Continuous self-similarity in parametric piecewise isometries
CContinuous self-similarityin parametric piecewise isometries
J. H. Lowenstein and F. Vivaldi † Dept. of Physics, New York University, 2 Washington Place, New York, NY10003, USA † School of Mathematical Sciences, Queen Mary, University of London, London E14NS, UK
Abstract
We exhibit two distinct renormalization scenarios in many-parameterfamilies of piecewise isometries (PWI) of a rhombus. The rotationalcomponent, defined over the quadratic field K = Q ( √ K . In each case the parameters range over a convexdomain.In one scenario the PWI is self-similar if and only if one parameterbelongs to K , while the other is free. Such a continuous self-similarityis due to the possibility of merging adjacent atoms of an induced PWI,a common phenomenon in the Rauzy-Veech induction for interval ex-change transformations.In the second scenario, the phase space splits into several disjoint(non-convex) invariant components. We show that each componenthas continuous self-similarity, but due to the transversality of the cor-responding foliations, full self-similarity in phase space is achieved ifand only if both parameters belong to K .All our computations are exact, using algebraic numbers. October 20, 2018 a r X i v : . [ m a t h . D S ] A ug Introduction
This work represents a first study of renormalization in many-parameterfamilies of planar piecewise isometries (PWI). These are maps of polygonaldomains partitioned into convex sub-domains —called atoms— in such away that the restriction of the map to each atom is an isometry. The first-return map to any convex sub-domain D is a new PWI, called the inducedPWI on D . If by repeating the induction we obtain a sub-system conjugateto the original one via a suitable group of isometries and homotheties, thenwe consider the original PWI to be renormalizable.Recent work on renormalization in two-dimensional parametric familiesconcerned one-parameter deformations of the translational part of a PWI.[10,16,23]. Induction is accompanied by a transformation s (cid:55)→ r ( s ), where s is the parameter and r is the renormalization function. Self-similarity thencorresponds to the periodic points of r .This setting is analogous to Rauzy-Veech induction for interval exchangetransformations (IET’s, see [21, 24, 26]), which are one-dimensional PWI’s.For an IET, the parameters are a vector of sub-interval lengths togetherwith a permutation, and the renormalization acts on the lengths via anintegral matrix. The fixed-point condition for self-similarity is an eigenvaluecondition for a product of matrices.An arithmetical characterisation of self-similarity is provided by theBoshernitzan-Carroll theorem [5], which states that if an IET is definedover a quadratic number field (meaning that all intervals’ lengths belongto that field), then inducing on atoms results in only finitely many distinctIETs, up to scaling. In the case of two intervals (a rotation), this theoremreduces to Lagrange’s theorem on the eventual periodicity of the continuedfractions coefficients of quadratic irrationals. However, unlike for continuedfractions, there are self-similar IETs over fields of larger degree [3]. It is alsoknown that in a uniquely ergodic self-similar IET, the scaling constant is aunit in a distinguished ring of algebraic integers [20].In two dimensions general results are scarce [18, 19]. All early results onrenormalization concerned specific models of PWI’s defined over quadraticfields (the field of a PWI is determined by the entries of the rotation matricesand the translation vectors defining the isometries) [1, 2, 12, 14, 22]. A moreintricate form of renormalization has been found in a handful of cubic cases[9, 15].The first results on parametric families concerned polygon-exchange trans-formations, due to Hooper [10] (on the measure of the periodic and aperiodicsets in a two-parameter family of rectangle-exchange transformations) and2chwartz [23] (on the renormalization group of a one-parameter family ofpolygon-exchange transformations). Subsequently, the present authors [16]studied two one-parameter families of piecewise isometries. Each familyhas a fixed rotational component defined over a quadratic field ( Q ( √
5) and Q ( √ s restricted to a suitable interval, there is an induced PWI on atriangular sub-domain (the so-called base triangle ), which reproduces itselfafter scaling and the reparametrisation s (cid:55)→ r ( s ). After an affine change ofparameter, the function r was found to be of L¨uroth type —a piecewise affineversion of Gauss’s map [4,8,17]. In the pentagonal model, the discontinuitiesof r accumulate at the origin; in the octagonal case one has r = f ◦ f , where f has two accumulation points of discontinuities. In both cases r is expandingand preserves the Lebesgue measure.Figure 1: One-parameter rhombus map of the pentagonal model of [16]. Therhombus and all directions are fixed. The parameter shifts the inner boundaries ofthe atoms in such a way that all three atoms retain a reflection symmetry.
The present work deals with two-parameter extensions of the one-parameterpentagonal model of [16] (figure 2). The roles of the parameters is madeexplicit in figure 3, where we employ a new co-ordinate system in orderto restrict the arithmetic to the quadratic field K = Q ( √ u i · x = b i , b i = b i, + b i, s + b i, s , b i,j ∈ Q ( √
5) 0 (cid:54) i (cid:54) s , s are the parameters. We use these co-ordinates for calculations,reserving the original co-ordinates for graphics.3 R R R R R ρ R R Figure 2:
Two-parameter rhombus map.
In section 2 below, we develop the necessary geometrical constructs:tiles, dressed domains, isometries, etc. In section 3 we review the renormal-ization theory of the base triangle developed in [16], with two additions. Weextend the isometries to the atoms’ boundaries (which will be needed to gluetogether atoms in section 4), and we introduce an improved renormalizationscheme whose renormalization function has finitely many singularities.The first renormalization scenario is established in section 4. There wemake three successive inductions on triangular sub-domains, to obtain aPWI whose eight distinct return orbits tile the rhombus, apart from theorbits of finitely many periodic domains. We identify a convex polygon Πin the t, s parameter space within which the induced map has only threedistinct isometries. By merging neighbouring atoms which share the sameisometry, we recover the one-parameter PWI discussed in the previous sec-tion. As a result, self-similar dynamics occurs if and only if s ∈ K , and since t is arbitrary, the self-similarity constraint corresponds to a foliation of Π.The appearance of a hidden reduction of the number of atoms, whichreveals itself only after induction, is present in a simpler form in the Rauzy-Veech induction of interval exchange transformations. In section 5 we showthat the occurrence of free parameters in renormalizable IETs can be under-stood in terms of the properties of the associated translation surfaces [11].In particular, there are self-similar IETs without free parameters, as long asthe number of intervals is greater than three.4 − t β+β s + t −β s − t R R ρ R R R R R Figure 3:
The two-parameter map conjugate to the rhombus map of figure 2.
In view of this analogy, we conducted an extensive search for a non-degenerate two-parameter family, using two-dimensional sections of a three-parameter system of the 2 π/ Acknowledgements:
JHL and FV would like to thank, respectively, theSchool of Mathematical Sciences at Queen Mary, University of London, andthe Department of Physics of New York University, for their hospitality.
Throughout this paper, we let α = √ , ω = ( α + 1) / , β = ω − = ( α − / . (1)The arithmetical environment is the quadratic field Q ( ω ) with its ring ofintegers Z [ ω ], given by Q ( ω ) = { x + yα : x, y ∈ Q } , Z [ ω ] = { m + nω : m, n ∈ Z } . (2)5he number ω , which is the fundamental unit in Z [ ω ] (see [6, chapter 6]),will determine the scaling under renormalization. The number β = ω − A tile X with n edges is a convex polygon defined by the half-plane condi-tions u m i · x < b i (excluded edge)or u m i · x (cid:62) b i (included edge) i = 1 , . . . , n, (3)where x = ( x, y ), b i ∈ R , and the u m are the vectors u m = (cid:18) cos 2 πm , sin 2 πm (cid:19) m ∈ { , . . . , } . (4)For the i th edge, defined by u m i · x = b i , we introduce an index (cid:15) i , where (cid:15) i = − X , and (cid:15) i = 1 if it is excluded. We thenrepresent X as a triple of n -vectors X = [( m , . . . , m n ) , ( (cid:15) , . . . , (cid:15) n ) , ( b , . . . , b n )] . (5)We shall assume that n is minimal, namely that X is not definable by fewerconditions.A tiling X is a set of disjoint tiles, X = { X , . . . , X N } and is associated with a domain X (union of tiles)X = N (cid:91) k =1 X k . Note that a domain need not be convex, or even connected. Note further thatthanks to (3), if a pair of tiles have disjoint interiors but share a commonboundary segment, that segment belongs to one and only one tile of thepair. This allows the possibility of gluing together adjacent tiles withoutdisturbing the inclusion relation of the respective edges.6 .2 Similarity group
The transformation properties of planar objects are provided by a group G which comprises the rotations and reflections of the symmetry group of theregular pentagon (the dihedral group D ) together with translations in K and real scale transformations.We adopt the following notation: U m : reflection about the line generated by u m . R m : rotation by the angle 2 mπ/ T d : translation by d ∈ K . S η : scaling by η ∈ R + .We write X ∼ Y to indicate that X is similar to Y , i.e., that X = G ( Y ) forsome G ∈ G . As G is a group, this is an equivalence relation. Within G wedistinguish two important subgroups: the isometry group I generated by ro-tations, reflections, and translations, and the dynamical group I + , generatedby rotations and translations. A dressed domain is a triple X = (X , X , ρ ) , (6)where X = { X , . . . , X N } is a tiling of the domain X , and ρ = { ρ , . . . , ρ N } ,where ρ i ∈ J + is an orientation-preserving isometry acting on the tile X k .Under the action of G ∈ G , a dressed domain X transforms as G ( X ) = G (X , X , ρ ) = ( G (X) , { G ( X ) , . . . , G ( X k ) } , G ◦ ρ ◦ G − )where the conjugacy acts componentwise. To emphasize the association ofa mapping ρ with a particular dressed domain X , we use the notation ρ X .Let X = (X , X , ρ X ) be a dressed domain, and let Y be a sub-domain ofX. We denote by ρ Y the first-return map on Y induced by ρ X . We call theresulting dressed domain Y = (Y , Y , ρ Y ) a dressed sub-domain of X , andwrite Y (cid:47) X . (7)The dressed sub-domain relation (7) is scale invariant , namely invariantunder an homothety. Indeed, if S η denotes scaling by a factor η , then in the7ata (5) specifying a tile, the orientations m k remain unchanged, while thepentagonal coordinates b k scale by η . Moreover, the identity S η T d R n = T η d R n S η . shows that the piecewise isometries ρ scale in the same way. We concludethat the subdomain relation (7) is preserved if the dressed domain parame-ters are scaled by the same factor for both members. In this article we consider continuously deformable dressed domains X = X ( s ) called parametric dressed domains , depending on a real parametervector s = ( s , . . . , s p ).These are domains whose tiles X k and image tiles ρ k ( X k ) depend on s only via the coefficients b i , while the parameters n , m i and (cid:15) i remain fixed[see (5)]. We shall require that the b i ’s be affine functions of s , . . . , s p , withcoefficients in Q ( ω ). Algebraically, this is expressed as b i ∈ S S = Q ( ω ) + Q ( ω ) s + · · · + Q ( ω ) s p , (8)where s , . . . , s p are regarded as indeterminates. The set S is is a ( p + 1)-dimensional vector space over Q ( ω ) (a Q ( ω )-module).The condition (8) gives us affine functions b i : R p → R b i ( s , . . . , s p ) = b i, + b i, s + · · · + b i,p s p b i,j ∈ Q ( ω ) . (9)We define the bifurcation-free set Π ( X ) to be the maximal open set suchthat all of the edges of all X k ( s ) have non-zero lengths. Note that othertypes of bifurcations may occur if X is embedded within a larger domain(see section 2.6.) A parametric dressed domain X ( s ) is said to be renormalizable over an opendomain Π ⊂ R p if there exists a piecewise smooth map r : Π → Π such thatfor every choice of s ∈ r − (Π) the dressed domain X ( s ) has a dressed subdo-main Y similar to X ( r ( s )) which satisfies the recursive tiling property. Thefunction r depends only on s , a requirement of scale invariance. In general,we have Y = Y i ( s ) ( s ), where i is a discrete index. The set r − (Π) neednot be connected (even if i is constant), each connected component being a8ifurcation-free domain of Y . (To extend the renormalization function r tothe closure of Π, one must include bifurcation parameter values, as in [16].)If s = s is eventually periodic under r , then we say that X ( s ) is self-similar . A self-similar system has an induced sub-system which reproducesitself on a smaller scale under induction.Let a parametric dressed domain X ( s ) have induced X j ( s ), such that,for j = 1 , . . . , N we have: ( i ) X j is renormalizable over a domain Π j ; ( ii )the X j recursively tile X ; ( iii ) the Π j have non-empty intersection Π. Thenwe still consider X renormalizable over Π.The definition of renormalizability given above is tailored to our model; itis not the most general possible, and it is local in parameter space. We allow Y to depend on a discrete index (as in Rauzy induction for interval-exchangetransformations —see section 5) to obtain a simpler renormalization function r (section 3). We only require X to be eventually renormalizable, and weallow X to have sub-domains with independent renormalization schemes(which is a common phenomenon, see section 6). For computations, we use the Mathematica R (cid:13) procedures listed in the Elec-tronic Supplement [7]. All computations reported in this work are exact,employing integer and polynomial arithmetic, and the symbolic representa-tion of algebraic numbers.The geometrical objects defined in section 2.1 require arithmetic in a bi-quadratic field, since only the first component of the vectors u m is in Q ( ω ).To circumvent this difficulty, we conjugate our PWI to a map of a squarewhere the clockwise rotation 2 π/ Z [ ω ] (cid:18) − β (cid:19) where β was defined in (1). (This is still a PWI with respect to a non-Euclidean metric.) In the new co-ordinates, the vectors u m become { (1 , , (0 , , ( − , β ) , ( − β, − β ) , ( β, − } which belong to Z [ ω ] . With this representation, all of our calculations canbe performed within the module S defined in (8). We shall still displayour figures in the original coordinates, where geometric relations (especiallyreflection symmetries) are more apparent to the eye.In constructing a return map orbit of a domain X ( s ) by direct iteration,one determines inclusion and disjointness relations among domains, which9equire evaluations of inequalities (3). Since the latter are expressed byaffine functions of the parameter s in some polytope Π, it suffices to checkthe inequalities on the boundary of Π. All these boundary values belong tothe field Q ( ω ), and the inequalities can be reduced to integer inequalities.Typically, X will be immersed in a larger domain Y (an atom, say).Therefore, in addition to the intrinsic bifurcation-free polytope Π( X ) definedin section 2.4, one must also consider the polytope Π( X , Y ) defined by theinclusion X ( s ) ⊂ Y ( s ), as well as intersection of these polytopes.The recursive tiling property defined in section 2.5 is established byadding up the areas of the tiles of all the orbits, and comparing it with thetotal area of the parent domain.With these techniques, we are able to establish rigorously a variety ofstatements valid over convex sets in parameter space. The base triangle is the simplest one-parameter renormalizable piecewiseisometry associated with rotations by 2 π/
5; it is self-similar precisely forparameter in the quadratic field Q ( √ B prototype is the following dressed domain (see figure4): ρ B B B B Figure 4:
Base triangle prototype. = ( B, ( B , B , B ) , ( ρ , ρ , ρ ))where B = [(1 , , , ( − , , , ( τ − ω , , ,B = [(0 , , , (1 , , , (0 , , ω − ωτ )] , (10) B = [(1 , , , , , ( − , , , − , , ( τ − ω , ω − ωτ, , ω − ωτ, ,B = [(1 , , , ( − , , − , ( τ − ω , , ω − ωτ )] . The dynamics is given by a local reflection of each atom about its ownsymmetry axis, followed by a global reflection about the symmetry axis of B , which can be written as: ρ = T ( ωτ − ω, − ω + ω τ ) R ρ = T (0 ,ω τ − ω ) R (11) ρ = T ( ωτ − ω,ω τ − ω ) R . Here we have chosen a coordinate system such that the peak of the isoscelestriangle is at the origin and the altitude of the atom B is the parameter τ , which varies over the interval (0 ,
1) without the occurrence of a bifur-cation. This parameter (together with time-reversal invariance) determinesthe scale-invariant properties of the dressed domain, since it is related tothe ratio η of altitudes of B and B by the formula η = τω − τ . As τ varies from 0 to 1, η increases from 0 to β .The edges of the domain B are included or excluded as stipulated insection 2; a vertex joining two included edges is included, but is excludedotherwise. The renormalizability analysis will also require a second basetriangle ˜ B , differing from B by a change of sign of all edge coordinates andtranslation vectors, as well as of the respective (cid:15) i . The dressed domains B and ˜ B are G -inequivalent: not only do they have different boundary con-ditions, but their interiors differ by a rotation by π , not an element of thesimilarity group.The renormalizability analysis for the base triangle is summarized in thefollowing lemma: Lemma 1
Let B be as above. The following holds:(i) For < τ < β , B has a dressed subdomain B ∼ B which is scaled bya factor (1 − τ ) / ( ω − τ ) and has shape parameter r ( τ ) = ω τ . ii) For β < τ < β , B has a dressed subdomain B ∼ ˜ B which is scaled bya factor τ / ( ω − τ ) and has shape parameter r ( τ ) = ω ( β − τ ) .(iii) For β < τ < , B has a dressed subdomain B ∼ ˜ B which is scaled bya factor τ / ( ω − τ ) and has shape parameter r ( τ ) = ω (1 − τ ) . ββ β β τ r ( τ ) Figure 5:
Renormalization function r ( τ ) for base triangles. The renormalization function r has three branches (see figure 5). In cases(ii) and (iii) one induces on the atom B , over two disjoint bifurcation-freeparameter ranges. Since the size of B vanishes as τ approaches 0, in therange (i) we induce on the triangle [(1 , , , ( − , , , ( β τ − , , B i , together with a finitenumber of periodic tiles, completely tile the triangle B . For ˜ B , the pre-scriptions (i)-(iii) hold with the roles of B and ˜ B exchanged. The inductionrelations are represented as the graph in figure 6.As in [16], the proof of Lemma 1 is by direct iteration, as discussed insection 2.6. The main difference in the two computational algorithms liesin the procedures used to verify inclusion and disjointness of tiles (see [7]).Specifically, checking the sub-polygon relation X ⊂ Y requires verifying thatno included vertex of X has landed on an excluded edge of Y . Similarly, to12 B ~ τ > β τ < β τ < β Figure 6:
Renormalization graph for base triangles. A directed link from X to Y indicates that X has an induced dressed subdomain equivalent to Y , subject tothe the indicated parameter constraint. decide that X and Y are disjoint, one must check that no included vertexof either tile lies on an included edge of the other.With reference to lemma 1, we remark that the base triange B , withits definition extended to the two-atom limiting cases τ = 0 ,
1, is in factrenormalizable also at the parameter values 0 , β , β,
1, with r ( τ ) = 0 in allthese cases (see [7] for the calculations). These additional parameter valuesare needed to make B renormalizable over the whole interval [0 , ∩ Q ( ω ).We shall use this property in sections 4 and 6. We now turn to the two-parameter rhombus map introduced in section 1—see figures 2 and 3. In suitable coordinates, the dressed domain is givenby R = ( R, ( R , . . . , R ) , ( ρ R , . . . , ρ R )) , with (see figure 2) R = [(0 , , , , ( − , − , , , ( − t, − s, − t, − s )] ,R = [(0 , , , ( − , − , , ( − t, − s, − s )] ,R = [(0 , , , , , , ( − , − , − , , , , ( − t, − t, − − s, − t, − s, − s )] ,R = [(0 , , , (1 , , − , (1 − t, − − s, − t )] , (12) R = [(0 , , , , ( − , − , − , , ( − t, − s, − − s, − t )] ,R = [(1 , , , , ( − , , , , ( − s, − t, − t, − − s )] ρ R = T (0 , R , ρ R = T (0 , R , ρ R = T (0 , R , (13) ρ R = T (1 , R . ρ R = T (1 , R . The corresponding bifurcation-free parametric domain Π( R ), defined insection 2.4, is found to be the triangle with vertices at (0 , − /α, − /α ),13nd ( β/α, − β/α ). On the boundary of Π( R ) given by with s = t , the dresseddomain R collapses into the one-parameter pentagonal model of [16], andhence is self-similar for all s ∈ Q ( ω ) within a suitable interval. R R R R R A Figure 7:
The first two steps of the triple induction R (cid:46) R (cid:46) R (cid:46) A . The thirdstep produces the dressed domain A shown in figure 8. Our goal is to determine a parametric dressed domain A with bifurcation-free subdomain Π( A ) ⊂ Π( R ) over which R is renormalizable. To this end,we choose a specific parameter pair close to the s = t boundary: ( s , t ) =( − / , − / R , followedby two inductions on sub-triangles, as shown in figure 7. The last inductionproduces the dressed domain A , shown in figure 8, which is given by: A = ( A, ( A , . . . , A ) , ( ρ A , . . . , ρ A )) , (14)with A = [(2 , , , (1 , − , , ( − − s, β − s, − s )] A = [(1 , , , ( − , , − , ( β − t, − s, αβ − t )] ,A = [(1 , , , , ( − , , , − , ( β − t, αβ − t, − s, αβ − s )] ,A = [(0 , , , , (1 , , − , − , (1 − s, β − t, αβ − t, β − s )] ,A = [(4 , , , , (1 , , − , − , ( αβ − t, β − t, αβ − s, β − s )] ,A = [(2 , , , , , (1 , − , , , −
1) (15)( − − s, β − t, αβ − s, − s, − β − t )] , = [(2 , , , , (1 , , , − , ( − − s, − β − t, − s, − β − s )] ,A = [(2 , , , , (1 , − , , , ( − − s, β − s, αβ − s, β − t )] ,A = [(2 , , , (1 , , , ( − − s, − β − s, − s )] ,ρ A = ρ A = ρ A = ρ A = T (8 β , − β ) R ,ρ A = ρ A = ρ A = T (2 , β ) R ,ρ A = T (2 − β , − β ) R . (16) ρ A A
14 325 67 8 12 43 5 67 8
Figure 8:
The dressed domain A , with its 8 atoms numbered as in (15). Theboundaries of the composite atoms C , C , C are coloured red, green, and blue,respectively. We find that Π( A ) is the triangle with vertices( − α , − α ) , ( 12 (11 − α ) ,
14 ( −
25 + 11 α )) , ( 12 (11 − α ) ,
12 (11 − α )) , shown in figure 9. One verifies that Π( A ) is adjacent to the line s = t andthat ( s , t ) lies in its interior.Using direct iteration, we verify that for all ( s, t ) ∈ Π( A ) the returnorbits of the eight atoms of A , together with those of 13 periodic tiles,completely tile the rhombus R (see figure 10).A decisive simplification of the analysis results from the observation thatthe four atoms A , . . . , A of A are mapped by the same isometry, andhence, with regard to the first-return map to A , can be merged into a singletriangular tile, A . Similarly, atoms A , A , A can be merged into a single15 = t st Figure 9:
The parametric domain Π( A ). reflection-symmetric pentagon, A . The mergers have been suggested inthe shading of the tiles in figure 8. The dressed domain thus simplifies into C = ( C, ( C , C , C ) , ( ρ C , ρ C , ρ C )) def = ( A, ( A , A , A ) , ( ρ A , ρ A , ρ A )) (17)Moreover, one verifies that over Π( C ) = Π( A ), we have C ∼ B . The intrinsicshape parameter of C can be calculated from the ratio η C of the altitude of C to that of C : τ C = ω η C η C = ω ( αs + β ) . (18)As we transverse Π( C ) from left to right, s increases from 2 /α − − α ) /
2, with τ C increasing from 0 to 1.The issue of recursive tiling is now rather subtle. The rhombus is cer-tainly tiled by the return orbits of C , C , C , and the 13 periodic tiles whicharose in the induction on A . (see figure 10). However, the return paths arenot the same for all tiles. In (say) C , the tiles A , A , A , A have fourdistinct 78-step return paths, which go their separate ways, but recombineeventually to form an atom of the dressed domain with a unique isometry.The coincidence of the return times is not necessary to the recombination,as these times could differ by any integer multiples of 5. As a result, thepartition of C into A , A , A , A is relevant to the recursive tiling of theoriginal rhombus, but not to dynamical self-similarity. (We shall encounteragain the same phenomenon —recombination with different return times—in section 6.) 16 R Figure 10:
Tiling of R by return orbits of the 8 atoms of A (coloured) and 7periodic tiles (grey). Note that A , A , A , A , which comprise C , have distinctreturn orbits. The parameter pairs ( s, t ) ∈ Π( C ) corresponding to self-similarity for therhombus map R are now determined by the self-similarity of the induceddressed subdomain C . In turn, the latter are the values of s for which the basetriangle is self-similar, namely τ C ( s ) ∈ (0 , ∩ Q ( ω ), while t is unconstrained.Thus, by (18), R is renormalizable in Π( C ) if and only if( s, t ) ∈ Π( C ) ∩ ( Q ( ω ) × R ) . This is the main result of this section.17
Continuous self-similarity in Rauzy induction
Recombination of atoms, and the resulting appearance of a free parameterin self-similarity may seem a coincidental feature of planar PWI’s. This phe-nomenon is in fact common in the Rauzy-Veech analysis of renormalizableinterval exchange transformations (IET’s) [21, 24, 25].We fix a half-open interval Ω = [0 , l ) and a partition of Ω into n half-opensub intervals Ω i . An IET is a piecewise isometry of Ω which is a transla-tion on each Ω i . We represent it as a pair ( π, Λ), where Λ = ( λ , . . . , λ n )is the vector of the lengths of the sub-intervals and π is the permuta-tion of { , . . . , n } such that the intervals in the image appear in the order π (1) , . . . , π ( n ).We assume that π is irreducible in the sense that { , . . . , k } is mappedinto itself only if k = n . If we fix n , then ( π, Λ) is a parametric PWI,with discrete and continuous parameters π and Λ, respectively. IET’s whichdiffer only by an overall translation or scale transformation are consideredequivalent.The Rauzy-Veech induction on ( π, Λ) consists of inducing on the larger ofthe two intervals Ω (0) = [0 , l − λ n ) and Ω (1) = [0 , l − λ π − ( n ) ), denoted by type0 and type 1 induction, respectively (the case | Ω (0) | = | Ω (1) | is excluded fromconsideration, as in this case the map is not minimal). Induction correspondsto a map ( π, Λ) (cid:55)→ ( π (cid:48) , Λ (cid:48) ). Letting( a i ( π ) , A i ( π ) − Λ) = ( π (cid:48) , Λ (cid:48) ) i = 0 , A i ( π ) − is an n × n integral matrix (see [20] for explicitexpressions for a i and A i ( π )).The permutations π of n symbols are then represented as the verticesof the Rauzy graph . Each vertex has two outgoing and two incoming edges,associated with a i and a − i , respectively, for i = 0 ,
1. The
Rauzy classes
312 321 2310 1 01 0 1
Figure 11:
Rauzy graph for n = 3. All permutations are degenerate, and theirtranslation surface is a torus. Any renormalizable IET with three intervals willhave a free parameter. π , π , . . . , π p .Transversing such a circuit produces an induced IET which is a rescaledversion of the original one. Its length vector is an eigenvector of a productof matrices A i k ( π k ) − , k = 1 , . . . , p , with a scale factor given by the corre-sponding eigenvalue. In figures 11 and 12 we display the Rauzy graphs for n = 3 and n = 4 [25, section 6]. Figure 12:
Rauzy graph for n = 4. All permutations of the upper component aredegenerate. No permutation of the lower component has this property. Two of the three irreducible permutations of the Rauzy graph for n = 3(figure 11) and all of the permutations for the first class for n = 4 (fig-ure 12) contain a consecutive pair: ( . . . , j, j + 1 , . . . ). For those IET’s, theconsecutive intervals Ω j and Ω j +1 have the same translation vector, andhence the interval Ω j ∪ Ω j +1 may be merged into a single interval of length λ (cid:48) j = λ j + λ j +1 . The n -interval IET is thus equivalent to an n − λ j ∈ [0 , λ (cid:48) j ].Accordingly, we say that a permutation is degenerate if it has consecutivepairs or if it acquires this property after a single induction. In the lattercase, consecutive atoms of the child IET have distinct return paths in thetiling of the parent. 19uch a degeneracy is best understood by representing an IET as Poincar´esection of a flow on a translation surface [11, 24, 25]. The latter is a poly-gon with 2 n sides ( n is the number of intervals), labelled according to theordering of the intervals before and after the permutation. The sides thatcorrespond to the same interval have equal length and are parallel, and theyare to be identified. A rectilinear flow on the plane will develop conicalsingularities on the surface, in correspondence to the vertices of the 2 n -gon. While the translation surface is not unique, its genus and singularities(given by the total angle 2 π ( m + 1) at the identified vertices) depend onlyon the Rauzy class. The removable singularities ( m = 0) correspond to de-generate permutations, and they signal the appearance of free parametersin renormalizability. Since the translation surface does not change underinduction, these structures depend only on the Rauzy class to which thepermutation belongs. Furthermore, for any n (cid:62) n, . . . , n − ,
1) and ( n, n − , . . . , . . . , n, k + 1 , . . . , k ) for some k < n −
1, and all fourof its neighbours in a Rauzy graph have consecutive pairs, thanks to therelations a (( ...n, k + 1 , ..., k )) = ( ..., , k + 1 , k + 2 , ..., k ) ,a − (( ...n, k + 1 , ..., k )) = ( ..., n − , n, ..., k ) ,a (( ...n, k + 1 , ..., k )) = ( ..., n, k, k + 1 , ... ) ,a − (( ...n, k + 1 , ..., k )) = ( ..., n, ..., k, k + 1) . The n = 3 class provides the simplest illustration of this phenomenon.20 Weakly-discrete self-similarity
The analogy with Rauzy induction suggests that there might exist two-parameter planar PWI’s which do not admit self-similarity with free param-eters, meaning that both parameters would be algebraically constrained. Weshall exhibit a weak form of this property, resulting from the coexistence oftwo systems with continuous self-similarity. ρ R R R R R R Figure 13:
Four-atom rhombus PWI.
Our starting point is the four-atom, three-parameter PWI of the 2 π/ s, t, u ismade clear in the conjugate system of figure 14, where the rhombus appearsas a unit square. (A similar strategy can be pursued for the five-atom familyof figures 2 and 3, but we found that the four-atom family is somewhat easierto work with.)In convenient coordinates, the dressed domain is R = ( R, ( R , . . . , R ) , ( ρ , . . . , ρ )) , with R = [(0 , , , , ( − , − , , , ( − t, − s, − t, − s )] ,R = [(0 , , , , ( − , − , − , , ( − t, − t, − s, − s )] , {{ s − t s + β t − u R R R R R Figure 14:
Dressed domain R . Dependence of atoms on parameters s, t, u isshown. R = [(0 , , , , (1 , , , − , (1 − t, − s, − s, − t )] ,R = [(0 , , , , ( − , − , − , , ( − t, − s, − − s + u, − t )] , (19) R = [(0 , , , , (1 , , , − , (1 − t, − t, − − s + u, − s )] ρ = T (0 , R , ρ = T (0 , R ,ρ = T (1 , − u ) R ρ = T (1 , − u ) R . (20)From figure 14, we see that the bifurcation-free domain Π( R ) ⊂ R is thepolytope bounded by the planes s − t = 0, s − t = 1, u = 1, s + β t = 0, u − s − β t = 0, β − β s − t = 0, and 1 + β s + t − u = 0.This system has a simple one-parameter subsystem on the line L definedby s − t = β , u = β . We shall consider a two-parameter perturbation of thissubsystem in the plane u = β which intersects L . (We have also consideredother planes, obtaining other manageable examples: see remarks at the endof this section.)Setting u = β , the parameter polytope reduces to the hexagonal domainshown in figure 17 (left). As done in section 4, we choose a parameter pairclose to L lying within such a domain: ( s , t ) = (2 / , / R , we obtain the parametric dressed domain F shownin figure 15. One readily verifies that the return orbits of the eight atomsof F completely tile R , so that the renormalizability of R will follow from22igure 15: Induced dressed domain F . that of F . We find that the complete tiling of F by renormalizable dressedsub-domains, given in figure 16, requires the return orbits of three dressedtriangles F , F , F , plus seven periodic tiles P i (five regular pentagons, onetrapezoid, and one rhombus).Letting Π ∗ ( F ) = (cid:92) i =1 Π( F i ) (cid:92) i =1 Π( P i ) (21)we find that Π ∗ ( F ) is the quadrilateral with vertices (cid:0) β , (cid:1) , (cid:0) β + β /α, β /α ) (cid:1) , (cid:0) β + β /α, β /α (cid:1) , (cid:0) β , β /α (cid:1) , shown in figure 17 (right). Note that one of the bounding edges of theparameter domain coincides with the line L which was the starting point ofour perturbative exploration.The dressed domains F and F are base triangles equivalent to theprototype B , with respective shape parameters τ = ω α ( s − β ) , τ = ω αt. Examination of F shows that its atoms with four and five sides share thesame isometry (in spite of having different return paths, and even differentreturn times on the rhombus), and hence can be merged for the purpose oftesting renormalizability. After the merger, F is also equivalent to B , withshape parameter τ = ω α ( s − β ) . F F Figure 16:
Tiling of F by return orbits of F (red), F (green), F (blue), andseven periodic tiles (grey). Since each F i is renormalizable for τ i ∈ Q ( √ F (hence R ) is renormalizable when all three shape parameters are in Q ( √ s, t ) is constrained to belong to Π ∗ ( F ) ∩ Q ( √ .Each of the three renormalizable dressed domains F i provides a sequenceof nested coverings of a distinct invariant component of the exceptionalset complementary to all periodic points of the rhombus. The number ofdistinct ergodic components of the exceptional set is thus at least three. Forthe model of section 4, on the other hand, we believe that there is a singleergodic component.In closing, we summarize briefly the results of our explorations of thethree-parameter space of the four- or five-atom rhombus maps.Manageable renormalizations are likely to be found in systems specifiedby parameters of small height. [The height H ( ζ ) of the algebraic number ζ =( m/n )+( m (cid:48) /n (cid:48) ) ω is defined as H ( ζ ) = max( | m | , | n | , | m (cid:48) | , | n (cid:48) | )]. Such was thecase for the domain R , and the two-parameter restriction u = β describedabove. We have considered other planes of small height: s − t − ωu + β = 0, s − t + u − N disjoint dressed domains, each tiled by the return orbits of a single basetriangle (provided we ignore the “decorations” produced by the common24 .20 − − t s Π ( R ) Π ( F ) Π ( F ) Π ( F ) Π ( F ) Π ( F ) s-t = β .39 .40 .41 .42 .43 .44 st ∗ Figure 17:
Left: The bifurcation-free domains Π( R ) and Π( F ) for the parameters s and t , with u = β . Right: Detailed view of Π( F ), showing the trapezoidal domainΠ ∗ ( F ) of equation (21). The latter is constructed as the intersection of Π( F ) (semi-transparent green trapezoid), Π( F ) (semi-transparent blue triangle), and Π( F )(semi-transparent red square). The seven periodic tiles P i do not contribute anyadditional constraints. The unperturbed one-parameter model corresponds to the(yellow) line s − t = β , which lies along the south-east boundary of Π( F ) .boundaries of merged atoms), plus a finite number of periodic tiles.For the plane u = 0 of the five-atom map, the case N = 1 appears to bethe norm, so that in those models the exceptional set is likely to be uniquelyergodic. Elsewhere, a proliferation of ergodic components is typical (albeitnot universal). Notably, we have found no example of a rigidly self-similar,single component piecewise isometry with two or more parameters. Whethersuch a dynamical system exists at all remains an important open question. References [1] R. Adler, B. Kitchens and C. Tresser, Dynamics of non-ergodic piece-wise affine maps of the torus,
Ergod. Th. and Dynam. Sys. (2001)2559–999.[2] S. Akiyama and H. Brunotte and A. Peth˝o and W. Steiner, Periodic-ity of certain piecewise affine integer sequences, Tsukuba J. Math. (2008) 197–251.[3] P. Arnoux P. and J. Yoccoz, Construction de diffeomorphismes pseudo-Anosov, C. R. Acad. Sci. Paris (1981) 75–78.[4] J. Barrionuevo, R. M. Burton, K. Dajani and C. Kraaikamp, Er-godic properties of generalised L¨uroth series,
Acta Arithm.
LXXIV (4) (1996) 311–327.[5] M. D. Boshernitzan and C. R. Carroll, An extension of Lagrange’s theo-rem to interval exchange transformations over quadratic fields,
Journald’Analyse Math´ematique (1997) 21–44.[6] H. Cohn, Advanced number theory , Dover, New York (1980). (Firstpublished as
A second course in number theory
John Wiley and Sons,New York (1962).)[7] J. H. Lowenstein and F. Vivaldi, Electronic supplement to this ar-ticle, https://nyu.box.com/s/idygjzsyt2f5f24tzezvfcn7fi00rpi9 (2015).[8] J. Galambos,
Representations of Real Numbers by Infinite Series , Lec-ture Notes in Math. 502, Springer, Berlin (1982).[9] A. Goetz and G. Poggiaspalla, Rotation by π/ Nonlinearity (2004)1787–1802.[10] W. P. Hooper, Renormalization of polygon exchange maps arising fromcorner percolation, Invent. Math. , (2013) 255-320.[11] M. Kontsevich and A. Zorich, Connected components of the modulispaces of Abelian differentials with prescribed singularities, Invent.Math. (2003) 631–678.[12] K. L. Kouptsov, J. H. Lowenstein and F. Vivaldi, Quadratic rationalrotations of the torus and dual lattice maps,
Nonlinearity (2002)1795–1842.[13] J. H. Lowenstein, Pseudochaotic kicked oscillators , Higher EducationPress, Beijing and Springer-Verlag, Berlin (2012).2614] J. H. Lowenstein, S. Hatjispyros and F. Vivaldi, Quasi-periodicity,global stability and scaling in a model of Hamiltonian round-off,
Chaos (1997) 49–66.[15] J. H. Lowenstein, K. L. Kouptsov and F. Vivaldi, Recursive tiling andgeometry of piecewise rotations by π/ Nonlinearity (2004) 371–395.[16] J. H. Lowenstein and F. Vivaldi, Renormalization of one-parameterfamilies of piecewise isometries, subm. to Dynamical Systems arXiv:1405.7918 (2014).[17] J. L¨uroth, Ueber eine eindeutige Entwickelung von Zahlen in eine un-endliche Reihe,
Math. Ann. (1883) 411–423.[18] G. Poggiaspalla, Auto-similarit´es dans les syst`emes isom´etriques parmorceaux, PhD thesis, Universit´e de la M´editerran´ee, Aix-Marseille II,(2003).[19] G. Poggiaspalla, Self-similarity in piecewise isometric systems, Dynam-ical Systems , (2006) 147–189.[20] G. Poggiaspalla, J. H. Lowenstein and F. Vivaldi, Geometric represen-tation of interval exchange maps over algebraic number fields, Nonlin-earity (2008) 149–177.[21] G. Rauzy, ´Echange d’intervalles et transformations induites, ActaArith. , (1979) 315–328.[22] R. E. Schwartz, Outer billiards on kites , vol. 171 of Annals ofMath. studies, Princeton University Press, Princeton (2009).[23] R. E. Schwartz,
The Octagonal Pet , Mathematical Surveys and Mono-graphs, Volume 97, American Mathematical Society, (2014).[24] W. Veech, Gauss measures for transformations on the space of intervalexchange maps,