 2: Piecewise isomorphism: Cut a polygon X into pieces like jigsaw puzzle and reconstruct X in a different manner. Example 3: Outer billiard: A point x and a convex polygon D is given. T (x) is the reflection of x with respect to the right most point of D visible from x. We wish to know why the Pisot number plays the role. Self-inducing structure is modelled by Substitutive dynamical system. Typeset by FoilTEX 5

7 Pisot conjecture Put A = {0, 1,..., k 1} and consider a substitution, for e.g.: σ(0) = 01, σ(1) = 02, σ(2) = 0 having a fixed point x = Let s be the shift acting on A N : s(a 1 a 2... ) = a 2 a We have a topological dynamics X σ = {s n (x) n = 0, 1,... } where s acts. The incidence matrix M σ is the k k matrix whose (i, j) Typeset by FoilTEX 6

8 entry is the number of i occurs in σ(j): for e.g., M σ = If this is primitive, (X σ, s) is minimal and uniquely ergodic. The cylinder set for a word w is [w] = {x X σ x has prefix w}. Then the measure is given by µ([w]) = Frequency of the word w in the fixed point x We wish to discuss the spectral property of isometry U s : f f s on L 2 (X σ, µ). Typeset by FoilTEX 7

9 I start with an important fundamental result (under some mild conditions): Theorem 1 (Host [8]). Eigenfunctions of U s are chosen to be continuous. The dynamical system has pure discrete spectrum if measurable eigenfunctions of U s forms a complete orthonormal basis. von Neumann s theorem asserts that a dynamical system is pure discrete iff it is conjugate to the translation action x x + a on a compact group. Let Φ σ be the characteristic polynomial of M σ. The substitution is irreducible if Φ σ is irreducible. We say σ Typeset by FoilTEX 8

10 is a Pisot substitution if the Perron-Frobenius root of M σ is a Pisot number. Conjecture 1 (Pisot Conjecture). If σ is a irreducible Pisot substitution then (X σ, s) has purely discrete spectrum. In this talk, I shall try to explain the necessity of above two assumptions, i.e.. Pisot property and Irreducibility. Typeset by FoilTEX 9

11 Why Pisot? A reason is hidden in the proof of Theorem 2 (Dekking-Keane [7]). (X σ, s) is not mixing. The fix point x of σ naturally gives rise to a 1-dimensional self-affine tiling T by giving length to each letter. The lengths are coordinates of a left eigen vector of M σ. Taking closure of translations of R we get a different dynamical system (X T, R). Pursuing this discussion we arrive at: Theorem 3 (Bombieri-Taylor [3], Solomyak [12]). (X T, R) is not weakly mixing if and only if the Perron Frobenius root of M σ is a Pisot number. Typeset by FoilTEX 10

12 This is a dynamical version of Hardy s theorem. Now we have two systems: (X σ, s) and (X T, R). Spectral property of (X σ, s) is rather intricate. Clark-Sadun [4] showed that if we ask whether pure discrete or not, two dynamics behave the same. Typeset by FoilTEX 11

13 Solomyak showed an impressive result: A self-affine tiling dynamical system is pure discrete iff density T (T β n v) 1, where β is the Perron-Frobenius root of M σ and v be a return vector. It is equivalent to overlap coincidence. We can also define tiling dynamical system (X T, R d ) in higher dimensional case with an expanding matrix Q which satisfies Pisot family condition. We have to take d-linearly independent return vectors over R. Typeset by FoilTEX 12

14 Algorithm for pure discreteness Overlap coincidence gives an algorithm to determine pure discreteness. However this is difficult to compute, because it depends on topology of tiles. Recently with Jeong-Yup Lee, we invented an easy practical algorithm [1]. We also implemented a Mathematica program. It works for all self-affine tilings, including non-unit scaling and non lattice-based tilings. What we need is a GIFS data of a self-affine tiling. Typeset by FoilTEX 13

15 Why irreducible?: An example by Bernd Sing. Tiling dynamical system of substitution on 4 letters: 0 01, 1 0, 0 01, 1 0 is not purely discrete (c.f. [11]). This substitution is a skew product of Fibonacci substitution by a finite cocycle over Z/2Z. Typeset by FoilTEX 14

16 Naive Pisot Conjecture in Higher dimension Assume two conditions: 1. Q satisfies Pisot family condition. 2. Congruent tiles have the same color. Then the tiling dynamical system (X T, R d ) may be pure discrete. Pisot-family condition Meyer property In 1-dim, irreducibility implies the color condition. Typeset by FoilTEX 15

17 Example by the endomorphism of free group (Dekking [5, 6], Kenyon [9]) We consider a self similar tiling is generated a boundary substitution: θ(a) = b θ(b) = c θ(c) = a 1 b 1 acting on the boundary word aba 1 b 1, aca 1 c 1, bcb 1 c 1, Typeset by FoilTEX 16

18 representing three fundamental parallelogram. Associated the tile equation is αa 1 = A 2 αa 2 = (A 2 1 α) (A 3 1) αa 3 = A 1 1 with α i which is a root of the polynomial x 3 + x + 1. Typeset by FoilTEX 17

19 Figure 1: Tiling by boundary endomorphism The tiling dynamical system has pure point spectrum. Typeset by FoilTEX 18

20 Example: Arnoux-Furukado-Harriss-Ito tiling Arnoux-Furukado-Harriss-Ito [10] recently gave an explicit Markov partition of the toral automorphism for the matrix: which has two dim expanding and two dim contractive planes. They defined 2-dim substitution of 6 polygons. Let α = a root of x 4 x The multi Typeset by FoilTEX 19

21 colour Delone set is given by 6 6 matrix: {} {z/α} {z/α} {} {} {} {} {} {} {z/α} {z/α} {} {} {} {} {} {} {z/α} {z/α} {} {} {} {} {} {} {z/α + 1 α} {} {} {} {} {} {} {} {(z 1)/α + α} {} {} and the associated tiling for contractive plane is: Typeset by FoilTEX 20

22 Figure 2: AFHI Tiling Our program saids it is purely discrete. Typeset by FoilTEX 21

23 A counter example!? Figure 3: Fractal chair tiling by C.Bandt [2] Typeset by FoilTEX 22

24 Fractal chair tiling turned out to be not purely discrete! An overlap creates new overlaps without any coincidence. One can draw an overlap graph. Figure 4: Overlap graph of fractal chair Typeset by FoilTEX 23

25 However, one can construct another tiling which explains well this non pureness. Figure 5: Tiling from overlaps Typeset by FoilTEX 24

26 The 2nd tiling associates different colors to translationally equivalent tiles. The situation is similar to Sing s example. References [1] S. Akiyama and J.Y. Lee, Algorithm for determining pure pointedness of self-affine tilings, to appear in Adv. Math. [2] C. Bandt, Self-similar tilings and patterns described by mappings, The Mathematics of Long-Range Aperiodic Order (Waterloo, ON, 1995) (R. V. Moody, ed.), NATO Adv. Sci. Inst. Ser. C, Math. Phys. Sci., pp Typeset by FoilTEX 25

27 [3] E. Bombieri and J. E. Taylor, Quasicrystals, tilings, and algebraic number theory: some preliminary connections, Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp [4] A. Clark and L. Sadun, When size matters: subshifts and their related tiling spaces, Ergodic Theory Dynam. Systems 23 (2003), no. 4, [5] F. M. Dekking, Recurrent sets, Adv. in Math. 44 (1982), no. 1, [6], Replicating superfigures and endomorphisms of Typeset by FoilTEX 26

28 free groups, J. Combin. Theory Ser. A 32 (1982), no. 3, [7] F.M.Dekking and M. Keane, Mixing properties of substitutions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 42 (1978), no. 1, [8] B. Host, Valeurs propres des systèmes dynamiques définis par des substitutions de longueur variable, Ergodic Theory Dynam. Systems 6 (1986), no. 4, [9] R. Kenyon, The construction of self-similar tilings, Geometric and Funct. Anal. 6 (1996), Typeset by FoilTEX 27

29 [10] E. Harriss Sh. Ito P. Arnoux, M. Furukado, Algebraic numbers, free group automorphisms and substitutions of the plane, To appear in Trans. Amer. Math. Soc. (2010). [11] B. Sing, Pisot substitutions and beyond, Phd. thesis, Universität Bielefeld, [12] B. Solomyak, Dynamics of self-similar tilings, Ergodic Theory Dynam. Systems 17 (1997), no. 3, Typeset by FoilTEX 28