Tangency Properties of a Pentagonal Tiling Generated by a Piecewise Isometry.

Transcription

1 Tangency Propertes of a Pentagonal Tlng Generated by a Pecewse Isometry. Marcello Trovat and Peter Ashwn Mathematcs Research Insttute School of Engneerng, Computer Scence and Mathematcs Unversty of Eeter Eeter EX QE, UK August 6, 7 Abstract Pecewse Isometres (PWIs) are known to have dynamcal propertes that generate nterestng geometrc planar packngs. We analyse a partcular PWI ntroduced by Goetz that generates a packng by perodcally coded cells, each of whch s a pentagon. Our man result s that the tangency graph assocated wth ths packng s a forest (.e. has no nontrval cycles). We show however that ths s not a general property of PWIs by gvng an eample that has an nfnte number of cycles n the tangency graph of ts perodcally coded cells. Introducton Pecewse sometres (PWIs) are natural generalsatons of nterval echange transformatons to dmensons hgher than one. They appear n the theory of dgtal flters, Hamltonan systems and polygonal dual bllards, and have been etensvely studed usng technques from dynamcal systems and ergodc theory; see for eample []. In [8], a systematc study of Eucldean PWIs s proposed, focusng on the geometry and symbolc dynamcs generated by pecewse sometrc systems. Other work [9, ] eamnes nterestng eamples of PWIs defned by pecewse rotatons on polygonal regons n the plane, whle n [6], t s shown that general PWIs defned on a fnte unon of polytopes have dynamcs wth zero topologcal entropy. In general a PWI wll defne a partton of phase space nto a number of perodcally coded cells (consstng of conve polygons or dscs) and a remander, referred to as the eceptonal set. There have been several conjectures as to the nature of these perodc cells and the eceptonal set that have only been settled for a very lmted set of eamples where one can perform a detaled renormalsaton of the phase space as n [9, ]. In [6] a general framework s ntroduced n whch pecewse sometrc systems can be fully or partally renormalsed, but ths only characterses a small regon n phase space for most PWIs. A partcular conjecture s that typcal PWIs have an eceptonal set wth postve measure but open dense complement []. Roughly speakng ths suggests that the dynamcally defned packng of phase space for typcal PWIs s loose n that t contans enough gaps for a set of postve measure to be n ts complement. Some work towards ths conjecture s [] where t s shown that for typcal PWIs n a partcular one-parameter famly, the tangency graph assocated wth ths packng s trval,.e. has no tangences. On the other hand, [7] shows that certan famles of dsc packng wth nfnte numbers of tangences (Arbelos packngs) cannot be generated by PWIs because of ssues of fnte generaton of the tangency drectons.

2 Ths paper contnues ths lne of nvestgaton by lookng at a case where there are nontrval tangences. We consder a partcular PWI ntroduced by Goetz [8, 9] on an sosceles trangle wth rotatons by multples of π/5. For ths PWI t s known from prevous work that ts perodc cells are a full measure set consstng of an nfnte number of pentagons whose szes depend on the scalng propertes assocated wth ths PWI. The man goal of secton s show that the tangences between these pentagons are, although nontrval, qute sparse (we only count tangences between mdponts of sdes of the pentagons as these represent tangences of nscrbng dscs that are stably perodcally coded). More precsely we show that the tangency graph s such that every verte s ncdent to four or fve other vertces and s a forest,.e. an nfnte collecton of trees wth non nontrval cycles. Ths s shown by eamnaton of standard and not-so-standard renormalsatons of the dynamcs. Nonetheless, we show that lack of cycles s not a necessary condton for tangency graphs of perodc cells for PWIs; n secton, we ntroduce a related PWI, the Pe, whose tangency graph contans an nfnte number of cycles. We start wth some defntons and propertes of PWIs. Defnton. Let X R m be a bounded regon and defne M to be a partton of X f t s a collecton {M,... M n } of dsjont open bounded conve sets such X = n = M. We call an element M j of a partton an atom. A pecewse sometry n R n s a par (T,M), where T : X X s a map such that ts restrcton T M to each atom M, =,... n s an sometry. If T M s a rotaton for all =,...,n, then T s called a pecewse rotaton. We wll be eclusvely concerned wth a case m =, meanng we can dentfy X wth a subset of the comple plane C and rotatons can be wrtten R(z) = z + w wth,w C and =. The tnerary of a pont X s the sequence of atoms vsted by the forward orbt. Ths gves a way of codng the orbt of a pont X. We defne a cell as the set of ponts wth the same tnerary; those that have eventually perodc tnerary are called perodcally coded cells or for short perodc cells. It s easy to see that aperodcally coded cells are conve and have zero measure; hence they must be at most a lne segment. The eceptonal set s the set of all ponts that are not perodcally coded. If T : X X s a pecewse sometry and S any polygonal subset of X (ths need not be an atom) we defne the map that T nduces on S, T S : S S to be T S () = T k() () where k() N s frst return to S,.e. the smallest value such that T k() () S. The nduced map of a PWI wll be a PWI tself f k() s bounded on S. The followng result relates perodc cells of the nduced map to the orgnal. Lemma. Suppose (T,X) s a pecewse sometry and X be any atom such that T X s a pecewse sometry. Let P be a perodc cell for T X. Then P s a perodc cell for T. Proof. If P s a perodc cell for the codng of T X = T wth perod p, there must be a k such that T(P) = T k (P). Smlarly T (P) = T k +k (P), S (P) = T k +k +k (P) and so on. Thus we have P = T p (P) = T k +k +k + +k p (P) and hence P s a perodc cell for T. A standard argument (e.g. [, 7]) shows that for typcal pecewse rotatons the perodc cells wll be dscs because the nduced map on the cell s typcally an rratonal rotaton of a conve nvarant set. Ths means that the set of perodc cells for a typcal PWI conssts of an nvarant set of non-overlappng dscs; a dsc packng. In our case ratonal rotatons mean that the perodc cells are polygons. For an nvarant dsc packng we say two dscs are tangent f ther closures have a pont n common. Ths defnes a graph where vertces are the centres of dscs and there s an edge precsely when two dscs are tangent. Smlarly, we say two pentagonal cells are tangent f ther nscrbed dscs are tangent.

3 α+α +α α+α +α M T T( M ) M T( ) M α α Fgure : Acton of the pecewse rotaton T on X = M M. T acts by a clockwse rotaton by π/5 on M and an antclockwse rotaton by π/5 on M. We say a PWI T : X X has a renormalsaton to X X f the map nduced by T on X s conjugate to T by a smlarty,.e. f there s a smlarty π wth π(x ) = X such that T X () = π T π().. An Eample of a Pecewse Isometry In ths secton we descrbe the eample of Goetz n [] llustrated n Fgure. Let α = e π/5 be a prmtve ffth root of unty (note that α 5 = and α + α + α + α + = ). Defne a trangle M wth vertces (,α + α + α, ), a trangle M wth vertces (,,α ); one can verfy that these descrbe ponts as shown n Fgure. Defnton. We defne a PWI on X = M M by T : X X where T M = R, T M = R () where the rotatons R (z) = α z + (α + α + α ) and R (z) = α z + α. Ths pecewse sometry has a clear self smlar structure of cells that arses from the teraton of the map as depcted n Fgure. The algebrac structure of the map allows a full understandng of ts dynamcs as charactersed by the followng theorem of Goetz. Theorem. [9] Let T be the pecewse rotaton (). Let E = α, F = α + α + α, then trangle EF s the unon of an nfnte number of perodc cells, an eceptonal set R EF of non-eventually perodc ponts, and a set of zero Lebesgue measure. The eceptonal set has zero Lebesgue measure and ts Hausdorff dmenson les strctly between and. A comparable result s obtaned by Kahng [] for a smlar map on the torus. In [9], the numercal value of the Hausdorff dmenson of the eceptonal set s gven as.676. However, by consderng the equaton k s + (k ) s = assocated wth t (where k s the nverse of the golden rato, see [7] for more detals), we note that the correct value s log /log k =.. Theorem mples n partcular that the number of non-eventually perodc ponts s uncountable as, f that was not the case, then ther Hausdorff dmenson would be zero. The man result we wll prove n ths paper s the followng. Theorem. The tangency graph of the perodcally coded pentagonal cells of the pecewse rotaton T s a forest.

4 Fgure : The self smlar structure of the perodc cells generated by the teraton of the map T on the trangle X. Note that all perodc cells are regular pentagons. Consequences of ths nclude the observaton that the tangency graph s bpartte smply because every tree s bpartte. We wll gan a full descrpton of ths forest by applcaton of a number of dfferent renormalsatons. Renormalsaton and tangency propertes of the cells nduced by T We now dscuss some dynamcal propertes of the map T by consderng dfferent renormalsatons. The frst s the obvous renormalsaton consdered by Goetz [9]; the second s not so obvous but s needed n the proof of Theorem. Theorem. The map T has a renormalsaton to M. Proof. We defne a smlarty π : C C by π(z) = (α + α + α)(z + ) so that π(m ) = X. One can verfy that the nduced map T = T M s a pecewse sometry wth atoms (see Fgure ) M and M that can be renormalsed to M and M respectvely, va the map π. In fact the return map to M on M s gven by T M = R R R. Smlarly the return to M on M, ths s gven by T M = R. By drect calculaton one can verfy that the vertces of M under T M and M under T M map around such that T = π T π.. Renormalsaton to Trangles We partton the trangle X nto subsets consstng of 5 pentagons and smaller trangles X obtaned by renormalsng the orgnal trangle X such that the mdponts of the bases of the trangles X are the tangency ponts of the crcles nscrbed n the pentagons (see Fgure ). Let π : X X be a smlarty such that π (X ) = X and π s surjectve. In partcular X s scaled by a factor whch s γ k where γ = ( + 5)/ s the golden rato and for some

5 α+α+α α+α+α M M +α+α +α+α M M M M α α =γ α α =γ α α Fgure : The return map nduced on the atom M. In partcular we llustrate M and M on the left and ther mages for the nduced map on the rght. postve nteger k. Observe form Fgure that X = = X and wll show n Lemma 6 that we can obtan perodc cells by,..., under repeated applcaton of smlartes: π n π ( j ), () π n where k, k n and j {,,,,}. We say that perodc cells of ths type have level n. Theorem 5. The trangle X can be parttoned nto the dsjont unon of fve pentagons, =,, and eleven trangles X, =,, such that the map T can be renormalsed to each X usng a smlarty π. The nduced map on each pentagon k s perodc, wth the cells mapped around by T( ) =, T( ) =, T( ) =, T( ) =, and T( ) =. Hence the return to each k s a rotaton of order fve. Proof. From the defnton of the pecewse sometry T, one can verfy that the tnerary of the trangles X for =,... s as shown n Fgure 5, where X Y means T(X) = Y and T s surjectve and X Y means T(X) Y wth T not surjectve. Consder the return to X : note that T (X ) = X ; on the net terate, one part of X returns to X whle the other part returns after a further 7 terates. Hence T X s a pecewse sometry wth two atoms, namely the premages of T(X 7 ) and T(X ) X (see Fgure 6). One can verfy that ths map s conjugate to the orgnal map T va a smlarty π : X X such that π (X ) = X. Also, one can verfy that T( ) = meanng that T s a rotaton of order fve, for =,...,. = 5

6 Fgure : Partton of the trangle X nto trangles X,X,...,X (of equal sze), trangles X 5,...X (of equal sze but smaller than the prevous trangles) and 5 pentagons,...,. Note that the sdes of X for =,,,, are scaled by γ and the sdes X for = 5,6,..., are scaled by γ wth respect to those of X, where γ s the golden rato. X X X7 X6 X 9 X X X X X 5 X 8 Fgure 5: Dagram showng the tnerary of the eleven trangles X of Theorem 5 under the map T. 6

7 T T T T T T U Fgure 6: The tnerary of the trangles X and X under the PWI defned n Defnton, defnng the return map to X X. 7

8 Lemma 6. Let T be the pecewse sometry defned n Defnton. Then all perodc cells are pentagons of the form: π k π ( j ), () π k for l, l k and j. Proof. Suppose T k ( j ) = j and let π = π π π k and P = π ( j ) = π k π k π ( j ), for l, l k and j. Let B = π (X), then T B = π T π and note that T k B(π ( j )) = (π T π) (π T π) (π T π) π ( j ) = π T k ( j ) = π ( j ). But j s a perodc cell for T, so P = π ( j ) s perodc wth perod k for T B. By Lemma we have that P must also be perodc for T. Let A be the area of X; recall from () that all the pentagons,,..., are of level and those obtaned by scalng them n-tmes are sad to be of level n. By Theorem 5, the area of all the pentagons at level s gven by Aβ for < β < and smlarly the area of those at level are gven by A( β)β (as the area left s A( β) and ths area s covered by smlar copes of X). So the total area covered by the pentagons up to and ncludng level n s Aβ + A( β)β + A( β) β + + A( β) n β Therefore the total area covered by these cells s Aβ ( β) k = k= Aβ = A, () ( β) meanng that they have full measure X and there are no further conve non-empty perodc cells.. Propertes of the Tangency Graph Ths secton leads up to the proof of Theorem. We frst ntroduce some more smlartes as follows: Defnton. Defne smlartes Π, Π and Π, such that and Π and Π, such that Π (X X 5 X 6 ) = Π (X X 7 X 8 ) = Π (X X X 9 ) = X. Π (X X 5 X X 6 X ) = Π (X 7 X 8 X X X ) = X. Note that we can wrte Π ( ) = Π ( ) = and Π ( ) = Π ( ) =. Lemma 7. Each of the smlartes Π, Π, Π, Π and Π can be chosen such that there s a renormalsaton,.e. f X = Π (X) where {,,,, }, then T = T X satsfes T = Π T Π. 8

9 9 Π (X) Π (X) Π (X) Π (X) 6 (a) 6 (b) 9 Π (X) (c) Fgure 7: The representaton of the mages of X under the renormalsaton defned n Defnton, where (a) refers to the smlartes Π, Π, Π, (b) to Π and (c) to Π. 9

10 π( ) π( ) 6 π( ) π( ) Fgure 8: The perodc cells tangent to are determned by renormalsng to the trangles X,X,X and X, as descrbed n Lemma 8. Proof. Note from Theorem 5, the trangle X can be renormalsed to each X va a smlarty π for =,..., as graphcally descrbed n Fgure 5. Each smlarty n Defnton acts on unons of some X for =,...,. One can thus eplctly fnd the return maps assocated to the above smlartes. Lemma 8. The only perodc cells tangent to are π ( ), π ( ), π ( ) and π ( ) as shown n Fgure 8. Proof. As shown n Fgure 8, the only cells tangent to are those that are located on the base of the trangles X,X,X and X such that the mdponts of the bases of the trangles touch the mdponts of the edges of. We know that the all such trangles are defned as X = π (X) for =,,,. We have that X, =,,, have pentagonal cells on ther bases gven by π ( ) for =,,,. Lemma 9. The only cells that are tangent to are Π ( ), for =,,. Moreover Π ( ) = Π ( ) =. Proof. From Fgure the only pentagonal cells that are tangent to are those at the mdpont of the base of the trangles (X X 5 X 6 ), (X X 7 X 8 ) and (X X X 9 ). These are renormalsed to X va the smlarty Π for =,,. Therefore the tangent cells to are Π ( ), for =,,. Lemma. Let and be the pentagonal cells as depcted n Fgure ; then they are of the form = Π ( ) and = Π ( ). Furthermore the only perodcally coded cells tangent to and are Π (π ( )), Π (π ( )), respectvely, for. Proof. By Defnton we have that Π ( ) = for =,. By Lemma 8, the cells tangent to are π ( ), for =,,,. Therefore the only tangent cells to j, are Π j (π ( )) for j =,, and.

11 Lemma. Let be the pentagonal cell depcted n Fgure ; then t can be wrtten as Π ( ). The only perodc cells tangent to are of the form Π ( Π ( ) ) for =,,. Proof. From Defnton, t s clear that Π ( ) = Π ( ) = and that Π ( ) =. We can mmedately see that = Π ( ), and by Lemma 9, the only pentagonal cells tangent to are Π ( ), for =,,. Therefore the only perodcally coded cells tangent to are of the form Π ( Π ( ) ) for =,,, the result follows Defnton. Let P be any pentagonal cell defned by the pecewse sometry T and Q s a pentagon of smaller sze, such that the dscs nscrbed nsde P and Q are tangent. Then Q s sad to be a chld of P. We wll now show the followng result Theorem (The Four Chldren Theorem). Any perodc cell for T has four chldren ecept those of the form (Π )k ( ) for k =,,,... that have three chldren. Note that the pentagons of the form (Π )k ( ) for k =,,,... are precsely those cells that are tangent to two boundares of X. Proof. We prove the statement by nducton. As for the frst nductve step, by Lemmas 8, 9, and, we know that all the perodc cells tangent to the cells wth level are tangent to at most one bgger pentagonal cell. Ths means we know that the cells tangent to them (.e. cells of level ) can be descrbed as follows: Cells tangent to Π ( ): (whch are tangent to by Lemma 9), for =,, they are of the form Π (πj ( )) for j. Cells tangent to Π are of the form Π ( )): (whch are tangent to by Lemma ), for =,, they j ( ))) for j. (Π (Π (π Cells tangent to π ( ): (whch are tangent to by Lemma 8), for they are of the form πj (π ( )), for,j, and Cells tangent to Π l (π ( )): (whch are tangent to l by Lemma ), for, l =, they are of the form Π l (π (πj ( ))) for,j, l =,. All the above renormalsatons result n chldren that are not of the form (Π )k ( ) for k =,,,... meanng they wll also have four chldren. Thus we know that the frst nducton step s satsfed. Assume the theorem s true for all the cells wth level n. Consder any cell T of level n + : by Lemma 6, t can be wrtten as π k π k π ( j ), for l, l k and j. If T s tangent to at most one boundary of X, then by Lemma, t can be wrtten as one of the followng π n+ π n π ( ) π n+ π n π Π j ( ) for j =,

12 Therefore the cells of level n + tangent to t are of form of one of the followng π n+ π n π (π k ( )) π n+ π n π Π j (π k ( )) for j =, and k. Alternatvely f T s tangent to two boundares of X, then T s of the form (Π )k+ ( ) and the cells tangent to t are of the form (Π )k+ (Π ( )) for =,,. Theorem. Let G be the connected component of the tangency graph contanng. Then G s a tree where the verte correspondng to has degree and all the other vertces have degree 5. Proof. Obvously s a cell that has level ; by Lemma 8, the cells tangent to are π ( ) for and these have level. We can now attach to each of the cells π ( ) for, four cells of level. Ths s possble because each X, for =,,,, can be renormalsed to X va the smlarty π ; then the cells of level are of the form π π ( ) for,. We can contnue the above reasonng and the cells wthn G wll be of the form π k π k π ( ) (5) where l, for l k. Now suppose there s a cycle C n the connected component of the tangency graph G ; wthout loss of generalty, pck a cell n the cycle that has smallest area of all n the cycle and ths must have at least two neghbours that are equal or larger. Ths contradcts the four chldren result for ths cell and so there can be no cycles. Theorem. Let G,G,G,G be the connected components of the tangency graphs contanng,, and respectvely. Then they are all trees such that G and G where the vertces correspondng to and have degree ; whereas G and G have degree at the vertces to and. All other vertces n any of the above graphs have degree 5. Proof. We frst consder G. The pentagonal cell s tangent to perodc cells such that they are located on the mdpont of one of the edges of the trangles X,X and X as depcted n Fgure. These cells of level can be wrtten as π ( ),π ( ) and π ( ). By Defnton, t s easy to see that π ( ) = Π ( ) for =,,. Then by Theorem, all the cells wthn G are of the form π k π k π ( ) for l, and l k. Therefore all the cells wthn G are of the form ( ) Π π k π k π ( ) where =,,, l, and l k. Therefore G s a tree. Now consder G and G. Note that both G and G have the vertces correspondng to and wth degree whereas all the others have degree 5. Recall Π and Π from Defnton. Note that X X 5 X X 6 X s equal to the atom M. Then clearly Π ( ) =, for =,. Thus the cells wthn G and G are of the form ( ) Π π k π k π ( ),

13 for =,, l, and l k. Therefore, by Theorem, G and G are trees. Fnally consder G. Clearly G has the verte correspondng to wth degree and all the other vertces have degree 5. Furthermore the cells wthn G are of the form ( Π Π (π k ) π k π ( )) where =,,, l, and l k. Therefore, by Theorem, G s a tree, and the result follows. We can now conclude wth the proof of our man result Proof of Theorem. By Theorems and, we know that G j s a tree for j. Note that each of the cells j for j, are cells of level. Defne G () j to be the tangency graphs of the cells wthn π (G j ), for j and. Note that some of the cells are already contaned n G j for j. Contnue the above reasonng n order to get G (n) j. Then defne the followng G = n G (n) j and note ths s a (not necessarly dsjont) unon of trees by Theorems and. Pck any connected component G of G and consder the largest cell n G. Ths must be a pentagon of the form Π ( j ) for some j and a smlarty Π. However the tangency graph from ths pentagon wll be of the form Π (G j ) whch s a tree. Hence G s a dsjont unon of trees. By Theorem, the tangency graph s a unon of trees; hence t s bpartte (see e.g. []). A Pecewse Isometry wth Cycles n the Tangency Graph In the prevous secton we analysed the tangency graph assocated wth a partcular eample of a pecewse sometry, showng that t s a forest. It s natural to ask whether ths s a general property for any pecewse sometry. As we wll see n ths secton, the answer s negatve; n fact a countereample s the Pe. We note that n ths case, Theorem, no longer holds. Defnton 5. Let X be the trangle n Defnton generatng the self smlar structure depcted n Fgure. Defne the Pe as the PWI shown n Fgure 9 actng on the unon of sometrc copes of X. Note that snce the Pe s a unon of copes of T actng on copes X, the perodcally coded cells are just copes of those for T as depcted n Fgure. It s clear from Fgure, that although one can verfy that the tangency graph of the cells of the Pe s bpartte, t s by no means a forest and contans an nfnte number of cycles: Theorem 5. Consder the Pe PWI as defned n Fgure 9. The tangency graph G (Pe) contans an nfnte number of cycles of length. Proof. Note that each of the slces of the Pe s the sosceles trangle depcted n Fgure. In partcular the atoms n Fgure 9 are of the form M a and M b where a,b {,,...,9}; t s clear that the former correspond to the atom M and the latter to M as shown n Fgure. Therefore A s a pecewse sometry. In Fgure we clearly see the self smlar structure of the perodc cells of the Pe. Note that ths has a seres of tangences between each crcle nscrbed n the bggest pentagonal cell of each slce (.e. the pentagon P as shown n Fgure ). Ths s eplctly ndcated n Fgure by the shaded pentagons. The result follows.

14 M 9 M M M M 8 M 9 M M 8 M M M 7 M 7 M M 6 M 6 M 5 M 5 M M M A A(M ) A( M 9 ) A( M ) A( M 9 ) A( M 8 ) A( M 8 ) A( M 7 ) A( M ) A( M ) A( M ) A( M ) A( M M ) ) A( A( M 7 ) A( M ) 6 A( M 6 ) A( M 5 ) A( M ) A( M ) A( M 5 ) Fgure 9: The pecewse sometry actng on the Pe as defned n Defnton 5. Note the each slce s an sosceles trangle dentcal to the orgnal trangle X depcted n Fgure. The acton of A on the atoms M and M for =,...,9, generates a self-smlar structure as shown n Fgure.

15 Fgure : The self smlar structure of the perodc cells of the Pe defned n Defnton 5. Note that the shaded cells form a tangency cycle of length. Other Geometrc Propertes of the Tangency Graph PWIs have been etensvely nvestgated n terms of ther dynamcal propertes: they however generate nterestng geometrcal confguratons whch could be analysed per se. In partcular ther dynamcal propertes can be used to emphasse some propertes otherwse dffcult to pnpont by usng other technques. In fact n sectons. and, we have consdered some topologcal propertes of the tangency graphs: n order to prove Theorem, we have eploted the dynamcs assocated wth the tangency graph. In secton., we wll nvestgate some further fractal propertes of the tangency graph; n partcular we wll see that we can produce dfferent fractal confguratons such that ther Hausdorff dmensons are dfferent from the dmenson of the eceptonal set consdered by Goetz [9].. Geometrc Propertes of the Edges of The Tangency Graph In secton. we have nvestgated some combnatoral propertes of the tangency graph assocated to the cell structure generated by the pecewse sometry (): however ths graph s embedded n the plane and has a well defned geometrc structure. Furthermore t has the followng property (see [5] for more detals): Lemma 6. Let G be the tangency graph assocated to the pecewse sometry (). Then G s an algebrac graph,.e. the length of ts edges s an algebrac number. Proof. Snce the vector between the vertces of any tangent pentagons s n Q[α], then the length of a vector v,j between P and P j s d,j = v,j = (v,j v,j ) / Whch s clearly real. Then from Proposton. page n [], d,j s algebrac as Q[α] s a fnte etenson of Q; n other words t s the root of a polynomal n Q[]. In fact there s a polynomal D() Q[] 5

16 (a) (b) Fgure : (a) Constructon of the set of lmt ponts Λ of a component of the tangency group tangency graph assocated to (). (b) The Cantor set Λ obtaned by removng the sdes of all the perodcally coded cells that le on the base of the trangle X. such that ( d,j ) s a factor of D(). Then f we consder z = then (z d,j ) = (z d,j )(z + d,j ) s a factor of D(z ) Q[z]. Hence the result follows.. Some Fractal Propertes of the Adjacency Graph In [9] the Hausdorff dmenson of the eceptonal set s nvestgated as reproduced n Theorem ; however ths s not the only nterestng geometrc subset defned by the pecewse sometry (). We compute the Hausdorff dmenson of the lmt ponts of the tangency graph and we wll nvestgate the Cantor set generated by removng the edges of the pentagons that le on one sde of another pentagon. Consder the tangency graph assocated to the crcle packng generated by nscrbng crcles nsde the pentagonal cells generated by the pecewse sometry n Defnton, and pck a connected component of ths. The centres of the tangent pentagons have a lmt Λ. On the other hand, f we eamne one sde of a pentagon and eamne all parts that are not shared wth a sde of another pentagon, ths defnes another set Λ ; see Fgure. For these Cantor sets one can compute dm H (Λ ) = log log γ =.96, dm H(Λ ) = log log γ =.68. by notng that n the frst case the dmenson s satsfes γ s = where γ = + 5 whle n the second case t satsfes γ s =. 5 Conclusons PWIs have been deeply nvestgated manly usng technques taken from dynamcal systems. In ths paper we have analysed some combnatoral propertes assocated wth the tangency graph for a specfc eample of a PWI: our man goal s to consder a property that s not usually lnked wth dynamcs. In fact t suggests an nterestng relatonshp between geometrc dynamcal systems and other topcs such as packngs and tlngs, geometrc graph theory and algebra. In partcular the dynamcal renormalsaton structure assocated wth the PWI T s essental n analysng the combnatoral propertes of ts tangency graph. It would be natural to try to etend our approach to more general pecewse sometrc systems, however there are several problems. For general PWIs, the eact scalng propertes of the cells are not well understood. In [] a pecewse sometry actng on a sosceles trangle wth angles (π/7,π/7,π/7) s studed: t s by no means obvous whether t s possble to eplot the same method as n ths paper to nvestgate the propertes related to ts tangency graph. 6

17 In fact there s generally a lack of general results when dealng wth PWIs. However n [6], a more general renormalsaton scheme s ntroduced whch could help to analyse more eamples and gve more a useful nsght nto some general propertes of PWIs. In partcular, what s the general structure of the tangency graph assocated wth general PWIs? In ths paper we eamne two eamples: the frst has no cycles n ts tangency graph, whereas the second has an nfnte number of them. In [], Ashwn and Fu show that the tangency graphs assocated wth some PWIs are trval (.e. have an empty edge-set). Ths suggests that n general PWIs tend to have graphs wth dsconnected components. Is therefore the Pe one of the few eamples where graphs ehbt a more comple structure? How do perturbatons of a PWI nfluence ts tangency graph? Would we always have a graph that s very poorly connected? In ths paper we have defned the tangency graph wth respect to the tangences between dscs (or crcles) nscrbed nsde the perodc cells. Is t possble to broaden ths defnton n order to embrace other tangency propertes or to nfer other dynamcal or geometrc propertes of the eceptonal set from ths? Acknowledgements We thank Ngel Byott and Arek Goetz for ther comments. References [] Agarwal, P and Pach, J. Combnatoral Geometry. New York: Wley-Interscence, (995). [] Ashwn, P. and Fu, X.-C. Tangences n nvarant dsc packngs for certan planar pecewse sometres are rare. Dyn. Syst. 6, (), 5. [] Ashwn, P. and Fu, X.-C. On the geometry of orentaton preservng planar Pecewse Isometres. J. Nonlnear Sc., (), 7. [] Bollobás, B. Graph Theory. An Introductory Course. Sprnger-Verlag, (979). [5] Bressaud, X and Poggaspalla, G. A tentatve classfcaton of bjectve polygonal Pecewse Isometres. Epermental Mathematcs 6, (7) 77. [6] Buzz,J. Pecewse Isometres have zero topologcal entropy. Erg. Th. Dyn. Sys., (), [7] Falconer, K. Fractal Geometry. John Wley & Sons, (99). [8] Goetz, A. Dynamcs of Pecewse Isometres. Illnos J. Math., (), [9] Goetz, A. A self-smlar eample of a pecewse sometrc attractor. Dynamcal Systems (Lumny-Marselle), (998), [] Goetz, A. Return Maps n Cyclotomc Pecewse Smlartes. Dynamcal Systems., (5), [] Goetz, A and Poggaspalla, G. Rotatons by π/7. Nonlnearty 7, (), [] Lang, S. Algebra (Thrd Edton). Addson-Wesley Publshng Company., (99). 7

18 [] Katok, A. and Hasselblatt, B. Introducton to the Modern Theory of Dynamcal Systems. Encyclopeda of Mathematcs and Its Applcatons., Cambrdge Unversty Press, (995). [] Kahng, B. The unque ergodc measure of the symmetrc pecewse toral sometry of rotaton angle θ = kπ/5 s the Hausdorff measure of ts sngular set. Dynamcal Systems 9, (), 5 6. [5] Maehara, H. Dstance graphs and rgdty. Contemporary Mathematcs, (). [6] Poggaspalla, G. Self-Smlarty n Pecewse Isometrc Systems. Dynamcal Systems, () [7] Trovat, M. Ashwn, P. and Byott, N. Some Tangency Propertes of Geometrc Packngs Induced by Pecewse Isometres: the Arbelos. Preprnt, Unversty of Eeter, (7). 8