Impulsive motion on synchronized spatial temporal grids

We introduce a family of kinetic vector fields on countable space-time grids and study related impulsive second order initial value Cauchy problems. We then construct special examples for which orbits and attractors display unusual analytic and geometric properties.

1. Introduction. The paper is a contribution to the research in two related though distinct fields. In loose terms, the first field is the study of evolution processes involving an increasing number of particles. The second field is the study of the motion of boundaries and interfaces of planar open domains with increasing lengths.
More specifically, we introduce a new class of second order ODEs and related initial value problems, which describe the non autonomous evolution in continuous time of an increasing number of particles subject to a forcing vector field obtained as the superposition of a smooth force field and of a concentrated force field on a countable grid in the space-time cylinder. As the number of particles increases to infinity, a limit system is obtained which is continuous in space. However, in our applications, such a limit system cannot be described by a differential equation. In other words, the discrete objects underlying our ODEs are not automata in the sense of physics, because the number of particles becomes infinite in the limit, and at the same time they are not numerical approximations of differential equations, because the limit system is not differential. In this collocation between discrete and continuous structures lies the main novelty of our study, as part of the first field of research mentioned before.
In the second field of research mentioned before, the main contribution of this paper consists in the construction of open simply-connected planar domains of finite area, topologically bounded by oriented Jordan curves of any given Hausdorff dimension between 1 and 2. Such Jordan curves are the common boundary of two adjacent open domains, hence, in physical terms, the are interfaces. Such a family of curves is constructed by means of an ODEs evolution problem of the kind mentioned before, with the forcing vector field being interpreted now as a highly irregular concentrated vector curvature field. The construction of such curvature field is obtained by combining the action of a family of similarities with a suitable 6070 UMBERTO MOSCO finite group of rotations in the plane. Similarities and rotations can be given the role of control variables in a control problem aimed at designing optimal interfaces of the kind explained before. In this perspective, our study is a contribution to the mathematical modeling of small cells with highly rippled boundaries.
All complex nonlinear systems loosely described before combine short-range spatial interactions with fast time observations, with equations that take place on a sequence of increasing synchronized finite spatial-temporal grids of decreasing spatial and time sizes. In this perspective, it is also the purpose of this paper to open new lines of study for various synchronized systems that occur in the applications.
A few examples will be discussed in Section 5.
We now describe the structure of the paper in some more detail.
We introduce a family of impulsive initial value problem of second order: with x ∈ R 2 and − → v 0 ∈ R 2 given initial data and g(y, t) ∈ L(R 2 , R 2 ) a highly irregular non-autonomous vector field on the cylinder (y, t) ∈ R 2 × [0, +∞). The vector field g : where G ∞ × T ∞ is a discrete, countable subset of the space-time cylinder R 2 × [0, +∞). More precisely, g is of the type g : R 2 × [0, +∞) → C with g(P, t) = γ(P, t), if (P, t) ∈ S g(P, t) = (0, 0), if (P, t) ∈ R 2 × [0, +∞) \ S and γ a vector field γ : S → C with domain S. The field g : R 2 × [0, +∞) → C in (1.1) is thus very irregular, both in space and time. Problem 1.1 describes an impulsive motion in R 2 , driven by a non autonomous force field γ of purely discrete nature, supported on the set S.
Our main motivation for studying such a kind of highly singular, impulsive evolution equations comes from fractal theory, in particular from the classical constructions of fractals based on countable iterations of a finite family of similarities in the plane, e.g. see [6]. The description of these classical constructions as an evolution ODE in continuous time is new in the fractal literature to date.
We give various examples of this kind of problems. The force fields γ are constructed by exploiting suitable symmetry and similarities maps in the space R 2 . The actions of these maps in space is synchronized with the ticking of time, short steps in space being accompanied by fast ticks in time. Space-time synchronization is regulated by a set of multi-indices (words) W ∞ = {nκi/n : n ∈ {0, N}, κ ∈ {0, 1, . . . , K − 1}, i/n ∈ {0, 1, . . . , N − 1} n } where i/n = i 1 . . . i n and K ≥ 2, N ≥ 2 are two integers, ordered lexicographically. Both the discrete grids G ∞ × T ∞ as well as the set S and the field γ inherit synchronization from the set W ∞ . In these examples, the trajectories traced in R 2 by the solutions as time runs in [0, +∞) have interesting geometric properties. They produce orbits and attractors with fractal features. Section 2 is dedicated to introduce our general impulsive initial value problems and provide a definition of weak solution. Theorem 2.1 gives the existence and uniqueness of the weak solution for general impulsive problems. Theorem 2.2 and Theorem 2.3 deal with the special case of purely impulsive problems. Theorem 2.2 gives the special expression taken by the weak solution in the purely impulsive case. Theorem 2.3 shows that the problem in continuous time considered in Theorem 2.2 can be equivalently formulated as a countable set of vector inequalities, solvable iteratively.
Section 3 is dedicated to the study of fully discrete impulsive problems on synchronized space-time grids. By this we mean that not only the time variable is discretized, as done in the theory developed in Section 2, but now also the space variable is discretized and required to belong to a discrete space grid. The nonautonomous problems in this section are thus formulated on synchronized discrete space-time grids. Such a fully discrete theory allows for the applications to particle systems, as mentioned before. Theorem 3.1, establishes the existence and uniqueness of the solutions in this case.
The rest of the paper is dedicated to supply special examples of the general theory developed in Section 2 and Section 3. Our main application is given in Section 4. It consists in providing an explicit construction of a family of vector force fields to which the impulsive theory applies. The construction is based on special families of similarities and rotations of the plane. Our main result is given in Theorem 4.1. This result applies to the construction of the boundaries of open set with large Hausdorff dimensions, as illustrated before.
Analytic and geometric features of the problems treated in 4.1, as well as further applications and developments are described in the final Section 5.
As the proof of Theorem 4.1 involves rather complicated, though elementary, computations, we put these in the Appendix.
2. Impulsive initial value problems. In this section we prove three main results. The first result, Theorem 2.1, is about the existence and uniqueness of the weak solution of a second order Cauchy initial value problem for a vector force-field in the plane resulting from the superposition of a continuous (smooth) component and of a (discontinuous) impulsive component. The second result, Theorem 2.2, refers to the special case of a purely impulsive force-field, for which the continuous component vanishes. The third result, Theorem 2.3, shows that the purely impulsive Cauchy problem can be equivalently formulated as a countable family of vector equations for the force vectors, which can be solved iteratively.
The problems in this section, formally stated, are of the following kind   is a non autonomous vector field, possibly very irregular in both space and time variables, and where are assigned initial value conditions. The impulsive character of this problems is due to a suitable discretization of the time variable. We fix two integers and we define T ∞ to be the set of all mod-K rationals: We write where for every n ∈ {0, N} (2.9) To simplify notation, we also write and, occasionally, when i 1 = · · · = i n = 0 we write i/n = i 1 . . . i n = 0 n . The set T ∞ is an ordered set with the order relation induced on T ∞ by the lexicographic order (≺, , ) of the set of multi-indices The multi-index following nκ i/n in the lexicographic order of W is denoted by nκ i/n+, the one preceding nκ i/n = 0 0 0 n is denoted by nκ i/n−. The time that follows τ nκ i/n in T ∞ is τ nκ i/n+ , and the time that precedes τ nκ i/n > 0 is τ nκ i/n− . We have ∆τ = τ nκ i/n+ − τ nκ i/n = N −n (2.12) for all nκ i/n ∈ W ∞ . With the set T ∞ we associate the space Y T ∞ (0, +∞) of vector functions y : [0, +∞) → R 2 , defined according to the Definition 2.1. Y T ∞ (0, +∞) is the space of all vector functions y : [0, +∞) → R 2 which have the following properties: : (i) y(t), for t ∈ [0, +∞), is continuous on (0, +∞) and right-continuous at t = 0 with value denoted by y + (0), y + (0) = y(0) ; : (ii) y(t) has a continuous derivativeẏ in each open interval (τ nκi/n , τ nκi/n+ ) with τ nκi/n ∈ T ∞ ;ẏ is right-continuous at each τ nκi/n ∈ T ∞ , that is, y(τ nκi/n ) =ẏ + (τ nκi/n ) wherė y(τ nκi/n + ) − y(τ nκi/n ) ; y possesses the left-limiṫ y(τ nκi/n ) − y(τ nκi/n − ) at each τ nκi/n ∈ T ∞ , τ nκi/n > 0, the limits being taken in R 2 ;
Property (iii) forẏ can be stated more explicitly as follows. In each open interval (τ nκi/n , τ nκi/n+ ), the R 2 -valued functionẏ of (ii) has a R 2 -valued bounded continuous derivativeÿ and the identity is satisfied for every real-valued differentiable function φ with support in (0, +∞). We assume that a vector field is given, which is the superposition of a bounded, Lipschitz vector field and of a vector field supported in time on the discrete set T ∞ . We point out that, because of the discrete nature of the time-set T ∞ and of the absence of any regularity assumption of γ = γ(y, t) in the space-variable y ∈ R 2 , the vector field g allows for sharp variations of the field at concentrated instants of time, as well as for very sharp discontinuities in space. Special examples of such irregular vector fields will be given later on.
We give a precise meaning to problem (2.2) by the following definition This definition is validated by the following result, which show that a weak solution of problem (2.2) does in fact exists and is unique.
In the rest of this paper we shall focus on the special case of purely impulsive problems, that is, on the case where the field g in (2.14) consists only of an impulsive component, supported on a subset of R 2 × [0, +∞), and g 0 = 0. Such special, synchronized vector fields are called kinetic vector field and will be defined precisely later. Clearly, in a purely impulsive problem, the intermediate solutions y yg 0 ,τ nκi/n , evolving between two consecutive times of T ∞ , vanish, thus the solution over the whole time range [0.+∞) takes a simplified form, as we now describe in more detail.
We assume that we are given a subset We define a kinetic vector field to be a map that assigns a vector γ(P, τ ) ∈ C to each space-time location (P, τ ) ∈ S for (P, τ ) = (x, 0) and satisfies the property where, according to (2.7) and (2.12), ∆τ = τ nκi/n+ − τ nκi/n = N −n for τ ∈ [Kn, K(n + 1)). Note that, by its very definition, a kinetic vector field should be more accurately defined as the triple {S, x, γ}, to make clear that the field is only Such a canonical extension, more accurately denoted by g = g {S,x,γ} , will also be named a kinetic vector field on R 2 × [0, +∞). We now assume that a kinetic vector field g = g {S,x,γ} is given, and that, additionally, a second vector is also assigned. With the data {S, x, γ, − → v 0 }, and the associated vector field g = g γ at hand, we consider the problem formally stated as Problem (2.28) is a special impulsive initial value problem of the kind of problem (2.2) introduced before. Therefore, we can define a weak solution for Problem (2.28) by just applying the Definition 2.2 given for the problems (2.2).
In order to complete the proof of the Theorem we need, preliminarily, a family of recursive identities over the sequence of time intervals [nκi/n, nκi/n+) defined by (2.7). We obtain these identities, stated below in (2.31), as follows.
Before stating our third result, we set some notation for functions y in the func- and the vectors The result that follows shows that the purely impulsive Cauchy problem (2.28) in continuous time can be equivalently formulated as a countable set of vector inequalities, solvable iteratively. y(τ nκi/n+ ) = y(τ nκi/n ) + t nκi/n ∆τ, ∆τ = N −n .
Note. The vectors γ(y(τ nκi/n ), τ nκi/n ), which are the data of the system (2.34), are associated with the function y and the given vector field γ, and the vectors a nκi/n also depend on y, therefore the system to be solved in {y(τ nκi/n )} has an implicit dependence on y.
Remark 2.1. From the expression (2.29) of the solutions y(t), we see that a trajectory y = y(t) changes its direction at a location y(τ nκi/n ) if and only if the three locations y(τ nκi/n− ), y(τ nκi/n ), y(τ nκi/n+ ) do not lie on the same straight line of R 2 , in which case the two consecutive vectors t nκi/n− and t nκi/n are not collinear and a τ nκi/n = 0. In the opposite case, that is when the three locations y(τ nκi/n− ), y(τ nκi/n ), y(τ nκi/n+ ) do lie on the same straight line of R 2 , the two consecutive vectors t nκi/n− and t nκi/n and the vector a nκi/n also lie on the same line. In this case there will be no directional change of the vector derivative of y(t) at the location y(τ nκi/n ). However, at such a location we may still have a change of the scalar derivative, and, since the time intervals τ nκi/n+ − τ nκi/n = τ nκi/n − τ nκi/n− = N −n are all equal, that will happen if and only if |y(τ nκi/n+ ) − y(τ nκi/n )| = |y(τ nκi/n ) − y(τ nκi/n− )|, in which case the two vectors t nκi/n− and t nκi/n , though being aligned, have different magnitudes and |a τ nκi/n | = 0. Finally, it can be easily checked that if the derivative dy(t)/dt has a vector-jump at the time t = τ nκi/n > 0, then the jump is the vector a τ nκi/n = [y(τ nκi/n+ ) − 2y(τ nκi/n ) + y(τ nκi/n− )] N −n , nκi/n ∈ W ∞ (2.40) 3. Kinetic vector fields on synchronized grids. The objective of this section is to perform discretization not only in the time variable, as done in Section 2, but in both time and space variables simultaneously, what introduces synchronization on the discrete space-time grids. We accomplish this goal by replacing the spacetime cylinder R 2 × [0, +∞) with a discrete subset G ∞ × T ∞ ⊂ R 2 × [0, +∞) in such a way that while the evolution takes place on G ∞ × T ∞ spatial steps become shorter and shorter and, simultaneously, time ticking becomes quicker and quicker. Synchronization of this sort plays an important role in the constructions carried out in this work. We put ourselves in the setting and notation of Section 2. In particular, we consider the set W ∞ in (2.11) and the map W ∞ → T ∞ that associate the time τ nκi/n with any (multi-) index nκi/n ∈ T ∞ . We recall that i/n = i 1 . . . i n ∈ {0, 1, . . . , N − 1} n .
We proceed by discretizing space. In addition to the integers parameters K ≥ 2 and N ≥ 2 fixed in Section 2, we now fix the real parameters and (3.42) We assume that a map P : W ∞ → R 2 is given, where | · | is the Euclidean distance of R 2 and Under these assumptions, we introduce the countable set in the Euclidean plane R 2 and the countable space-time grid in the space time cylinder R 2 × [0, +∞). We see that as n increases and we move from one time period [Kn, K(n + 1)) to the next, spacial steps |P nκi/n+ − P nκi/n | and time intervals τ nκi/n+ − τ nκi/n become simultaneously smaller and smaller. We point out that we have not required the map P : W ∞ → R 2 in (3.43) to be injective. As a consequence, a point P ∈ G ∞ can be obtained as a point P = P nκi/n given by the map P for possible infinitely many nκi/n ∈ W. The times τ n κ i/n ∈ T ∞ , associated with such recurrent multi-indices nκi/n in W leading to the same P ∈ G ∞ , are the local times at P and they form the set T ∞ P := {τ = τ n κ i/n ∈ T ∞ : P n κ i/n = P } . (3.50) The graph of the multi-valued map P → T ∞ P is the countable subset S P = {(P, τ ) : P = P nκi/n , τ = τ nκi/n ∈ T ∞ P , nκi/n ∈ W ∞ } (3.51) of R 2 × [0, +∞) contained in G ∞ × T ∞ . We point out that the projections P ∈ G ∞ and τ ∈ T ∞ of a point (P, τ ) ∈ S P are simultaneously determined by the same multi-index nκi/n of W. This is how synchronization is incorporated in the set S P . Since the grids G n are monotone increasing in n, for a given P ∈ G ∞ there exists a smallest n ≥ 0 such that P ∈ G n for all n ≥ n . (3.52) As P ∈ G n , there exists a smallest index i/n = i 1 i 2 . . . i n ∈ W in the lexicographic order of W, such that P = P n κ i/n .
In turn, the index n κ i/n uniquely determines the time τ = τ n κ i/n ∈ T ∞ P . (3.53) The time τ = τ n κ i/n is the lowest term of the sequence T ∞ P , it is the initial local time at P . We point out that all the (finitely many) intermediate times τ ≺ τ nκi/n ≺ τ of T ∞ lying between two successive terms τ ≺ τ of the sequence T ∞ P are also local times, however not at P , but at different points Q = P of G β,∞ . Indeed we have The general setting for the synchronized problems in this section is the same as for Theorem 2.3, with the only change that the set S is now specified to be the set S P associated with the map P, as explained before. Accordingly, the kinetic vector field g = g {S,x,γ} is specified to be the vector field g = g {S P ,x,γ P } , obtained as the canonical extension of a given kinetic vector field that assigns a vector γ(P, τ ) ∈ C to each space-time location (P, τ ) ∈ S P for (P, τ ) = (x, 0) and satisfies the property (P, τ ) ∈ S P implies (P + γ(P, τ )∆τ, τ ) ∈ S P (3.56) with ∆τ = N −n for τ ∈ [Kn, K(n + 1)). Since the kinetic field γ depends on the map P, then also the field g depends on P. In a more precise notation, we write g = g {S P ,x,γ P } = {S P , x, γ P }. We assume that, in addition to such a field g, we are given a vector − → v 0 ∈ R 2 such that (x + − → v 0 , 1) ∈ S P (3.57) With this field g and the vectors x and − → v 0 , we consider the purely impulsive Cauchy problem of the kind considered in Theorem 2.3 (ii) a function y ∈ Y T ∞ (0, +∞) is the solution of problem (2.28) if and only if (y(τ nκi/n ), τ nκi/n ) ∈ S P for every nκi/n ∈ W ∞ and the system of equation (2.34) is satisfied in the space R 2 × T ∞ , the vectors γ nκi/n being given by γ nκi/n = γ P (y(τ nκi/n ), τ nκi/n ) for all nκi/n ∈ W ∞ \ {00∅} (3.59) and the vectors a nκi/n by (2.33);
in the lexicographic order of W ∞ .
where P is the given map (3.43).
The equivalent system of equations (2.34) is intrinsic to the discrete kinetic field {S P , x, γ P }, as the vectors P(nκi/n) stay in the domain S P of the map P. The solution y = y(t), as given in the Corollary 3.1, is a parametric equation of the geometric polygonal curve that interpolates the vertices P(nκi/n) in R 2 . This curve, instead, is not intrinsic to {S P , x, γ P }, because in each time interval (τ nκi/n τ nκi/n+ ) it moves into the surrounding space R 2 , away from the chord segment connecting the two vertices P(nκi/n) and P(nκi/n+).
Theorem 3.1 covers a variety of interesting situations, brought to light by appropriate choices of the kinetic field {P, x, γ P }. The section that follows is dedicated to some examples of kinetic vector fields {P, x, γ P } which by integration with Theorem 3.1 give origin to trajectories that display peculiar geometric and analytic properties. Symmetry and similarity are the basic transformations that lead to these interesting objects. 4. Symmetry and similarity. In this section we construct special grids and define on these grids special maps P of the kind considered in Section 3. The grids G ∞ × T ∞ are obtained by combining the action of symmetries with the action of a family of similarities in the Euclidean space R 2 . To keep our examples simple, we choose the similarity maps to be those occurring in the classic v.Koch fractal curves [8]. Alternative choices could be also, for example, Peano [23], Hilbert [5] and Polya [24] curves.
This monotonicity property (4.84) follows from the relation V β,n ⊂ V β,n+1 for all n ≥ 0 (4.86) which is easily proved by remarking that for 0 ≤ n < n + 1 we have i/(n + 1) = The second property is a consequence of ψ β i/n being contractive of a factor (β/4) n for n ≥ 1 in the Euclidean distance of R 2 , as already remarked.
The Cauchy initial value problem for the kinetic field g β with initial condition x and − → v 0 is now We have the following result. nκi/n ≺ nκi/n+ denoting two consecutive indices in the lexicographic order of W ∞ . Moroever, (i) y β (τ nκi/n ) = P β nκi/n for every nκi/n ∈ W ∞ ; (ii) y β (τ nκi/n ) ∈ G β,∞ for every nκi/n ∈ W ∞ ; (iii) the closure the grid G β,∞ enjoys symmetry and self-similar invariance .
Proof. Theorem 3.1 applies to the problem at hand, therefore the first part of the statement of theorem follows from Theorem 3.1 once we prove that the function y β ∈ Y T ∞ satisfies the equations a β nκi/n = γ β nκi/n for all nκi/n ∈ W ∞ \ {00∅} (4.107) where a β nκi/n are given by (2.33) with y replaced by y β . In order to verify the identities (4.107), we first compute the vectors t β nκ i/n for all indices nκi/n, then the vectors a β nκi/n for all indices nκi/n = 00∅. The computations are executed in the lexicographic order for the indices nκi/n ∈ W. Since they are elementary, but rather lengthy, we put them in the final Appendix in the form of a sequence of six lemmas.
In order to complete the proof of Theorem 4.1 we must verify the properties listed in the second part of the statement. Property (i) follows immediately from the expression (4.105) of the solution y β . Property (ii): by (4.82) and (4.83) we have z κ • ψ β i/n (0) ∈ G β,n for every n and every κ, hence, by (4.88) and (i), P β nκi/n = z κ • ψ β i/n (0) ∈ G β,∞ for every nκi/n ∈ W ∞ , what proves (ii). Property (iii): by (4.88) and (4.84), the set G β,∞ is the set-increasing union of the sets G β,n , n ∈ {0, N}, therefore, in order to prove (iii) it suffices to prove that G β,n ⊂ [−L, L] × [−L, L] for every n. From the expression of the similarity maps ψ β i , i = 0, .., 3, in (4.73) and seq, we find the values in the plane of the variable w, and by iterating the maps ψ β i we find that the points ψ is the closure of V β,∞ in R 2 , and The second proposition says that the invariance property of the discrete set G β,∞ is inherited by its closure We proceed with the proofs.
Proof of Proposition 4.2. We start by proving that for K β = V β,∞ we have We have for every n implying that the first inclusion holds. On the other hand, for every n we have also holds. We now prove the identity is satisfied by the similarity maps occurring in Theorem 4.1. Once this identity is proved, we conclude from the first of the two inclusions proved before that and from the second that what leads to the identity By iterating this identity over the maps i 1 , . . . , i n we finally get the identity (4.111), concluding the proof of Proposition (4.2). The proof of Theorem 4.1 is now complete.

5.
Remarks. In this section we collect a few examples and remarks about the applications and research perspectives open by the results of this paper.
The orbits described by the trajectory traced in the plane by the solution y β (t) of Theorem 4.1 as t runs in [0, +∞), as mentioned in the Introduction, have a peculiar structure that depends critically on the parameter β. We summarize the main properties for the case 1 ≤ β < 2 and for the case β = 2 separately, by omitting proofs which will appear elsewhere [20].
The case 1 ≤ β < 2. For every n ∈ {0, N}, the orbit Γ β,n := {y ∈ R 2 : y = y β (t), 4n ≤ t < 4(n + 1), y β (4n) = x, y β (4(n + 1)−) = x} described by the trajectory y β (·) in the interval of time 4n ≤ t < 4(n + 1) is a closed Jordan curve homeomorphic to the boundary ∂D (see next point 2.). Moreover, Γ β,n is the boundary Γ β,n = ∂D β,n of an open connected domain D β,n ⊂ R 2 , with Γ 1,n = ∂D and D 1,n = D for every n. The attractor where the limit is taken in the Hausdorff metric of compact subsets of R 2 , is also a closed Jordan curve homeomorphic to the boundary ∂D and the boundary Γ β = The case β = 2. For every n ∈ {0, N}, the orbit Γ 2,n := {y ∈ R 2 : y = y 2 (t), 4n ≤ t < 4(n + 1), y 2 (4n) = x, y 2 (4(n + 1)−) = x} is a closed continuous curve with multiple points. In each time interval [4n, 4(n+1)) the trajectory visits every vertex once, any site of G 2,n on the boundary twice, any site of G 2,n in the interior at 4 different local times, moving along the segments connecting these sites twice, in opposite directions. The attractor where the limit is taken again in the Hausdorff metric of compact subsets of R 2 , is the full domain Γ 2 = D. By approximating the value 2 with smaller values 2 − , > 0, the multiple points of the case β = 2 split into suitable quadruplets of the case 1 < β < 2, the approximation acting as a singularity resolution for curves.

Connections with PDEs.
As already noted in the Introduction and in the preceding Point 1, the orbits and attractors given by Theorem 4.1 for 1 ≤ β < 2 are in fact closed Jordan curves topologically homeomorphic to the boundary ∂D, that decompose the plane into an inner and an outer open domain. This is an important property, because it shows that these curves are in fact oriented interfaces. However, these interfaces have a quite unusual metric behavior, as it can inferred by noticing that for every segment [a, b] ⊂ Γ β,m ⊂ Γ β,n , m ≤ n, the ratio between the length of the arc connecting a ] connecting a to b in Γ β,m tends to ∞ as n → +∞. This is in sharp contrast with the case of smooth curves, for which the arc/chord ratio is finite and tends infinitesimally to 1 as the chord-length tends to zero. More details on the metric properties of the curves generated by the solutions of Theorem 4.1 are given in [20].
Interfaces occur in many applications of PDEs boundary value problems, in particular is so-called transmission problems for second order operators with transmission conditions of first order or of second order. A survey of recent results for interfaces of pre-fractal and fractal type can be found in [18]. The results of this paper open new perspectives to this kind of PDE applications.
3. Mathematical models of rippled cells.
The constructions described in Point 1 and Point 2 can be generalized to more general geometries and dynamics. The domain D can be chosen to be a K ≥ 2 sided regular polygon inscribed in a circle of radius R = L √ 2 and the symmetry maps can be related to the rotational symmetry of the polygon. The similarities can be chosen to be suitable N ≥ 2 contractive maps depending on a parameter 1 ≤ β ≤ 2, possibly with different contractive factors. An interesting example would be the Pólya curve, [24], [11].
Orbits and attractors can also be confined in a narrow ring-like neighborhood of the unit circle with arbitrarily small transversal diameter, leading to examples of mathematical models for very small cells with very rich and rippled boundaries. See Figure 2 In the theory developed in this paper the motion in time of the orbits toward their attractor is regulated by a very irregular curvature field, given by the kinetic vector field g β . This motion, however, has opposite features with respect to the classic geometric motion of curves by curvature, because in our case the length of the curves increases and becomes asymptotically infinite. Further generalizations of the models described so far take into account alternating similarity families in the construction of the kinetic fields, subjecting the choice of the similarities, considered to be a control variable, to the minimization of a suitable objective functional along the trajectories. This opens the way to a new kind of optimal switching control problems governing the growth of curves. Optimal design of this kind can be done for the boundaries and interfaces of fractal type recently studied in [3], [10], [18], [22]. In this context, stochastic perturbations could be introduced, by adding to the deterministic evolution equations a second order term consisting of a small parameter times the discrete Laplace operator on the similarity grids occurring in Point 1. Laplace operators on fractals go back to the early work on diffusions on fractals in [12], [2], [9] and to the analytic work in [7]. Discrete versions on the spatial grids of this paper can be constructed according to [17]. By suitably scaling the convergence of to zero in terms of the vanishing grid size, the stochastic optimal problem can be expected to converge to the deterministic one. Such non-local stochastic perturbation for discrete deterministic control problems were studied in [13] and [4], see also [1]. The convergence tools that allow for such applications are related to the so-called M-convergence and order-M-convergence in [14], [16]. This kind of switching deterministic and stochastic control problems is new in control theory.
Recently, in [21] and [19] a fully-discrete self-organized-criticality model of sandpile type has been introduced, which involve Euclidean synchronized space-time lattices of the kind described in Point 1. This work can be generalized to the more general countable grids G β × T ∞ described in Point 1, with a continuous spatial limit of Hausdorff dimension between 1 and 2. This adds universality to the selforganized-criticality paradigm.

6.
Appendix. This Appendix contains the computations of the vectors t β nκ i/n for all indices nκi/n and of the vectors a β nκi/n for all indices nκi/n = 00∅, which have been omitted in the proof of Theorem 4.1. The computations are executed in the lexicographic order for the indices nκi/n ∈ W. They are summarized in the six lemmas that follow.
To simplify the notation we omit the superscript β and write t nκi/n in place of t β nκi/n and a nκi/n in place of a β nκi/n , as well as P nκi/n in place of P β nκi/n for the points defined in (4.81). Similarly, we write θ for the angle θ β of (4.72). The proofs of the six lemmas that follow are rather lengthy, though elementary, and for sake of brevity are omitted. The lemmas hold for every value of the parameter 1 ≤ β ≤ 2.
The statement of this lemma can be simplified by using the reduced indices introduced before in our definition of the field γ β . In fact, when n ≥ 2 the last two expressions in Lemma 6.4 can be unified in a single expression, as we now show. We first observe that the cumulative range of the indices occurring in the last two formulas of Lemma 6.4 can be equivalently described as the set of all indices n =