Geometric properties of a class of piecewise affine biological network models

Transkript

1 J. Math. Bol. (2005) Mathematcal Bology Dgtal Object Identfer (DOI): /s Etenne Farcot Geometrc propertes of a class of pecewse affne bologcal networ models Receved: 13 Aprl 2005 / Revsed verson: 9 September 2005/ Publshed onlne: 28 December 2005 c Sprnger-Verlag 2005 Abstract. The purpose of ths report s to nvestgate some dynamcal propertes common to several bologcal systems. A model s chosen, whch conssts of a system of pecewse affne dfferental equatons. Such a model has been prevously studed n the context of gene regulaton and neural networs, as well as bochemcal netcs. Unle most of these studes, nonunform decay rates and several thresholds per varable are assumed, thus consderng a more realstc model. Ths model s nvestgated wth the ad of a geometrc formalsm. We frst provde an analyss of a contnuous-space, dscrete-tme dynamcal system equvalent to the ntal one, by the way of a transton map. Ths s smlar to former studes. Especally, the analyss of perodc trajectores s carred out n the case of multple thresholds, thus extendng prevous results, whch all concerned the restrcted case of bnary systems. The pecewse affne structure of such models s then used to provde a partton of the phase space, n terms of explct cells. Allowed transtons between these cells defne a language on a fnte alphabet. Some words are proved to be forbdden n ths language, thus mprovng the nowledge on such systems n terms of symbolc dynamcs. More precsely, we show that tang these forbdden words nto account leads to a dynamcal system wth strctly lower topologcal entropy. Ths holds for a class of systems, characterzed by the presence of a splttng box, wth addtonal condtons. We conclude after an llustratve three-dmensonal example. 1. Introducton Many bologcal systems may be descrbed as assembles of smlar consttuents evolvng n parallel, and nteractng n a structured way. The structure of nteractons s currently modeled by an orented graph, whose vertces represent elements n the system. Each edge represents a drect acton of ts ntal vertex on ts termnal vertex. Although very complex, and studed n ts own rght [1], ths structure s statc, and does not suffce to understand the behavour of the whole system. Moreover, t may evolve tself as the elements n the system are changng wth tme. Thus, dynamcal models are necessary n order to capture sgnfcant aspects of complex bologcal phenomena. Chosng to restrct our attenton on determnstc models, two man types of formulaton can be dstngushed : models wth E. Farcot: COMORE INRIA, U.R. Sopha Antpols, 2004 route des Lucoles, BP93, Sopha Antpols, France e-mal: etenne.farcot@sopha.nra.fr Key words or phrases: Gene and neural networs Pecewse-affne dynamcal systems Symbolc dynamcs

2 E. Farcot dscrete state space [28, 40, 8], and models wth contnuous state space, formulated as ordnary dfferental equatons [5, 36, 37]. Snce the latter lead to very complex nonlnear dynamcs n hgh-dmensonal spaces, and the frst one only provdes large scale qualtatve nsghts about the phenomenology of the systems, ntermedary formulaton are often consdered. Namely, systems of pecewse affne dfferental equatons are more tractable than nonlnear smooth ones, due to ther underlyng dscrete structure, whle they yeld fner nformaton than purely dscrete representatons. Furthermore, they seem well suted to expermental data, whch s often quanttatve wth non neglgble uncertanty,.e. data s partally qualtatve. The lterature about the pecewse lnear approach of complex nonlnear phenomena s huge, and we shall only menton typcal wors n the feld of bology, thus gnorng many aspects such as control theoretc ssues, or the many examples occurng n the context of automatcs, electrc and electronc crcuts, or embedded software. The man reason for excludng all these very actvely studed problems, s that they generally deal wth complex couplngs, whch would not be sutably descrbed by the class of models consdered n ths paper. Actually, the specfcty of ths class les n the fact that the proper lnear terms are uncoupled, and that nteractons are only present n the pecewse constant terms of the equatons. Ths may sound qute restrctve n regard to the much more general class of pecewse lnear dfferental equatons. Anyway, a lot of phenomena n bology are characterzed by strongly localzed couplng, that s by nteractons of an almost on-off nature. Ths ncludes swtchng networs le gene transcrptonal regulaton networs [6, 7, 18, 19, 26], neuron networs [17, 16, 30], as well as metabolc and chemcal pathways [20], whch all are currently studed examples n mathematcal models of bologcal dynamcs. Neural, metabolc and gene networs models can generally be put n the form of a system of pecewse affne dfferental equatons, wth a dagonal matrx as proper lnear term. Ths class of model has been nvestgated n tself, wthout specally focusng on one of the dfferent phenomena we just mentoned [10, 23, 38, 17]. Such a class could be called contnuous-tme swtchng networs, but ths does not llustrate the pecewse lnearty of the equatons, and would enclose a broader range of models. The term Glass networs has been proposed n [10], whch seems approprate, snce Leon Glass s the frst author to have explctly proposed ths model, emphaszng ts usefulness as tractable ndcator of the qualtatve propertes of nonlnear bologcal systems. Although the equatons studed here are more general than what s usually called a Glass system, ths term wll sometmes be used n the sequel. We wll use the term bnary systems when referrng to systems wth only one threshold per varable (.e. two dscrete states, whence the appellaton). The man contrbuton of ths paper les n the fact that worng hypotheses are lghtened as compared to prevous studes, such as [10,16,17,19 22,26,30, 32]. Namely, several thresholds are allowed for each varable n the system all through ths wor, and degradaton rates are not supposed unform for most of our results. Thus we deal wth a more realstc model than what s usually done, whch leads to mathematcal complcatons. To crcumvent these dffcultes, we

3 Geometrc propertes of a class adopt a geometrc pont of vew on the dynamcs. Ths approach proves useful wth respect to the analyss of perodc orbts, as well as the symbolc dynamcs approach. On the other hand, a notceable restrcton of the present wor s that t concerns networs wthout autoregulaton. Although severe n terms of bologcal plausblty, ths assumpton s made n all studes cted above. Actually, solutons are not well defned n systems wth autoregulaton. Two technques can be found n the lterature to face ths dffculty. The frst one conssts n studyng a smooth dynamcal system wth sgmods, whch tends to the pecewse affne system n the lmt of nfnte steepness. The analyss can be then carred out usng sngular perturbaton technques [36, 37]. The second way to handle ths stuaton, developed n [6,23], s to use the Flppov noton of soluton for dfferental equatons wth dscontnuous rght-hand sde. Such solutons are defned by the way of dfferental nclusons,.e. they are set-valued. As both technques are stll a current research topc, t seems reasonable to exclude the stuatons where they are requred. Furthermore, even wth ths smplfyng assumpton, there s stll a number of open problems concernng the phase portrats of these hgh-dmensonal dynamcal systems. In short, the dffcultes concern the number and localzaton of attractors, as well as ther formal descrpton, especally n the case of chaotc dynamcs [2,9,11,19]. Such problems are not only of mathematcal nterest, and ther nterpretaton mproves the understandng of complex bologcal networs. For example, t s commonly assumed, snce early suggestons of Waddngton and Delbrüc, that the attractors of a dynamcal system le those we study here are the mathematcal verson of modes of behavour of lvng cells. In partcular, cell dfferencaton s nterpreted as the coexstence of multple attractors wth large basns of attracton (see the chapter XII of [39] for a dscusson about ths dea and ts orgns). Then, dfferent cell types correspond to cells havng reached dfferent basns durng the development of an organsm. In the present paper, we manly focus on the formal descrpton of attractors, n a qute general and abstract settng. The hgh number of dmenson of systems le gene regulaton networs requres theoretcal tools and results, and must be essentally qualtatve. In ths drecton, the comparson of pecewse affne and dscrete systems seems well adapted, and s a major motvaton of the wor presented n ths paper. In secton 2, we present the model that motvated ths study, and show how t can be reduced to a dscrete-tme dynamcal system. The latter s rgorously defned n secton 3. Then, secton 4 s manly devoted to the study of perodc orbts, about whch prevously nown propertes are generalzed to the multple thresholds context. Fnally, secton 5 concerns symbolc dynamcs of the studed system. The latter s defned formally, and topologcal entropes of dfferent codngs are compared. As a man result, t s shown that the topologcal entropy of purely dscrete models s strctly greater than that nduced by a codng of pecewse affne dynamcs. Ths nequalty holds regardless of precse parameter values, and for a large class of systems characterzed by the presence of a splttng box. Ths result s llustrated on a three-dmensonal example n secton 5.3.

4 E. Farcot 2. Model descrpton 2.1. Equatons The general model studed here s a system of dfferental equatons of the form : dx = Ɣ(x) x, (1) dt where x R n, Ɣ : R n R n s pecewse constant, and R n n + s dagonal. We note Ɣ = (γ 1...γ n ), where γ : R n R, and = λ. Each coordnate x of vector x represents a characterstc quantty of the th member n a fnte populaton of n nteractng elements. For example, x s the concentraton of a proten whose producton s nduced from gene, orx s the voltage of a sngle neuron. Each quantty n ths nd of model s bounded, so that the doman n phase space where dynamcs must be confned wll be the cube U = [0, 1] n. Non-dagonal elements of beng zero, ths matrx only descrbes lnear degradaton of every component n the networ. Thus, s not related to couplng or autoregulaton, but rather to the fact that bologcal systems are usually dsspatve,.e. state space s globally contracted by the flow. Then, each dagonal element of s supposed postve. On the other hand, Ɣ descrbes couplng n the system. As t s pecewse constant, the doman of nterest U has to be parttoned. Ths wll be done wth n- rectangles,.e. products of n bounded ntervals. Such regons wll be called boxes, as usually done n the lterature. In each box, Ɣ taes a constant value,.e. the producton rate (resp. actvty) of all genes (resp. neurons) n the networ s constant. Hence a box s a regon n phase space where the dynamcs s approprately descrbed by a smple lnear system. Ths choce of a rectangular partton may seem arbtrary, or amed at smplfyng the analyss and computatons. In fact t s bologcally relevant, snce gene actvaton rates are nown to evolve n a swtch-le manner, that s they undergo sudden changes when some regulatng proten reaches a threshold value. Neuronal response to synaptc entres are also of swtchng nature. In ths latter case, the sudden changes usually appear when a certan lnear combnaton of the x s reaches a threshold. However, a smple change of varable leads to formulaton (1), where each threshold s related to a sngle varable, as explaned n [29]. These thresholds yeld the boundares of boxes ; we note and number them ={θ j j N p } n each drecton, wth the notaton N p ={1 p }. For the sae of bologcal consstency, when (1) models a gene networ, the θ j s are often gven n an unordered way, thus focusng on the nteractons between genes : θ j s the threshold at whch gene actvates (resp. nhbts) gene j by ncreasng (resp. decreasng) γ j s value when crossng ths threshold from left to rght [38, 40]. However, there s always a permutaton σ : N p N p, such that the θ σ (j) are n ncreasng order. Snce we care more about phase space geometrc structure

5 Geometrc propertes of a class than about dependence between genes (.e. the nteracton graph), we wll assume from now on, that the sets are ordered : θ 1 <θ 2 < <θ p. Then, boxes are explctly wrtten as : B a = B a1... a n = n [θ,a,θ,a +1], (2) =1 where the subscrpt a, belongs to the fnte set A = n N p (3) =1 Snce ths subscrpt a unquely determnes a box, and snce Ɣ s constant n each box, t wll be convenent to consder ths mappng as Ɣ : A R n. The set A wll sometmes be called an alphabet n the sequel. It wll be convenent to note elements of A as strngs of the form a = a 1...a n, nstead of vectors. One advantage of usng symbols n the dscrete set A s that t underscores the qualtatve nature of the model. Each symbol a can be seen as a dscrete state n whch all characterstc quanttes of the system are constant. As such, t leads to an automaton model that wll be explaned more deeply n secton 5. Moreover, as a fnte subset of the lattce N n, t nherts nce geometrc features of ths lattce. Especally, symbols n A correspond to boxes n U, whle straght lnes jonng these symbols correspond to facets of boxes. Hence, the geometrc structure of A s somehow dual to that of the partton of U. Ths s partcularly strng n R 2,as llustrated on fgure 1. In arbtrary dmenson, ths also has practcal consequences that wll be exploted n secton 5. x θ θ θ 21 θ 11 θ 12 θ 13 x 1 Fg. 1. An example of partton nto boxes, when U = [0, 1] 2. There are three thresholds n both drectons, hence A = N 3 N 3. After a rescalng, ths set can be superposed on the box partton, so that neghbourng boxes correspond to symbols dfferng by ±1 n a sngle drecton

6 E. Farcot 2.2. Flow A system of form (1) nduces a flow that can be explctly wrtten. In a gven box B a, Ɣ s a constant vector, thus the flow nsde ths box s : ( ) ϕ a (t, x) = x(t) = f + e t γ (x f) where f =. (4) λ =1...n Here, the vector f s called a focal pont, because t s obvously attractng n the above equaton. Hence, dependng on ts poston wth respect to B a, t wll be an asymptotcally stable steady state, or the trajectory wll encounter the boundary of the box. In the latter case, Ɣ s assgned a new value accordng to certan rules that wll be made precse, and one constructs a new pece of the trajectory by contnuty. Some results about polytopes wll be useful afterwards. In the rest of the paper we manly lean on [24,41] for such nd of propertes. The boundary of B a s formed by -faces, whch are -dmensonal rectangles, for {0...n 1}. When the ntersecton happens at a n 1 face, or facet (or wall), there s at most one adjacent box and the new value of Ɣ s unambguous. In the case of lower dmensonal faces, there are several adjacent boxes n general. Let be the dmenson of the face encountered by ϕ a (t, x),.e. t s gven by n hyperplane equatons of the form x = θ j. There are 2 n adjacent boxes sharng ths face, ncludng B a, correspondng to the above/below poston wth respect to each of the defnng hyperplanes (on the boundary of the doman U, there are of course less adjacent boxes). In each of these neghbourng boxes, the flow has a fxed value, for whch the -dmensonal face can be ether attractng or repellng. The resultng flow s not straghtforwardly defned n such regons, as wll be made precse n secton 3.3. The general case requres the Flppov noton of soluton of a dfferental equaton wth dscontnuous rght-hand sde [6, 23]. Untl further clarfcatons (sec. 3.3), we can exclude all -dmensonal faces, for <n 1, le s done for example n [17, 16]. Wth ths procedure, one constructs contnuous trajectores as far as they do not cross any face of dmenson <n 1. There are stll some degenerate cases for whch ths constructon s not well defned. We wll mae two assumptons that allow us to neglect these cases. The frst one s : H1. a A, f (a) a A nt(b a). Here f s consdered as the map 1 Ɣ : A R n, and nt denotes the nteror. Ths assumpton means that the focal ponts all le nsde the doman U, and that none of them s on the boundary of a box. The frst aspect mples that U s postvely nvarant, and thus can be consdered as the only regon where relevant dynamcs tae place. The second one excludes (rare) cases whch would cause techncal complcatons wthout mprovng the model. The second hypothess concerns the case of autoregulaton. As mentoned n the ntroducton, ths assumpton s certanly too strong n regard to bologcal phenomena, but t may lead to the use of generalzed solutons n the sense of Flppov,.e. dfferental nclusons, to have a mathematcally rgorous defnton of the flow [6,23]. We neglect ths nd of dffculty, by restrctng the allowed

7 Geometrc propertes of a class dspostons of boxes and ther focal ponts. Observe that two boxes B a and B a are adjacent n a sngle drecton (.e. through a facet) f and only f a a 1 = 1, or equvalently f and only f there s some N n such that a a =±e (e beng the th vector of the canoncal bass of R n ). Usng a dscretzng operator d = (d 1...d n ) : a nt(b a) A, whch maps a pont lyng nsde a box to the subscrpt of ths box, we can wrte H2. N n, a, a A, a a =±e, )( ) (d (f (a)) a d (f (a )) a > 0, or (d (f (a)) a ) ( ) = 0 and d (f (a )) a (a a )>0, or the same wth a and a exchanged. In other words, H2 means that the th component of the vector feld does not change n sgn when crossng a wall n drecton. Autoregulaton (.e. ẋ depends on x ) s a necessary, but not a suffcent condton for ths confguraton to happen. We thus do not reject all forms of autoregulaton here. The stuatons avoded are schematcally represented n fgure Transton map Once the flow (4) s gven n a box B a, t s easy to compute the tme and poston at whch t ntersects the boundary of B a, f ever. The possblty for each facet to be encountered by the flow depends unquely on the poston of the focal pont : {x = θ,a } (resp. {x = θ,a +1}) can be crossed f and only f f <θ,a (resp. f >θ,a +1). Accordng to ths observaton, we note I + out (a) ={ N n f >θ,a +1}, and I out (a) ={ N n f <θ,a }, and I out (a) = I + out(a) I out(a). When t s unambguous, we wll omt the dependence on a, as we have already done mplctly wth the focal pont. Fg. 2. The two possble ambgutes due to autoregulaton, often called blac wall (on the left) and whte wall (on the rght)

8 E. Farcot Snce these functons wll be useful n the followng, we note : α (x) = f θ,a, and α + f x (x) = f θ,a +1. f x Now, n each drecton I out the tme at whch ϕ a (t, x) encounters the correspondng hyperplane, for x B a, s gven by : τ (x) = 1 ln ( α λ (x)) f Iout, and τ (x) = 1 ln ( α + λ (x)) f I out +. Ths dstncton between drectons n I out + and Iout shall occur qute often, leadng to unnecessarly cumbersome dscussons. Here ths dstncton may be avoded by observng that whenever I out, τ (x) = 1 ln(α (x)) where α (x) = mn(α λ (x), α+ (x)). (5) Ths can be checed by nspectng the range of α ± for all possble (see table (6)), and usng the fact that τ(x) must be nonnegatve. α (x) α+ (x) I out [α (θ,a +1), 1] ]0, 1] [1,α + (θ,a )] I + out [1,α (θ,a +1)] [α + (θ,a ), 1] ]0, 1] (6) I out and x <f [1, + [ ],α + (θ,a )] R I out and x >f ],α (θ,a +1)] R [1, + [ The drectons that are not n I out are of no use here, but wll be consdered n secton 3.1. Now notce that α s not defned f x = f, whch may only happen for I out. Tang the mnmum τ(x) = mn τ (x). (7) I out and renjectng t n equaton (4), we get the extng pont of B a for the ntal condton x. Snce ths process s ntended to be repeated along trajectores, x wll generally le on the boundary of the current box, except for the ntal condton, whch may however be chosen wthout loss of generalty on a wall. Then we get a transton map M a : B a B a, whch can be made explct, omttng a : Mx = ϕ (τ(x),x) = f + A(x)(x f). (8) Where A(x) s the dagonal matrx whose entres are e λ τ(x).ifq s an escapng drecton,.e. τ = τ q, we can also wrte the entres of A(x) as (α q (x)) λq, for = 1...n. λ

9 Geometrc propertes of a class We see here that M s nonlnear, but n the specal case where H3., j N n,λ = λ j,.e. λ R +, = λid. s fulflled, A(x) s proportonal to the dentty matrx as well, and M can be understood as a projectve transformaton (see e.g. [41] p.67). Actually, H3 mples that the flow n each box conssts of straght lnes drected towards f, and thus Mx s one of the ntersectons of the affne lne f + R(x f) wth B a. The set of all trajectores n B a s then the ntersecton of ths box wth the straght lnes composng the polyhedral cone f +{t(x f) t 0, x B a }. These nce aspects explan why ths last assumpton s currently done. Formally, (8) may be rewrtten qute smply under the latter assumpton : Mx = f + α(x)(x f), (9) where α(x) = max {α (x)}. (10) I out As α (x) only depends on coordnate x, we sometmes abusvely note α (x ). Unless ndcated, we suppose n the followng that assumptons H1 and H2 are satsfed. The unform decay rates assumpton H3 wll not be systematcally requred, unle most of the prevous wors concernng equatons of the form (1). In partcular, the term Glass networs usually refers to a system of the form (1), wth only one threshold θ dstnct from the doman s boundares, n each drecton, and the two hypotheses H1, H3 (see e.g. [10,27]). More or less explct versons of H2 are also very largely assumed n papers dealng wth such models. 3. Propertes of the transton map 3.1. Local propertes In each box, I out determnes all reachable boxes. Those are adjacent to the current box through walls supported by hyperplanes of the form {x = θ j }, for I out, and j {a,a +1} dependng on s belongng to Iout or I out. + We ntroduce the followng notaton for such walls : W + (a) ={x x = θ,a +1} B a and W (a) ={x x = θ,a } B a. (11) Then, each box can be parttoned n #I out regons from whch a sngle adjacent box s reachable. # denotes the cardnal. See fgures 3 and 4 for an llustraton. The subscrpts a wll be omtted n ths secton, where a sngle box B s consdered. Due to prevous dscussons, the only walls through whch trajectores may escape B may be put n the form W + (resp. W ), for I out + (resp. Iout). Conversely, any pont on a wall of the form W ±, I out, ± escapes mmedately, as can be seen from equaton (7), where escapng tme s clearly zero on the correspondng escapng wall. More precsely, at any ntal condton x, from equatons (4), (7), the drectons such that τ(x) = τ (x) are exactly those for whch Mx W ±. Accordng to ths, we can coarsely partton B nto two regons :

10 E. Farcot W + 1 f W Fg. 3. A box n R 3, wth two escapng walls : W + 1 and W + 2. Thus, I out = {1, 2}, and ± 1 =± 2 =+ Fg. 4. Partton of B n (on the left, seen from nsde B) and B out (on the rght, seen from outsde B). Dotted lnes on the unfolded representaton of B n relate ponts that are dentcal n R 3. The scale s arbtrary the outgong regon B out = I ± out the ncomng regon B n = B \ B out = W ± ={x B τ(x) = 0} I ± out W (W W + ). I out Where, for any set S, S denotes ts closure, and for ± {, +} the symbol denotes the opposte sgn. Thus, the ncomng and outgong regons are unons of walls, whch are closed and cover the boundary B.We can observe that B out B n whenever B out, and s the unon of some n 2-facets of B. Actually, B n always holds, due to I out ± W B n, and I out W W + B n, as follows from H1. Ths frst partton only allows a dstncton between escapng drectons and the others, and one gets : B out = f nt(b),

11 Geometrc propertes of a class whch we recall corresponds to f beng an asymptotcally stable equlbrum pont wth B contaned n ts attractng basn. Moreover, ths partton of B can lead to the transton map beng bjectve. Proposton 1. Assume H1 s satsfed for a system of form (1). Let M be the transton map n a box B, as defned n the prevous secton. Restrctng the doman and range of M to B n and B out respectvely, and abusvely eepng ts name unchanged, the followng s a homeomorphsm, provded B out : M : B n B out Proof. From ts defnton, M : x ϕ(τ(x),x), where ϕ s the flow assocated to an affne dynamcal system, and s contnuous f and only f τ s. Ths functon s defned as τ(x) = mn Iout τ (x). From equaton (5) each τ mght only be dscontnuous f α + (x) or α (x) s. From table (6), both are defned and contnuous on [θ,a,θ,a +1], for I out (furthermore, whch among α (x) and α+ (x) s the lowest does not depend on x for such ). Then, τ s contnuous as the mnmum of a fnte set of contnuous functons. Injectvty comes from the fact that x and Mx are always on the same orbt of ϕ, and from monotoncty (w.r.t. tme) of all coordnates ϕ of ths flow, along any orbt. Surjectvty wll come from the constructon of the nverse mappng, whch s defned on B out. Note ϕ(t,x) = ϕ( t,x) = f + e t (x f) the flow n reverse tme. From postvty of s entres, we get that all trajectores gven by ϕ dverge to + n each drecton, and hence leave B n fnte tme. For x <f (resp. x >f ), ϕ(t,x) may only encounter W (resp. W + ) n drecton. Ths can happen at tme τ (x) = λ 1 ln(α (x)) (resp. α+ (x)). Inspectng table (6) (usng the fact that I out + x <f ), we get a general expresson : τ (x) = 1 ln(max(α λ (x), α+ (x))) and τ(x) = mn ( τ (x)) N n for the extng tme from x, assocated wth ϕ. There s a problem when any x = f, where none of α ± (x) s defned. Actually, for x f (from left or rght), the quantty max(α (x), α+ (x)) always tends toward +. But for all j I out the correspondng τ j are always bounded (see table (6)), and t s assumed here that I out. Thus, when x f the mnmum τ cannot be gven by τ. Hence, M 1 x = ϕ( τ(x),x) = ϕ( τ(x),x) s a contnuous functon, whch s obvously defned on B out. Observe that the ncomng regon can now be defned as B n ={x B τ(x) = 0}. In the specal case where H3 s also fulflled, the nverse mappng can be expressed as : M 1 y = f + β(y)(y f). (12)

12 E. Farcot where N n,β (y) = max(α (y), α+ (y)), and β(y) = mn N n {β (y)}. (13) Ths results from the defnton, M 1 x = ϕ( τ(x),x), wth proportonal to the dentty Partton of boxes In ths secton, we shall analyze n some detal the possble confguratons that may happen at a sngle box, n terms of ncomng facets, outgong facets, and peces of trajectores between them. Thus, we consder a sngle box B, noted wthout subscrpt for sae of readablty. Moreover, we assume that ths box B has at least one escapng drecton,.e. I out, snce otherwse the analyss s of lttle nterest. Consderng a sngle box could seem much too local, as compared wth full orbts, crossng a possbly hgh number of successve boxes. But at ths local scale, we provde a geometrc descrpton of all possble stuatons, showng by the way that they form a combnatorally non trval set. Ths descrpton reles on a fner partton of B than that of the prevous secton. The proposed partton arses by dstngushng not only whch ponts escape n each drecton of I out but, on any ncomng wall, whch ponts are mapped onto a gven escapng wall. Conversely, those pont on an escapng wall that are mapped bacwards on a gven ncomng wall wll be dstngushed. Escapng walls are of the form W j ±, for j I out, where ± has a fxed value for each j. Thus, we note ± j the unque sgn such that j I ± j out. The somehow unusual symbol j s then defned as the opposte of ± j, and wll be useful on more than one occason. One may observe that ± j = sgn(d j (f (a)) a j ). Snce we deal wth a sngle box, a non ambguous and convenent notaton for thresholds n ths secton wll be : θ. = θ,a and θ +. = θ,a +1. It follows that, for I out, θ ± s the sngle threshold that may be reached n drecton. The sets we have descrbed n words can be formalzed : D j ± = W ± D j = W M 1 (W ± j j ) for N n \ I out,j I out. M 1 (W ± j j ) for, j I out. (14) and R j ± = M(W ± ) W ± j j for N n \ I out,j I out. R j = M(W ) W ± j j for, j I out. (15) To help memory, note that the letter D stands for doman, whle R stands for range (of M). When I out, both walls W + and W are subsets of B n. Accordngly, the superscrpt ± above means that two sets are defned. For I out, there s no possble ambguty n the superscrpts of walls : W ± B out and W B n.

13 Geometrc propertes of a class W + 1 f W W + 3 Fg. 5. A box n R 3, wth three escapng walls : W + 1, W + 2 vertces of D 22 and ther mages,.e. the vertces of R 22 and W + 3. Dotted lnes relate the Fg. 6. Partton of B n (on the left, seen from nsde the box) and B out (on the rght, seen from outsde the box). The scale s arbtrary, but the shape and poston of each set s well represented Examples of such sets are depcted n fgures 3 and 4, as well as fgures 5 and 6, on 3-dmensonal examples, wth the unform decay rate assumpton H3 (so that these regons are polytopes). The fact that these sets form a partton of B s due to M beng a bjecton. Actually, from ths we get that each pont on a wall W ± B n must be mapped on a wall n B out, of the form W ± j j, hence the sets Dj ε, wth ε {, +, } partton B n. Conversely each pont on B out has an antecedent on a wall of the form W ± B n, so that B out s parttoned by the sets Rj ε. We now consder some propertes of these sets. Frst, t appears that there s a smple relaton between sets n (14) and n (15).

14 E. Farcot Proposton 2. R ± j = M(D± j ) for N n \ I out,j I out. R j = M(D j ) for, j I out. Moreover, when H3 s true, the polytopal complexes formed respectvely wth all Dj ε and all Rε j are combnatorally somorphc. In partcular, each par of Dε j and Rj ε are combnatorally somorphc polytopes. Proof. Both equaltes follow from the fact that M : B n B out s a bjecton. Actually, the njectvty of M mples that M(A B) = M(A) M(B), for all subsets A and B n the doman of M. Surjectvty, on the other hand, mples M(M 1 (A)) = A for any subset A n the range of M. The concluson s a drect consequence of defntons (14) and (15). Under H3, we have already seen that all trajectores n a box are straght lnes, and that M s a projectve transformaton. Snce walls are polytopes ((n 1)-rectangles), and projectve transformatons transform polytopes nto polytopes [24, 41], the sets defned n (14) and (15) are polytopes. The combnatoral somorphsm means that the collecton of Dj ε and that of Rj ε have the same face structure, n terms of ncdence between faces of all dmensons. Ths can be seen by observng that M s the projectve mappng used n the constructon of the so called Schlegel dagram, whch preserves combnatoral structure (see [41], p ). More precsely, for a fxed j, the complex formed of the Rj ε and ther subfaces s precsely the Schlegel dagram of the polytope ) based on the face W ± j j. The polytopes B M 1 (W ± j j, and all Dj ε B M 1 (W ± j j partton B, and ther facets are exactly W ± j j ), j I out,, from whch the somorphsm s constructon follows. A more thorough dscusson of these constructons can be found n [14]. The sets that need a superscrpt + or are also smply related : Proposton 3. D j + = D j +(θ + θ ) e, where e s the th vector of the canoncal bass. Proof. Let x W, and defne x. = x + (θ + θ ). We show now that x belongs to Dj f and only f x belongs to D j +. x Dj f and only f the orbt startng at x escapes n drecton j,.e. τ(x) = τ j (x). Snce x and x have dentcal coordnates, except x = θ and x = θ +, wth I out (see (14)), the ext tme τ(x) s ndependent of x. Hence τ(x) = τ(x ) = τ j (x ), or equvalently Mx W ± j j,.e. x D j +. Now a result restrcts the possble transtons between ncomng and outgong walls. The term relnt denotes the relatve nteror : for any set S relnt(s) s the nteror taen n the smallest affne subspace contanng S. In the rest of the text, nteror wll mean relatve nteror. Lemma 1. There s at most one I out, such that relnt(d ) (and thus relnt(r )) s nonempty.

15 Geometrc propertes of a class Proof. Let N n such that relnt(d ). Equvalently, there exsts an x n W, such that Mx W ±. The frst condton mples x = θ, whle the second mples that τ(x ) = τ (x ) = τ (θ ) s a mnmum. x beng n the relatve nteror of D, ths mnmum s strct : I out,, τ (x )<τ (x ). (16) Now, from table (6) and monotoncty of α ± functons, t follows that τ (θ ) s the maxmum value of all τ (x ), for x [θ,θ + ]. Actually, α (x ) has a mnmum at x = θ, and τ decreases wth respect to α. Thus one gets : I out, τ (x ) = τ (θ ) = max τ (x ) x [θ,θ + ] From the latter and (16) we derve a necessary condton for relnt(d ) : I out,, max τ (x )< max τ (x ), x [θ,θ + ] x [θ,θ + ] whch obvously cannot be satsfed by more than one I out. Remar 1. From lemma 1, the number of drectons such that D has nonempty nteror s 0 or 1. The case where t s 0 s rare, n the sense that there s a measure zero set of parameters leadng to t. Ths set s gven by the equalty of all maxmal values of the τ functons (whch occur at threshold values of x ), for I out and #I out 2. When #I out {0, 1}, ths number s equal to the number of drectons wth nonempty relnt(d ). The prevous lemma ndcates that not all transtons are admssble for a fxed set of parameters. The next result shows that there are no other restrctons of ths nd. Lemma 2. For N n,j I out, j, and ε {, +, }, all D ε j, (resp. Rε j ), have nonempty relatve nteror. Proof. For I out, from proposton 3, Dj and D+ j are obtaned from each other by a translaton, so that t s suffcent to consder only one of them. Recall that s only defned for I out. A practcal conventon, for I out, wll be to defne as beng any of the two sgns, +. Then beng well-defned for N n, the result wll follow f we show that W M 1 (W ± j j ) (17) s of nonempty nteror, for N n and j I out. Agan, and whatever, θ s a generc notaton that wll be useful to avod enumeratng several analogous cases. The nteror of a set le (17) s defned by the equalty x = θ, and the nequaltes τ j (x j ) < τ (x ) for I out \{j}. From table (6) t appears that the range of the α functons always taes the form [α ± (θ ), 1], when I ± out. The upper value 1 s gven by α ± (θ ± ) = 1.

16 E. Farcot Thresholds beng dstnct n each drecton, the range of any α s of nonempty nteror. Then, each τ has also a range of nonempty nteror, gven by [0, 1 λ ln(α ± (θ ))]. If τ 1 = mn I out ln(α ± λ (θ )), one gets a postve length nterval [0,τ ], whch s contaned n the range of all τ, for I out. Snce #I out s fnte, we can choose ths number of ponts n the latter nterval : ϑ 1 > >ϑ #Iout. Then, we construct a pont x as follows: x = θ x j = τj 1 ( ) ϑ#iout, whch s possble by defnton of the ϑ s, x = τ 1 ( ) ϑp for Iout \{j}, where p s arbtrary n {1...#I out 1}, x ]θ,θ+ [ for N n \ I out., It s clear then, that ths pont belongs to the relatve nteror of the set (17). Remar 2. The fact that j, whch maes the dfference wth lemma 1, s mportant here. It s mplctly useful for t allows one to choose x j n the whole doman [θ j,θ+ j ], whle x s of fxed value θ. To conclude ths secton, we shall n fact provde an explct descrpton of the nonempty regons Dj ε and Rε j. Ths descrpton can be useful from an algorthmc pont of vew, snce t s a mnmal set of nequaltes, hence optmal n terms of memory resources. Proposton 4. For all but a zero-measure set of parameters (.e. thresholds and focal pont coordnates) the sets Dj ε defned n eq. (14), are bounded cells wth pecewse smooth boundary, whose (relatve) nteror s rredundantly defned by the followng nequaltes : x = θ ε, θ <x <θ +, (N n \ I out ) \{} α (θ λ )<α (x )<α j (x j ) j, I out \{, j} α j (x j )<1. Wth the nequaltes subject to addtonal condtons : If D jj, α j (θ j j )<α j (x j ) has to be added to the system above. If I out and D, α (θ λ ) 1 λ <α j (x j ) 1 j, has to be added. The exponent ε stands for f I out, and ε {+, } otherwse. Proof. The equalty satsfed by x ensures that Dj ε W ε. In all cases, the varables x, for I out, do not nfluence the ext tme. Hence, they are only submtted to the nequaltes θ <x <θ +. Recprocally, these λ

17 Geometrc propertes of a class nequaltes must be satsfed to ensure x B. Of course, when I out, x does not appear n these nequaltes. Varables x, for I out, must on the other hand satsfy two nds of constrants. Frstly, they have to be between thresholds θ and θ +. Snce the functons α are contnuous and monotone wth doman [θ,θ+ ] and range [α (θ ), 1], for all I out, these threshold nequaltes can equvalently be wrtten I out \{}, α (θ )<α (x )<α (θ ± ) = 1. (18) Secondly, for j, the correspondng ext tme τ (x ) must be hgher than τ j (x j ), snce Dj ε s part of M 1 (W ± j j ). Recall that ext tmes are defned n equaton (5) as τ (x ) = λ 1 ln(α (x )). Thus, τ (x )>τ j (x j ) leads to λ I out \{j}, λ α (x )<α j (x j ) j. (19) From α j (x j )<1, and the above, we deduce α (x )<1. Ths latter s hence removable from (18), for all x, I out \{j}, but must be mantaned for x j. For I out, and, j, the nequalty relatng x and the threshold θ admts on the other hand no concurrent nequalty from those of the form (19). Thus α (θ )<α (x ) has to be mantaned for all I out \{}. Now the last nequalty we need to dscuss s the one that gves a lower bound for α j (x j ). To acheve ths, one has to recall from the proof of lemma 1 that D admts I out \{}, α (θ λ ) 1 >α (θ λ ) 1, (20) as a set of necessary condtons. Ths lemma ensures moreover that, for almost all parameter values, there s exactly one such that D. Thus n the followng, we assume ths fact. We now consder two dstnct cases. If I out : the th varable does not appear n nequaltes (19). If D jj =, there s (for almost all parameter values) a I out \{j} wth D. It follows that α j (θ j j ) 1 λ j together mply α j (θ j j f D jj, n whch case t s unremovable. If I out : from (19), < α (θ λ ) 1, and then (18) and (19) )<α j (x j ). Ths latter nequalty has thus to be used only holds. Ths competes wth α (θ λ j ) λ <α j (x j ), (21) α j (θ j j )<α j (x j ), (22) appearng from (18). Now, f D for some, j, both lower bounds λ j of α j (x j ) are smaller than α (θ λ ), due to (20). From (19) ths quantty s a λ lower bound for α j (x j ) 1 j, and thus (21) and (22) are both redundant.

18 E. Farcot On the other hand, f ether D or D jj (or both when = j), one has an rredundant lower bound from (21) or (22), respectvely. Observe that n the specal case where H3 holds, all nequaltes n proposton 4 are affne, and thus the sets they descrbe are polytopes. We already new ths fact, but now the polytopes are explctly descrbed n terms of ntersectons of half-spaces. From a drect count of the nequaltes n proposton 4, one can nfer the corollary : Proposton 5. Any set Dj ε possesses 2 (n 1) facets (whch are here (n 2)-faces), except f I out or D =, and at the same tme D jj =, n whch case there are 2(n 1) 1 facets. These facets are obtaned by replacng one nequalty n the system by an equalty. The notons of -face, and facet, are usually defned for polytopes. In the corollary above, they are extended to our pecewse smooth cells n a straghtforward way. From proposton 2, the same holds of course for sets Rj ε. Furthermore, we provde an explct descrpton of the latter, whch s drectly derved from that of the sets Dj ε. Proposton 6. For fxed, j, and ε, a set Rj ε can be descrbed by an rredundant lst of nequaltes drectly obtaned from those of Dj ε, usng the translaton rules below. We call x a pont n Dj ε, and y a pont n Rε j. (n)equalty n D ε j (n)equalty n R ε j equalty x = θ ε y j = θ ± j j I out x <θ + y f <β (y ) λ λ (θ + f ) x >θ y f >β (y ) λ λ (θ f ) I out \{, j} λ λ α (x )<α j (x j ) j β (y )<β (θ ± ) α (x )>α (θ λ λ ) β (y )>β (y ) α j (x j )<1 β (y )>1 f D jj f D α j (x j )>α j (θ λ j λ j ) β (y ) α j (x j )>α (θ λ j ) <β j (θ ± j j ) λ β (y )<β (θ ± )

19 Geometrc propertes of a class Where the functons β are gven n equaton (13), used n the defnton of M 1. Proof. Let x Dj ε, and y = Mx Rε j = M(Dε j ). The hyperplane equaltes x = θ ε and y j = θ ± j j arse drectly from the defnton of Dj ε and Rε j. Now, from equaton (13) and the defnton of M 1, the two followng denttes are easly derved : λ λ x N n, = f + β (y ) (y f ), λ (23) λ y = f + α j (x j ) (x f ). From these one obtans : λ λ N n, α j (x j ) j = β (y ) λ λ. (24) The functons β are defned n equaton (13) for all N n. As equatons (23) show that θ <x <θ + s equvalent to θ + λ f λ >β (y ) (y f )>θ f, the case I out s proved. All other rules concern I out. Usng the notaton ntroduced at the begnnng of ths secton leads to : β (y ) = θ f y f. Then all these rules are obtaned usng equatons (23) and (24), from whch smple calculatons show that expressons on each sde of a symbol are equvalent. The last two rules are specal cases of those above them, but they have been wrtten apart snce they do not always apply. The proposton 5 can be checed on fgures 3 and 4. In these fgures, one has D 22. All Dj ε have 4 facets (here edges snce they are 2-dmensonal), wth the excepton of those wth no 2 among ther two ndces. Namely, D13 and D+ 13 have three edges. Of course, the same holds wth the sets Rj ε. Remarably, these fgures are only a partcular case, and constructng an example wth all possble nstances of Dj ε sets, n terms of facet numbers, would requre more than 3 dmensons. Actually, t should contan a nonempty D, wth 2(n 1) facets, as well as some D j, j, I out \{}, wth 2(n 1) 1 facets. Ths requres 3 escapng drectons, and corresponds to the example of fgures 5 and 6. Thus, to have at least one non-escapng drecton m, such that both D m ± have 2(n 1) facets and D mj ±, D± m have only 2(n 1) 1, one needs a fourth dmenson. Ths justfes the algebrac descrpton gven n propostons 4 and 6. Actually, proposton 5 gves a crteron to dstngush among confguratons that are not equvalent from a combnatoral pont of vew. The dscusson above proves that some of these confguratons only occur n hgher-dmensonal spaces, where vsualzaton s out of reach. More than the usefulness of an algebrac formulaton, ths shows that even when dealng wth a sngle box, the admssble behavours form a nontrval set. The relevance of ths secton s hence justfed a posteror.

20 E. Farcot 3.3. Extenson to the whole state space The precedng sectons allow a rgorous defnton of the transton mappng as an homeomorphsm, at the scale of a sngle box, provded ths latter admts escapng drectons. We qucly omtted ts superscrpt, but ths local map was ntally noted M a : Ba n Bout a. We now provde a precse defnton of the transton map on the whole state space. Although local maps are nvertble on boxes wth nonempty outgong doman, boxes wth no escapng drecton are on the other hand more problematc. As we wll see n ths secton, t s natural to map the boundary of such boxes to a sngle pont, whose premage wll then be the whole box boundary. Moreover, the whole doman boundary U s not reachable, due to H1. Hence a global mappng wll not be nvertble at all ponts n general, whch leads us to consder only forward terates of M n ths secton. Ths applcaton has to be terated on a B a, whch can naturally be seen as the underlyng set of a cubcal complex (see e.g. [41] for the defnton of a polyhedral complex, and replace polyhedras by n-rectangles), whose elements are faces of the form F = n =1 F where each F s of one of the followng forms : {θ,a }, {θ,a +1},or[θ,a,θ,a +1]. These F wll be called faces, or threshold (affne) subspaces n the followng. The dmenson of such a face s the number of F s that are not sngletons. H2 mples that any outgong facet W Bb out, for some b, s part of Ba n, for B a adjacent to B b at wall W. Actually, W Ba out would contradct the hypothess, and the outgong and ncomng regons of a box form a cover of ts boundary. Another possblty would be that W s the facet of no other box than B b, when t les on the boundary of the whole doman U. But H1 mples that n ths case W Bb n. Thus we get : a A B a = a A B n a. (25) Then any pont on a A B a belongs to Ba n, for some a A.If Bout a, then M a s well defned, but ths escapng regon may also be empty, whch we recall corresponds to f(a)beng an asymptotcally stable steady state lyng n nt(b a ).In such a case, all ponts n B a are n the basn of f(a), so that t seems reasonable to defne M a as the constant map wth mage f(a). Then {f(a)} has to be added to the doman of M a. These focal ponts beng (asymptotcally stable) equlbra, we put M a f(a)= f(a). After ntroducng the subset of termnal subscrpts : T. ={a A f(a) nt(b a )}={a A d(f (a)) = a}, one can defne local transton maps n all boxes : { M a : x Dom(M a f(a)+ A(x)(x f(a)) f a A \ T ) f(a) f a T (26)

21 Geometrc propertes of a class where the frst case s exactly dentcal to equaton (8). The doman Dom(M a ) = Ba n for a A \ T, and Dom(Ma ) = Ba n {f(a)} for a T. Thus, Dom(M a ) = B a {f(a)}. a A a A a T Yet, a global mappng can not be properly defned. Actually, f any x a A Dom(M a ) les n the doman of some local map M a, the choce of ths local map s not always unque. Ambgutes may happen at some face of codmenson 2 (n R n ) or more, snce they le at the ntersecton of at least two domans of the form above. Some of those lower dmensonal faces may however lead to a well-defned flow, as seen on a smple example. Consder a bnary system n the plane, wth all focal ponts lyng n the same box. Such a planar system has four boxes, ndexed by N 2 N 2 ={11, 12, 21, 22}. Let for example the focal ponts le n B 22 : the sngle pont wth coordnates (θ 11,θ 21 ) s a face wth codmenson 2, on whch all neghbourng flow lnes are drected towards B 22. Hence the flow s not ambguous on ths face, and leads to the asymptotcally stable steady state f(22). Other, less smple, examples also exhbt lower dmensonal faces wth a welldefned flow. A formal characterzaton of the faces on whch the flow s ambguous s not presently nown. We beleve that some combnatoral crteron, nvolvng the postons of all neghbourng focal ponts wth respect to neghbourng boxes, mght do the trc. Snce such a crteron s stll lacng, and n order to smplfy the analyss, we chose here to exclude all codmenson 2 faces from the analyss, along wth the subset from whch those faces can be reached. On such a doman, a global map can fnally be well defned, and wrtten for example wth the ad of characterstc functons (1 A (x) = 1 for x A, 0 otherwse) : Mx = a A 1 Dom(M a )(x) M a x. (27) And lettng F 2 be the unon of all threshold faces of codmenson 2 or more, the doman D on whch M and ts terates are well defned may be wrtten : D = Dom(M a ) \ M (F 2 ) (28) a A N The notaton M stands for the th terate of M defned n (27), and M (F 2 ) s then the premage of the set F 2. Ths requres mplctly that prevous terates are well defned,.e. belong to D as well. The excluded set s thus the unon of all fnte tme premages of codmenson 2 faces. As such, ts measure s zero n Dom(M a ), and the restrcton s not too strong. Remar 3. In the paragraph above, the term "fnte tme" s to be understood as "fnte number of terates". Ths dstncton must be consdered carefully, snce Edwards [10] has shown that some trajectores can cross an nfnte sequence of boxes n fnte tme. He showed that such trajectores always converge towards a face n the set we call F 2.

22 E. Farcot On the other hand, the doman D s not closed n a Dom(Ma ), and thus t s not compact. Actually, a sequence n D that converges to a pont n some codmenson 2 face s easly contructed. In partcular, some orbts n D may have an ω-lmt set that does not belong to D. Typcal examples of such orbts are those convergng towards a stable focus lyng at the ntersecton of two walls or more, wthout ntersectng any such lower dmensonal face n fnte tme (.e. a fnte number of teratons). Remar that D s not open ether, snce n general ts complement M (F 2 ) s not closed. Ths arses from the fact that the latter s an nfnte unon of closed sets. As there are only fntely many faces n M s doman, the only possblty for the prevous unon to be really nfnte, and to preclude closedness, s the exstence of a perodc or quasperodc unstable orbt O. Then there s a neghbourhood N = N (O) such that, for any x N, the trajectory ( M x ) escapes from N as +, and admts O as α-lmt set,.e. as set of lmt ponts when. Snce the solutons of the affne dfferental equatons n each box are contnuous wth respect to the ntal condton, there exst some ntal ponts x N whose trajectory ntersect F 2, as soon as one box around N admts two successors or more. Then, on such a trajectory, all ponts le n the complement of D. On the other hand, the sequence ( M x ) admts O as a lmt set, and O D. Hence, some subsequence of ( M x ) entrely les n the complement of D, and converges towards some lmt pont n D. It follows that the complement s not closed, and thus D s not open. We do not provde an explct example for such a confguraton here, but t s realzable. Actually, unstable lmt cycles are also realzable orbts, as wll appear more explctly n proposton 10. More concretely, the strange attractor presented n [33] explctly provdes an example of unstable lmt cycle. In the absence of confguraton of the form descrbed above, D s an open set. We plan to provde such an example n future wor. A last observaton s that D s not a connected space. Ths s clear from the fact that D s a subset of all walls n phase space, taen wthout ther boundares. Snce these open walls are dsjont, and D clearly has nonempty ntersecton wth all of them, t can not be connected. The connected components of D wll be made explct n secton 5. Despte ts somehow clumsy topology, the set D s almost (.e. up to a zeromeasure set) the largest one on whch all terates of M are well defned. The true largest doman on whch a global mappng s well-defned, along wth all ts terates, s not smple at all, as dscussed below eq. (26). Moreover, M s contnuous on the doman D, snce t s essentally a Dom(Ma ) wth all dscontnuty ponts removed. Now, (D, M) s a properly defned one-sded dscrete dynamcal system. The orbts n ths system are of the form {M x} N, for some x D. The terates of M are n fact compostons of local maps, whch depend on the sequence of walls that are crossed by the orbts. The next secton s devoted to the analyss of such terates, n partcular along cyclc sequences of walls.

23 Geometrc propertes of a class 4. Composte maps We call cycle maps the th terate of M along a cyclc sequence of walls, seen as a frst return map, defned on a subset of a wall. We deal n ths secton under assumpton H3, snce otherwse computatons rapdly become ntractable, and the lnear algebra tools we use here cannot be nvoed. The tas of descrbng cycle maps domans and fxed ponts has been nvestgated n early studes on systems of the form (1), manly [21,22,32]. All these prevous results, as well as some new ones can be found n a wor of R. Edwards [10], wth recent mprovements gven n [13] n terms of combnng multple loops. These studes all concern the case of a sngle threshold per drecton, whch s translated to zero : then, M s a fractonal lnear mappng,.e. a lnear mappng dvded by an affne 1-form. Composton preserves such mappngs. Gven a cyclc sequence of boxes, t s shown that the doman on whch a return map s well defned s a polyhedral cone. Furthermore, fxed ponts of the return maps are closely related to egenvectors of the lnear numerator of the return map. Although t s commonly thought that these results extend to the case of multple thresholds, ths has not been properly proved yet. Here, we propose such an extenson to the multple thresholds context. Ths leads to dealng wth fractonal affne mappngs nstead of fractonal lnear ones. In short, the man dfference wth prevous results s that one has to consder translatons of egenspaces, nstead of egenspaces themselves, and eep trac of all crossed thresholds along an orbt, snce they are not all the same. Ths not only complcates the formulae and computatons, but one also loses some nce features of the bnary case, as dscussed at the end of ths secton. Recall that snce H3 s assumed, n a box B a the mappng M = M a can be wrtten : ( ) Mx = f(a)+ α(x) x f(a) = f(a)+ θ ± ι ι f ( ) ι x f(a), (29) x ι f ι where ι s the escapng drecton for x, and θ ± ι ι s ether θι. = θ ιaι or θ ι + (the choce beng gven by the condton α(x) ]0, 1[, for example) Iterates of the transton map. = θ ιaι +1 A matrx formulaton of M s terates can be obtaned from the equaton above. Gven a wall W, l successve terates of the transton map can follow dfferent sequences of facets. Accordngly, W can be parttoned nto regons correspondng to specfc wall sequences. In partcular, f there s a perodc sequence of l crossed boxes, contanng W, one of these domans correspond to the correspondng loop. A necessary condton for a lmt cycle to happen s then naturally that ths doman be nonempty. More explctly, let a = a 0...a +1 be a fnte sequence of symbols n A, such that there may be some contnuous trajectory ntersectng successvely B a 0...B a +1. It follows that such a trajectory crosses successvely the walls W 0. = B out a 0 B n a 1,