Rectilinear Polygon Exchange Transformations via Cut-and-Project Methods

Transcription

1 Rectilinear Polygon Exchange Transformations via Cut-and-Project Methods Ren Yi (joint work with Ian Alevy and Richard Kenyon) Zero Entropy System, CIRM 1/39

2 Outline 1. Polygon Exchange Transformations (PETs) 2. Cut-and-Project Sets 3. (1) + (2) Rectilinear PETs (RETs) (Renormalization) 4. Multi-Stage RETs 5. Parameter Space of Multi-Stage RETs 2/39

3 What is a PET? (Polygon Exchange Transformation) 3/39

4 What is a PET? Let X be a polygon. Let A = {A k } N k=0 and B = {B k} N k=0 into polygons such that be two partitions of X B k = A k + v k, k = 0,, N. For each k, the polygons A k and B k are translation equivalent. A polygon exchange transformation (PET) is a dynamical system on X. The map T : X X is defined by x x + v k, x Int(A k ). 4/39

5 What is a PET? Figure: 1-dim example (interval exchange transformation) Figure: An example of PETs 5/39

6 What is a PET? 6/39

7 How to construct a PET? 7/39

8 How to construct a PET? Products of IETs Covering maps of some piecewise isometries (Goetz, Akiyama, Lowenstein, N. Bedaride, ) (studied by most of the participants!) Outer billiards on regular n-gons for n = 5, 7, 8, 12 (Tabachnikov, Bedaride & Cassaigne); outer billiards on kites (Schwartz) Multigraph PETs (R. Schwartz and R. Yi) 8/39

9 Rectilinear PETs (RETs) We consider the case when all A k A and B k B in the partitions are rectilinear polygons. Figure: An example of rectilinear PETs 9/39

10 Renormalization: An approach to study PETs 10/39

11 Renormalization Renormalization is a tool to zoom the space and accelerate the orbits of points along time. Let T : X X be a PET and Y be a subset of X. The first return map ˆT Y : Y Y is given by ˆT Y (p) = T n (p), where n = min{t k (p) Y k > 0}. Definition (Renormalization) A PET T : X X is renormalizable if there exists a subset Y X such that ˆT Y is conjugate to T by a affine map. 11/39

12 Renormalization An example of piecewise isometric maps and its renormalization (A. Goetz). 12/39

13 Cut-and-Project Method 13/39

14 Cut-and-Project Method For n 6, let M n be the matrix M n = n n + 1 Characteristic polynomial: q(x) = x 3 (n + 1)x 2 + nx 1 Eigenvalues: 0 < λ 1 < λ 2 < 1 < λ 3 The dominant eigenvalue λ 3 is a Pisot number. Eigenvectors (corresponding to λ i ): ξ i = (1, λ i, λ 2 i ). 14/39

15 Cut-and-Project Method For each matrix M n, H e := the expanding line H c := the contracting hyperplane π e := the projection of R 3 onto H e along H c π e : x x ξ 3 π c := the projection of R 3 onto H c along H e π c : x (x ξ 1, x ξ 2 ) 15/39

16 Cut-and-Project Method Let X be the unit square in R 2 centered at ( 1 2, 1 2 ). Define Λ X = {x Z 3 : π c (x) X)}. For p Λ X, define w p Λ X to be the point such that π e (w p ) π e (p) = min w Λ X {π e (w) π e (p) > 0} Let E = {w p p p Λ X }. E = N < 16/39

17 Rectilinear PET via Cut-and-Project Methods 17/39

18 Construction Recall that X is a unit square centered at ( 1 2, 1 2 ). Translation vectors on X: V = π c (E) We define an order on the elements in E by η i < η j if π e (η i ) < π e (η j ). 18/39

19 Construction The partition A = {A k } N 1 k=0 of X as follows: Let f v be the map of translation by v, i.e. f v : x x + v. A 0 = f 1 v 0 (X) X. A k = (f 1 v k (X) X) \ ( k 1 i=0 A i). 19/39

20 Construction An example of the rectilinear PET T n : X X constructed from cut-and-project method via matrix M n = for n = 6. 1 n n + 1 Each orbit of point x X corresponds to a lattice walk. 20/39

21 Renormalization Theorem Let Y X be the rectangle with top right vertex (1, 1) with width λ 1 and height λ 2 where 0 < λ 1 < λ 2 < 1 are two eigenvalues of M n. Let T n : X X be a rectilinear PET arisen from cut-and-project method via M n. Then ˆT n Y = ψ 1 T n ψ where ψ : X X is defined by ψ : (x, y) ( x + λ 1 1 λ 1, y + λ 2 1 λ 2 ). 21/39

22 Renormalization Figure: An illustration of renormalization. 22/39

23 Renormalization Proof Sketch: Define Λ Y = {(a, b, c) Z 3 π c (a, b, c) Y }. Note the fact that Λ Y Λ X. Define Ψ : Λ X Λ X to be the acceleration map defined by Ψ : a a 1 b 1 0 n b + 1 c 0 1 n + 1 c 0 We show that Ψ : Λ Y Λ Y corresponds to ˆT Y : Y Y. 23/39

24 Renormalization Proof Sketch: Return a a π c b = (x, y) π c Ψ b = (λ 1 x, λ 2 y) c c where 0 < λ 1 < λ 2 < 1 are eigenvalues of M n. First return The map Ψ preserves the order of the lattice walk {ω 1, ω 2, }, i.e. π e (ω i ) < π e (ω j ) π e Ψ(ω i ) < π e Ψ(ω j ). 24/39

25 Renormalization Underlying substitution symbolic dynamics: Set a = v 0, b = v 1 and c = v 3. substitution of Pisot type a abc, b abcb, and c (abc) n 3 bc for n 6 Incidence matrix: 1 1 n 3 W n = 1 2 n n 2 W n M n 25/39

26 Multi-Stage Rectilinear PETs 26/39

27 Multi-Stage Rectilinear PETs Let P n be the matrix of translation. Consider matrix products P 0 = P n1 P nk and P i = P n1 P ni for i 1. For each stage i 0, we want to construct a RET S i by via P i such that each S i has the same combinatorics as T 6. each S i is renormalizable, i.e. Ŝ i Yi = ψ 1 i S i+1 ψ i. 27/39

28 Multi-Stage Rectilinear PETs The matrix of translation P n = n 4 n 3 n 4 n 4 The matrix P n can be reduced to the incidence matrix W n. 28/39

29 Multi-Stage RETs (Construction) Figure: An illustration of renormalization. 29/39

30 Multi-Stage RETs (Construction) Construction: Let P = P n1 P nk. Non-zero eigenvalues: 0 < λ 1 < λ 2 < 1 < λ 3. Eigenvectors: ζ 1 = (x 0, x 1,, x 6 ) for x 1 x 0 = 1. ζ 2 = (y 0, y 1,, y 6 ) for y 1 y 0 = 1 Let V = {v 0,, v 6 } be the set of vectors given by v j = (x j, y j ). There is a RET S : X X with the set of translation vectors V = {v j } 6 j=0 if v j ( 1, 1) ( 1, 1). 30/39

31 Multi-Stage RETs (Construction) P = P n1 P nk and P i = P n1 P ni. P i ζ 1 = (x i 0,, xi 6 ) and P iζ 2 = (y0 i,, yi 6 ). Rescale x i 0 + xi 1 = 1 and yi 0 + yi 1 = 1. Translation vector for ith stage: v0 i = (x i 0, y0), i, v6 i = (x i 6, y6). i There is a multi-stage RET S : X X if at each stage i, all translation vectors v i j ( 1, 1) ( 1, 1). A product P is called admissible if there is a multi-stage RET. 31/39

32 Multi-Stage RETs (Construction) Figure: The multi-stage rectilinear PETs for P = P 7 P 8 P 6 P 7. 32/39

33 Multi-Stage RETs (Renormalization) Theorem (Renormalization Scheme) Let P = P n1 P nk be an admissible matrix product. Let S : X X be a a multi-stage RET determined by P. Then Ŝ Y0 = ψ 1 0 S 1 ψ 0 for the affine map ψ 0 : Y 0 X. Ŝ i Yi = ψ 1 i for the affine map ψ i : Y i X S i+1 ψ i 33/39

34 The Parameter Space of Multi-Stage RETs Figure: The parameter space of admissible rectilinear PETs in R 4 Theorem (in process) The parameter space M of admissible rectilinear PETs is a Cantor set in R 4. 34/39

35 The Parameter Space of Multi-Stage RETs Figure: P n M and Pn 1 M 35/39

36 The Parameter Space of Multi-Stage RETs We expect the dynamics on the parameter space to be a discrete horseshoe map /39

37 Domain Exchange Transformation We can construct domain exchange transformation on more general domains by the cut-and-project method. Figure: An example of circle exchange transformation by the cut-and-project method (Conjecture) The domain exchange transformations on convex domains via cut-and-project methods are renormalizable. 37/39

38 Future Directions PETs on general domains and their renormalizations Piecewise isometries arisen from cut-and-project methods associated to quartic polynomials Generalizations of the Three Gap Theorem Self-similar tilings from RETs? Generalizations of Rauzy inductions in PETs Complexity of the PETs 38/39

39 Thank you 39/39