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Global attractors and invariant measures for non-invertible planar piecewise isometric maps
Physics Letters A 371 (2007) 285–290 www.elsevier.com/locate/pla
Global attractors and invariant measures for non-invertible planar piecewise isometric maps Xin-Chu Fu a,b,∗ , Jinqiao Duan c a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China b Department of Mathematics, Shanghai University, Shanghai 200444, China c Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
Received 19 December 2006; received in revised form 28 May 2007; accepted 15 June 2007 Available online 19 June 2007 Communicated by A.P. Fordy
Abstract The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A. © 2007 Elsevier B.V. All rights reserved. MSC: 37D99; 37A05; 37B05 Keywords: Discontinuous map; Non-invertible discrete dynamical system; Piecewise isometry; Invariant measure; Global attractor
1. Introduction There are several situations in signal processing where systems are modelled by planar area-preserving discontinuous maps that are either exactly or close to being piecewise linear. We classify such area-preserving maps by the linear part of the map as being parabolic (two eigenvalues equal to one but only one eigenvector), elliptic (two eigenvalues are on the unit circle) or hyperbolic (no eigenvalues on the unit circle). The elliptic case can be transformed to a piecewise isometry, in particular, we can understand the dynamics of these maps by study of planar piecewise isometries [1–3]. The global attractors for parabolic maps can be decomposed into invariant straight line segments on which the map acts as a one-parameter family of general interval translation maps [4]. We do not con* Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China. E-mail addresses: [email protected] (X.-C. Fu), [email protected] (J. Duan).
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sider hyperbolic maps here as they can be understood to some extent by using techniques from smooth hyperbolic dynamical systems; see for example [5]. In particular the overflow oscillation problem for lossless digital filters [6–9], and bandpass sigma–delta modulator dynamics [10] are examples of maps that display nontrivial dynamics, where the usual techniques of smooth hyperbolic nonlinear dynamics do not work. For example, all Lyapunov exponents may be zero. These area-preserving piecewise linear discontinuous maps are also related to piecewise isometries. In this Letter we consider the dynamical behaviors of discrete systems generated by iterating planar piecewise isometries. The planar piecewise isometries are distance-preserving maps defined on a region M, which has a finite partition into convex components. These maps may be discontinuous and/or non-invertible. Although the Lyapunov exponents are zero for these maps [11–13], they may still have compact global attractors. It has recently been recognized [14,15] that even the definition of a global attractor is in fact quite subtle due to the lack of continuity of these maps.
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Roughly speaking, if a piecewise isometry is non-invertible there may be a subset of the phase space that contains the asymptotic dynamics for almost all initial conditions. The smallest such subset is called a global attractor. There has been some recent progress in the understanding of the dynamics of planar piecewise isometries. For example, the concept of attractors for planar piecewise rotations is discussed in [16]. In [4,17], the global attractor for the sigma–delta map and the extended overflow map and general definition of global attractors for piecewise isometries are discussed. Mathematical proofs for the sigma–delta global attractor are given in [18, 19]. In [20] new examples of global attractors arising in planar piecewise rotations and some subtleties on the definition and properties of piecewise isometric attractors are presented. We now recall the definition of planar piecewise isometries. Definition 1.1. We define a planar piecewise isometry (or PWI in short) f : M → M, M ⊆ R2 , to be a map on a finite partition {Mi }ni=1 of M into (possibly unbounded) convex polygonal atoms (i.e., Mi are convex sets and whose closures are polygons), such that M = ∪i Mi with Mi ∩ Mj = ∅ for i = j, and f restricted to each Mi is an isometry (which preserves Euclidean distance and thus preserves area). Denote by PWI(M) the set of all piecewise isometries defined on M. Remark 1.1. There are different ways to define a planar piecewise isometry. Another way of defining a planar PWI is referred, for example, to [2,20], where images of points on the boundaries of atoms Mi are not defined. This may be convenient when one considers an invertible piecewise isometry, as sometimes it may be impossible to join up the images on the whole boundary of each Mi . But in this Letter we discuss noninvertible PWIs, so we use the above Definition 1.1, as this may be helpful when we discuss some issues, such as the codings for the itineraries (e.g., [21,22]). But there is no essential mathematical difference between the two ways of defining PWIs, in the sense that the exceptional sets (see [20] for a definition) are the same up to a zero measure set. In this Letter, we investigate global attractors and invariant measures for a class of non-invertible planar piecewise isometries. In Section 2 we present basic definitions and basic dynamical behaviors of discrete systems defined by iterating planar piecewise isometries, and we propose the following definition for an global attractor A of a piecewise isometry f : it is a compact set which is Lebesgue mod 0 equal to the locally maximal invariant set of f over B for any B containing A. Then in Section 3 we characterize global attractors via invariant measures and positive continuous functions, and present the main result of this Letter (Theorem 3.4): A compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure (·) restricted to A is invariant while (·) restricted to B is not invariant for any measurable set B containing A with (B A) > 0. Finally, in Section 4, we discuss possible relevance of planar piecewise isometric maps to applications.
2. Preliminaries about piecewise isometric maps In this section, we present some basic definitions and properties for discrete dynamical systems generated by iterating planar piecewise isometric maps. What is an appropriate definition of a bounded global attractor for piecewise isometric maps? For nice topological structure, it would be useful to have a compact set, say, that contains the omega-limits for almost all initial points (i.e., a Milnortype attractor), even though this is not properly invariant. We should perhaps allow the possibility that some positive measure set goes off to infinity, and probably also allow attractors with one-sided basins. There are various possibilities here, for example, almost invariant sets, quasi-invariant set, and Milnor attractor. Note that it is not necessary to assume that the attractor has positive measure. We assume that M ⊆ Rn and (·) is the Lebesgue measure on M. We also assume that (M) > 0 (it can be infinite). As in [15,23] we introduce the notion “almost equal” =0 to mean equivalence of sets up to zero measure, i.e., U =0 V means that (U V ) = 0, where U V (U \ V ) ∪ (V \ U ) is the symmetric difference. Moreover, U ⊆0 V means that there is a U1 =0 U such that U1 ⊆ V . If V =0 ∅ we say V is almost empty. We say V is almost closed if V =0 V¯ , whereas V is almost open if V =0 int(V ). Suppose A is almost open. Since A Int(A) = Ac(Int(A))c , we have Ac =0 A¯ c , i.e., Ac is almost closed. Conversely, if A is almost closed then its complement is almost open. If f (V ) =0 V we say V is almost invariant under f ; as defined in [14,20], in the stronger case that V is almost invariant with f (V ) ⊆ V we say V is quasi-invariant under f (here we include the trivial case that V =0 ∅); in the case that f (V ) = V we say V is invariant under f . We denote the orbit through x by orb(x) = {f n (x): n ∈ Z}, the forward orbit orb+ (x) = {f n (x): n = 0, 1, 2, . . .} and the ω-limit by orb+ f k (x) . ω(x) = k0
Observe that although the orbit is always a countable union of zero-dimensional points, the closure of the orbit can in principle have any dimension between 0 and n. Note that although the Poincaré recurrence theorem holds for measure preserving maps on M (with (M) < ∞) that may be discontinuous everywhere, the ω-limit set is only invariant at points of continuity of the map. To see this, consider any x ∈ M and y ∈ ω(x). We can find a sequence {nk } such that d(f nk (x), y) → 0. If f is continuous at y, then d(f nk +1 (x), f (y)) → 0. As introduced in [14,20,23], we say f : M → M is almost invertible if f is invertible on M except on an almost empty subset of M, i.e., y ∈ M, #f −1 (y) = 1 =0 ∅. Suppose f (A) ⊆ A ⊆ M. We say f is almost invertible on A, we mean f |A is invertible on A except on an almost empty
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subset of A, i.e., y ∈ A, # f −1 (y) ∩ A = 1 =0 ∅.
Lemma 2.2. A global attractor for a planar PWI, if it exists, is unique up to an almost empty set.
A map f : M → M is called Lebesgue non-singular if ∀S ∈ B(M), (S) > 0 ⇔ (f (S)) > 0. Where B(M) is the σ -algebra of Borel subsets of M. When f : M → M is almost invertible and preserves Lebesgue measure, then f is Lebesgue non-singular, because we have (S) = (f (S)).
Proof. Suppose A1 , A2 ⊂ M are two global attractors for a planar PWI f , then from the definition above, A1 =0 Mf (A1 ∪ A2 ), A2 =0 Mf (A1 ∪ A2 ), therefore, A1 =0 A2 . That is to say, the global attractor for a planar PWI is unique up to an almost empty set. 2
Lemma 2.1. Let f : M → M be Lebesgue non-singular. Suppose A ⊆ M is quasi-invariant under f , and f is almost invertible on A, then there exists a subset A1 of A with A1 =0 A such that A1 is invariant and f is invertible on A1 . ˜ where Proof. Let A1 = A \ B, f −k f l (B) B˜ = k,l0
is the set of all points that hit B under forward and/or backward iteration, and B is the almost empty set on which #(f −1 (y)∩ A) = 1. So (f (B)) = 0 and (f −1 (B)) = 0, hence (f n0 f n1 · · · f nk (B)) = 0, ∀ni ∈ Z, k 0, therefore B˜ =0 ∅, so A1 =0 A. ˜ So f (x) ∈ f (A) ⊆ A. If / B. ∀x ∈ A1 , i.e., x ∈ A and x ∈ ˜ ⊆ B, ˜ that is a contradiction. So it ˜ then x ∈ f −1 (B) f (x) ∈ B, ˜ This gives f (A1 ) ⊆ A1 . must have f (x) ∈ / B. ˜ so #(f −1 (y) ∩ A) = 1, i.e., there ∀y ∈ A1 , we have y ∈ / B, ˜ then y = exists a unique x ∈ A, such that f (x) = y. If x ∈ B, ˜ ˜ ˜ so f (x) ∈ f (B) ⊆ B, again, this is a contradiction. So x ∈ / B, x ∈ A1 . This implies f (A1 ) = A1 . So A1 is invariant, and f is invertible on A1 . 2 Even though the Lyapunov exponents are zero for discontinuous parabolic [11] and elliptic [12,13] maps, the dynamics generated by these maps may have sensitive dependence on initial conditions due to the presence of discontinuities. However, for such maps on the plane there may still be compact global attractors, though due to the presence of discontinuities, that give rise to regions with multiple preimages. For example, the maps investigated by Goetz, etc. [1–3,24] can have quite complicated but bounded globally attracting subsets for the dynamics. It has been recognized [14,15] that the definition of a global attractor is surprisingly subtle if we follow the usual framework for defining an attractor, again, due to the presence of discontinuities of the map. But on the other hand, a piecewise isometry moves the whole block of each atom together, so in many cases, a global attractor for this kind of action is actually simply the result of the overlapping of all the atoms. So we give a simple definition as follows. Definition 2.1. For a planar PWI f : M → M with finite partition, we call A ⊂ M the global attractor for f , if (i) A is compact; (ii) A =0 Mf (B), ∀B with A ⊆ B ⊂ M, where i Mf (B) = +∞ i=0 f (B) is the local maximal invariant set of f over B.
Thus we will refer a global attractor for a planar PWI “the global attractor”. We remark here that there are many definitions of what an attractor should be. In [25] the authors define a new notion of attractor which they call minimal attractor, and six other popular definitions of attractors. So the issue of attractors in dynamical systems is not resolved, as there are many competing definitions. However, the above article is about continuous maps, thus it does not apply to piecewise isometries. 3. Global attractors and invariant measures After preliminaries in the last section, we now discuss the main issue of this Letter, the characterizations of global attractors for planar piecewise isometric maps by invariant measures and by positive continuous functions. Ergodic theory becomes useful in studying PWI dynamical systems, because ergodic theory is not concern with discontinuity and also because PWIs are area-preserving locally. Suppose that f : M → M, M ⊆ R2 , is a planar piecewise isometry with a finite partition, i.e., f ∈ PWI(M). Denote by M(M, f ) the set of all measures μ on M making f a measurepreserving transformation of (M, B(M), μ). Note that here μ is a measure on (M, B(M)) but not necessary a probability measure, it even can be an infinite measure. For A ∈ B(M), let μA (S) = (A ∩ S), ∀S ∈ B(M). Note that (·) is the 2-dimensional Lebesgue measure. So μA (·) is a measure on M. Lemma 3.1. If A ⊆ M is the global attractor for f ∈ PWI(M), and A ∩ B =0 ∅, then A ∩ f −1 (B) =0 ∅. Proof. Suppose there is A1 ⊆ A ∩ f −1 (B), s.t. (A1 ) > 0, then A1 ⊆ A, A1 ⊆ f −1 (B), namely, A1 ⊆ A, f (A1 ) ⊆ B. Note that f (A1 ) ⊆0 A, and (f (A1 )) = (A1 ) > 0, we have f (A1 ) ⊆0 A ∩ B, this implies (A ∩ B) > 0. This is a contradiction. Therefore A ∩ f −1 (B) =0 ∅. 2 Lemma 3.2. If A ⊆ M is the global attractor for f ∈ PWI(M), then μA ∈ M(M, f ). Proof. For any S ∈ B(M),
f −1 (S) = f −1 (A ∩ S) ∪ Ac ∩ S = f −1 (A ∩ S) ∪ f −1 Ac ∩ S , therefore,
A ∩ f −1 (S) = A ∩ f −1 (A ∩ S) ∪ A ∩ f −1 Ac ∩ S .
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From Lemma 3.1, A ∩ f −1 (Ac ∩ S) =0 ∅, hence A ∩ f −1 (S) =0 A ∩ f −1 (A ∩ S). Let f −1 (A ∩ S) = A1 ∪ S1 , where A1 ⊆ A, (f (A1 )) = (A ∩ S), and S1 ⊆ Ac , then A∩f
−1
(S) =0 A1 .
Note that (A1 ) = (f (A1 )), as A is almost invariant under f . Consequently we have A ∩ f −1 (S) = (A1 ) = f (A1 ) = (A ∩ S), that is to say, μA (f −1 (S)) = μA (S).
2
Remark 3.1. We may ask what will happen if A is a global attractor such that dim(A) < n (say for a PWI on Rn )? In such a case μA may be not a nontrivial measure; i.e., it could be identically zero for all sets. This case can be viewed as a degenerate one in our context, but nevertheless, it is an important case. To discuss this case, we may still choose a lower dimensional (Lebesgue or Hausdorff) measure to make μA a nontrivial invariant measure. The detailed discussion is similar to the non-degenerate cases. Note that in one-dimensional case, i.e., for IETs, the above situation cannot happen. Rather it can happen for interval translation maps, an example is given in [26], and a whole class of examples is studied in [27], where it is shown that the appropriate Hausdorff measure restricted to the attractor is invariant (but they do not know if it is 0, finite or infinite). Lemma 3.3. If A ⊆ M is compact, f ∈ PWI(M), μA ∈ M(M, f ), and for all measurable B with A ⊂ B ⊆ M, (B \ A) > 0, μB ∈ / M(M, f ), then A is the global attractor for f . Proof. We only need to show that for any A1 with A ⊆ A1 ⊂ M, the local maximal invariant set of f over A1 is almost equal to A, i.e., A =0 Mf (A1 ) B. ∈ M(M, f ), f −1 (A) =
From μA 0 A, so A is almost invariant, therefore A ⊆0 B. Suppose that A =0 B, without loss of the generality, we may suppose A ⊂ B and (B \ A) > 0. From [20], B is invariant, and by Lemma 2.1 μB ∈ M(M, f ), a contradiction. This confirms that A =0 Mf (A1 ). 2 From Lemmas 3.2 and 3.3 we have the following theorem: Theorem 3.4. (Characterizations of global attractors by invariant measures.) If A ⊆ M is compact and f ∈ PWI(M), then A is the global attractor for f if and only if μA ∈ M(M, f ), and / M(M, f ) for all measurable B with A ⊂ B ⊆ M, (B \ μB ∈ A) > 0. If A is the global attractor for f ∈ PWI(M); then f : (M, B(M), μA ) → (M, B(M), μA ) is invertible mod 0 ⇔ f˜−1 B(M) = B(M) [28]. Denote by C + (M) all the positively valued continuous functions ϕ : M → R+ , i.e., ϕ(x) > 0, ∀x ∈ M.
Lemma 3.5. Let f ∈ PWI(M), and A ⊆ M be compact, then μA ∈ M(M, f ) if and only if (ϕ − ϕ ◦ f ) dμA = 0, ∀ϕ ∈ C + (M). M
Proof. Let m be the measure on M such that m(S) = μA f −1 (S) , ∀S ∈ B(M). We have χS dm = χS ◦ f dμA ,
∀S ∈ B(M).
So if s is positive-valued, and is a linear combination of some characteristic functions, i.e., s is a simple function with positive coefficients, we have s dm = s ◦ f dμA . Therefore we have ϕ dm = ϕ ◦ f dμA
(3.1)
for all ϕ ∈ C + (M), as this can be done by choosing an increasing sequence of simple functions with positive coefficients which converges pointwise to ϕ. If μA ∈ M(M, f ), then μA (f −1 (S)) = μA (S), ∀S ∈ B(M), that is, m(S) = μA (S), therefore from (3.1), ϕ ◦ f dμA = ϕ dm = ϕ dμA . This implies (ϕ − ϕ ◦ f ) dμA = 0,
∀ϕ ∈ C + (M).
Conversely, if ∀ϕ ∈ C + (M), (ϕ − ϕ ◦ f ) dμA = 0,
from (3.1), ϕ dm = ϕ ◦ f dμA = ϕ dμA ,
∀ϕ ∈ C + (M).
The proof is done if we can show m = μA , and it suffices to show that m(B) = μA (B) for all bounded closed sets B ⊆ M. Let ε > 0. There exists an open set O with B ⊂ O and μA (O \ B) < ε. Define ϕ : M → R+ by ε x ∈ M \ O, 1+ε , ϕ(x) = d(x, M\O) 1 1+ε ( d(x,M\O)+d(x,B) + ε), x ∈ O, ε then ϕ is continuous, ϕ = 1+ε on M \ O, ϕ = 1 on B, and ε 0 < 1+ε ϕ(x) 1, ∀x ∈ M. So ∀ε > 0 we have ε m(B) ϕ dm = ϕ dμA μA (M \ O) + μA (O) 1+ε ε < (A) + μA (B) + ε, 1+ε
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hence m(B) μA (B); similarly, we can obtain that μA (B) m(B) by using symmetry. Therefore μA = m. That is to say, μA ∈ M(M, f ). 2 Therefore from Theorem 3.4 and Lemma 3.5 we have the theorem below: Theorem 3.6. (Characterizations of global attractors by positive continuous functions.) If A ⊆ M is compact and f ∈ PWI(M), then A is the global attractor for f if and only if (ϕ − ϕ ◦ f ) dμA = 0 and (ϕ − ϕ ◦ f ) dμB > 0 M
M
for all ϕ ∈ C + (M), and all measurable B with A ⊂ B ⊆ M, (B \ A) > 0. 4. Discussions
289
any measurable set B with A ⊂ B ⊆ M, (B \ A) > 0, we have / M(R2 , f ), due to the phase space reduction effect. While μB ∈ from Fig. 2, the set A cannot be further reduced to a smaller set, as it consists of infinitely many invariant disks [15]. Therefore μA ∈ M(R2 , f ). Thus by Theorem 3.4, the set A is the global attractor for the sigma–delta map. More generally, the dynamics of piecewise continuous or piecewise smooth maps is not Markov in general. This makes the dynamics of such maps much richer than their globally continuous or globally smooth counterparts. This Letter shows that it is important to understand the structure of invariant measures of piecewise continuous maps. Note that the Lebesgue measure restricted to a maximal invariant set is invariant under such a map, but in general it is not necessarily true that it is ergodic. In particular, it may be the case that this restriction is trivial in which case the empirical measures can still be defined. Moreover, by analogy with the interval translation maps [26,29], we think that there may be cases where a Hausdorff measure restricted to a maximal invariant set is invariant.
The main result (Theorem 3.4) of this Letter shows that invariant measures may be used to characterize global attractors of non-invertible planar piecewise isometric maps. Here we give a typical example to geometrically and numerically illustrate the characterization. We take the model for a bandpass sigma–delta modulator introduced by Feely and Fitzgerald [10] as an example. By an appropriate linear shear the model can be transformed into a piecewise isometric map defined on the whole plane. We call this piecewise isometry the sigma–delta map f [17]. Indeed f is non-invertible. It was conjectured that the sigma–delta map f has a bounded global attractor [17]. By using very complicated mathematical analysis, and assisted with numerical simulations, Deane [18], and Yu et al. [19] proved the conjecture. By using the main result Theorem 3.4 of this Letter, however, it is rather easy to show that the irreducible invariant set in Fig. 2 is the global attractor for the sigma–delta map. Since from Fig. 1, under the iteration of f , the set consisting of 8 trapezia reduces to the set consisting of 4 trapezia, and the latter further reduces to the set consisting of 2 trapezia, which we denote as set A. Then it can be shown geometrically that for
Fig. 1. The reduction of the phase space.
Fig. 2. The irreducible invariant set.
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Acknowledgements The research was supported jointly by NSFC Grant 10471087 and the Illinois Institute of Technology via a Visiting Professorship for X. Fu. This work is also partially supported by the NSF Grant DMS-0542450 and NSFC Grant 10672146. A part of this work was done while J. Duan was visiting the American Institute of Mathematics, Palo Alto, California, USA. The authors would like to thank Prof. Peter Ashwin and Dr. Miguel Mendes for their helpful discussions. The authors also thank the anonymous referees for their invaluable comments. References [1] A. Goetz, Illinois J. Math. 44 (2000) 465. [2] J. Buzzi, Ergodic Theory Dynam. Systems 21 (2001) 1371. [3] R.L. Adler, B. Kitchens, C. Tresser, Ergodic Theory Dynam. Systems 21 (2001) 959. [4] P. Ashwin, X.C. Fu, On the dynamics of some non-hyperbolic areapreserving piecewise linear maps, in: J.G. McWhirter, I.K. Proudler (Eds.), Proceedings of the Fifth IMA Conference on Maths in Signal Processing, in: Mathematics in Signal Processing, vol. V, Oxford Univ. Press, 2002, pp. 137–145. [5] S. Vaienti, J. Stat. Phys. 67 (1992) 251. [6] L.O. Chua, T. Lin, IEEE Trans. CAS 35 (1988) 648. [7] A.C. Davies, Philos. Trans. R. Soc. London A 353 (1995) 85. [8] L. Kocarev, C.W. Wu, L.O. Chua, IEEE Trans. CAS-II 43 (1996) 234. [9] P. Ashwin, W. Chambers, G. Petkov, Int. J. Bifur. Chaos 7 (1997) 2603. [10] O. Feely, D. Fitzgerald, Non-ideal and chaotic behaviour in bandpass sigma–delta modulators, in: Proceedings of Workshop on Nonlinear Dynamics of Electronic Systems, Sevilla, 1996.
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Global attractors and invariant measures for non-invertible planar piecewise isometric maps Xin-Chu Fu a,b,∗ , Jinqiao Duan c a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China b Department of Mathematics, Shanghai University, Shanghai 200444, China c Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA
Received 19 December 2006; received in revised form 28 May 2007; accepted 15 June 2007 Available online 19 June 2007 Communicated by A.P. Fordy
Abstract The authors investigate dynamical behaviors of discrete systems defined by iterating non-invertible planar piecewise isometries, which are piecewisely defined maps that preserve Euclidean distance. After discussing subtleties for these kind of dynamical systems, they have characterized global attractors via invariant measures and via positive continuous functions on phase space. The main result of this Letter is that a compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure restricted to A is invariant, while it is not invariant restricted to any measurable set B which contains A and whose Lebesgue measure is strictly larger than that of A. © 2007 Elsevier B.V. All rights reserved. MSC: 37D99; 37A05; 37B05 Keywords: Discontinuous map; Non-invertible discrete dynamical system; Piecewise isometry; Invariant measure; Global attractor
1. Introduction There are several situations in signal processing where systems are modelled by planar area-preserving discontinuous maps that are either exactly or close to being piecewise linear. We classify such area-preserving maps by the linear part of the map as being parabolic (two eigenvalues equal to one but only one eigenvector), elliptic (two eigenvalues are on the unit circle) or hyperbolic (no eigenvalues on the unit circle). The elliptic case can be transformed to a piecewise isometry, in particular, we can understand the dynamics of these maps by study of planar piecewise isometries [1–3]. The global attractors for parabolic maps can be decomposed into invariant straight line segments on which the map acts as a one-parameter family of general interval translation maps [4]. We do not con* Corresponding author at: Department of Mathematics, Zhejiang Normal University, Jinhua 321004, China. E-mail addresses: [email protected] (X.-C. Fu), [email protected] (J. Duan).
0375-9601/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2007.06.033
sider hyperbolic maps here as they can be understood to some extent by using techniques from smooth hyperbolic dynamical systems; see for example [5]. In particular the overflow oscillation problem for lossless digital filters [6–9], and bandpass sigma–delta modulator dynamics [10] are examples of maps that display nontrivial dynamics, where the usual techniques of smooth hyperbolic nonlinear dynamics do not work. For example, all Lyapunov exponents may be zero. These area-preserving piecewise linear discontinuous maps are also related to piecewise isometries. In this Letter we consider the dynamical behaviors of discrete systems generated by iterating planar piecewise isometries. The planar piecewise isometries are distance-preserving maps defined on a region M, which has a finite partition into convex components. These maps may be discontinuous and/or non-invertible. Although the Lyapunov exponents are zero for these maps [11–13], they may still have compact global attractors. It has recently been recognized [14,15] that even the definition of a global attractor is in fact quite subtle due to the lack of continuity of these maps.
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Roughly speaking, if a piecewise isometry is non-invertible there may be a subset of the phase space that contains the asymptotic dynamics for almost all initial conditions. The smallest such subset is called a global attractor. There has been some recent progress in the understanding of the dynamics of planar piecewise isometries. For example, the concept of attractors for planar piecewise rotations is discussed in [16]. In [4,17], the global attractor for the sigma–delta map and the extended overflow map and general definition of global attractors for piecewise isometries are discussed. Mathematical proofs for the sigma–delta global attractor are given in [18, 19]. In [20] new examples of global attractors arising in planar piecewise rotations and some subtleties on the definition and properties of piecewise isometric attractors are presented. We now recall the definition of planar piecewise isometries. Definition 1.1. We define a planar piecewise isometry (or PWI in short) f : M → M, M ⊆ R2 , to be a map on a finite partition {Mi }ni=1 of M into (possibly unbounded) convex polygonal atoms (i.e., Mi are convex sets and whose closures are polygons), such that M = ∪i Mi with Mi ∩ Mj = ∅ for i = j, and f restricted to each Mi is an isometry (which preserves Euclidean distance and thus preserves area). Denote by PWI(M) the set of all piecewise isometries defined on M. Remark 1.1. There are different ways to define a planar piecewise isometry. Another way of defining a planar PWI is referred, for example, to [2,20], where images of points on the boundaries of atoms Mi are not defined. This may be convenient when one considers an invertible piecewise isometry, as sometimes it may be impossible to join up the images on the whole boundary of each Mi . But in this Letter we discuss noninvertible PWIs, so we use the above Definition 1.1, as this may be helpful when we discuss some issues, such as the codings for the itineraries (e.g., [21,22]). But there is no essential mathematical difference between the two ways of defining PWIs, in the sense that the exceptional sets (see [20] for a definition) are the same up to a zero measure set. In this Letter, we investigate global attractors and invariant measures for a class of non-invertible planar piecewise isometries. In Section 2 we present basic definitions and basic dynamical behaviors of discrete systems defined by iterating planar piecewise isometries, and we propose the following definition for an global attractor A of a piecewise isometry f : it is a compact set which is Lebesgue mod 0 equal to the locally maximal invariant set of f over B for any B containing A. Then in Section 3 we characterize global attractors via invariant measures and positive continuous functions, and present the main result of this Letter (Theorem 3.4): A compact set A is the global attractor for a piecewise isometry if and only if the Lebesgue measure (·) restricted to A is invariant while (·) restricted to B is not invariant for any measurable set B containing A with (B A) > 0. Finally, in Section 4, we discuss possible relevance of planar piecewise isometric maps to applications.
2. Preliminaries about piecewise isometric maps In this section, we present some basic definitions and properties for discrete dynamical systems generated by iterating planar piecewise isometric maps. What is an appropriate definition of a bounded global attractor for piecewise isometric maps? For nice topological structure, it would be useful to have a compact set, say, that contains the omega-limits for almost all initial points (i.e., a Milnortype attractor), even though this is not properly invariant. We should perhaps allow the possibility that some positive measure set goes off to infinity, and probably also allow attractors with one-sided basins. There are various possibilities here, for example, almost invariant sets, quasi-invariant set, and Milnor attractor. Note that it is not necessary to assume that the attractor has positive measure. We assume that M ⊆ Rn and (·) is the Lebesgue measure on M. We also assume that (M) > 0 (it can be infinite). As in [15,23] we introduce the notion “almost equal” =0 to mean equivalence of sets up to zero measure, i.e., U =0 V means that (U V ) = 0, where U V (U \ V ) ∪ (V \ U ) is the symmetric difference. Moreover, U ⊆0 V means that there is a U1 =0 U such that U1 ⊆ V . If V =0 ∅ we say V is almost empty. We say V is almost closed if V =0 V¯ , whereas V is almost open if V =0 int(V ). Suppose A is almost open. Since A Int(A) = Ac(Int(A))c , we have Ac =0 A¯ c , i.e., Ac is almost closed. Conversely, if A is almost closed then its complement is almost open. If f (V ) =0 V we say V is almost invariant under f ; as defined in [14,20], in the stronger case that V is almost invariant with f (V ) ⊆ V we say V is quasi-invariant under f (here we include the trivial case that V =0 ∅); in the case that f (V ) = V we say V is invariant under f . We denote the orbit through x by orb(x) = {f n (x): n ∈ Z}, the forward orbit orb+ (x) = {f n (x): n = 0, 1, 2, . . .} and the ω-limit by orb+ f k (x) . ω(x) = k0
Observe that although the orbit is always a countable union of zero-dimensional points, the closure of the orbit can in principle have any dimension between 0 and n. Note that although the Poincaré recurrence theorem holds for measure preserving maps on M (with (M) < ∞) that may be discontinuous everywhere, the ω-limit set is only invariant at points of continuity of the map. To see this, consider any x ∈ M and y ∈ ω(x). We can find a sequence {nk } such that d(f nk (x), y) → 0. If f is continuous at y, then d(f nk +1 (x), f (y)) → 0. As introduced in [14,20,23], we say f : M → M is almost invertible if f is invertible on M except on an almost empty subset of M, i.e., y ∈ M, #f −1 (y) = 1 =0 ∅. Suppose f (A) ⊆ A ⊆ M. We say f is almost invertible on A, we mean f |A is invertible on A except on an almost empty
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subset of A, i.e., y ∈ A, # f −1 (y) ∩ A = 1 =0 ∅.
Lemma 2.2. A global attractor for a planar PWI, if it exists, is unique up to an almost empty set.
A map f : M → M is called Lebesgue non-singular if ∀S ∈ B(M), (S) > 0 ⇔ (f (S)) > 0. Where B(M) is the σ -algebra of Borel subsets of M. When f : M → M is almost invertible and preserves Lebesgue measure, then f is Lebesgue non-singular, because we have (S) = (f (S)).
Proof. Suppose A1 , A2 ⊂ M are two global attractors for a planar PWI f , then from the definition above, A1 =0 Mf (A1 ∪ A2 ), A2 =0 Mf (A1 ∪ A2 ), therefore, A1 =0 A2 . That is to say, the global attractor for a planar PWI is unique up to an almost empty set. 2
Lemma 2.1. Let f : M → M be Lebesgue non-singular. Suppose A ⊆ M is quasi-invariant under f , and f is almost invertible on A, then there exists a subset A1 of A with A1 =0 A such that A1 is invariant and f is invertible on A1 . ˜ where Proof. Let A1 = A \ B, f −k f l (B) B˜ = k,l0
is the set of all points that hit B under forward and/or backward iteration, and B is the almost empty set on which #(f −1 (y)∩ A) = 1. So (f (B)) = 0 and (f −1 (B)) = 0, hence (f n0 f n1 · · · f nk (B)) = 0, ∀ni ∈ Z, k 0, therefore B˜ =0 ∅, so A1 =0 A. ˜ So f (x) ∈ f (A) ⊆ A. If / B. ∀x ∈ A1 , i.e., x ∈ A and x ∈ ˜ ⊆ B, ˜ that is a contradiction. So it ˜ then x ∈ f −1 (B) f (x) ∈ B, ˜ This gives f (A1 ) ⊆ A1 . must have f (x) ∈ / B. ˜ so #(f −1 (y) ∩ A) = 1, i.e., there ∀y ∈ A1 , we have y ∈ / B, ˜ then y = exists a unique x ∈ A, such that f (x) = y. If x ∈ B, ˜ ˜ ˜ so f (x) ∈ f (B) ⊆ B, again, this is a contradiction. So x ∈ / B, x ∈ A1 . This implies f (A1 ) = A1 . So A1 is invariant, and f is invertible on A1 . 2 Even though the Lyapunov exponents are zero for discontinuous parabolic [11] and elliptic [12,13] maps, the dynamics generated by these maps may have sensitive dependence on initial conditions due to the presence of discontinuities. However, for such maps on the plane there may still be compact global attractors, though due to the presence of discontinuities, that give rise to regions with multiple preimages. For example, the maps investigated by Goetz, etc. [1–3,24] can have quite complicated but bounded globally attracting subsets for the dynamics. It has been recognized [14,15] that the definition of a global attractor is surprisingly subtle if we follow the usual framework for defining an attractor, again, due to the presence of discontinuities of the map. But on the other hand, a piecewise isometry moves the whole block of each atom together, so in many cases, a global attractor for this kind of action is actually simply the result of the overlapping of all the atoms. So we give a simple definition as follows. Definition 2.1. For a planar PWI f : M → M with finite partition, we call A ⊂ M the global attractor for f , if (i) A is compact; (ii) A =0 Mf (B), ∀B with A ⊆ B ⊂ M, where i Mf (B) = +∞ i=0 f (B) is the local maximal invariant set of f over B.
Thus we will refer a global attractor for a planar PWI “the global attractor”. We remark here that there are many definitions of what an attractor should be. In [25] the authors define a new notion of attractor which they call minimal attractor, and six other popular definitions of attractors. So the issue of attractors in dynamical systems is not resolved, as there are many competing definitions. However, the above article is about continuous maps, thus it does not apply to piecewise isometries. 3. Global attractors and invariant measures After preliminaries in the last section, we now discuss the main issue of this Letter, the characterizations of global attractors for planar piecewise isometric maps by invariant measures and by positive continuous functions. Ergodic theory becomes useful in studying PWI dynamical systems, because ergodic theory is not concern with discontinuity and also because PWIs are area-preserving locally. Suppose that f : M → M, M ⊆ R2 , is a planar piecewise isometry with a finite partition, i.e., f ∈ PWI(M). Denote by M(M, f ) the set of all measures μ on M making f a measurepreserving transformation of (M, B(M), μ). Note that here μ is a measure on (M, B(M)) but not necessary a probability measure, it even can be an infinite measure. For A ∈ B(M), let μA (S) = (A ∩ S), ∀S ∈ B(M). Note that (·) is the 2-dimensional Lebesgue measure. So μA (·) is a measure on M. Lemma 3.1. If A ⊆ M is the global attractor for f ∈ PWI(M), and A ∩ B =0 ∅, then A ∩ f −1 (B) =0 ∅. Proof. Suppose there is A1 ⊆ A ∩ f −1 (B), s.t. (A1 ) > 0, then A1 ⊆ A, A1 ⊆ f −1 (B), namely, A1 ⊆ A, f (A1 ) ⊆ B. Note that f (A1 ) ⊆0 A, and (f (A1 )) = (A1 ) > 0, we have f (A1 ) ⊆0 A ∩ B, this implies (A ∩ B) > 0. This is a contradiction. Therefore A ∩ f −1 (B) =0 ∅. 2 Lemma 3.2. If A ⊆ M is the global attractor for f ∈ PWI(M), then μA ∈ M(M, f ). Proof. For any S ∈ B(M),
f −1 (S) = f −1 (A ∩ S) ∪ Ac ∩ S = f −1 (A ∩ S) ∪ f −1 Ac ∩ S , therefore,
A ∩ f −1 (S) = A ∩ f −1 (A ∩ S) ∪ A ∩ f −1 Ac ∩ S .
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From Lemma 3.1, A ∩ f −1 (Ac ∩ S) =0 ∅, hence A ∩ f −1 (S) =0 A ∩ f −1 (A ∩ S). Let f −1 (A ∩ S) = A1 ∪ S1 , where A1 ⊆ A, (f (A1 )) = (A ∩ S), and S1 ⊆ Ac , then A∩f
−1
(S) =0 A1 .
Note that (A1 ) = (f (A1 )), as A is almost invariant under f . Consequently we have A ∩ f −1 (S) = (A1 ) = f (A1 ) = (A ∩ S), that is to say, μA (f −1 (S)) = μA (S).
2
Remark 3.1. We may ask what will happen if A is a global attractor such that dim(A) < n (say for a PWI on Rn )? In such a case μA may be not a nontrivial measure; i.e., it could be identically zero for all sets. This case can be viewed as a degenerate one in our context, but nevertheless, it is an important case. To discuss this case, we may still choose a lower dimensional (Lebesgue or Hausdorff) measure to make μA a nontrivial invariant measure. The detailed discussion is similar to the non-degenerate cases. Note that in one-dimensional case, i.e., for IETs, the above situation cannot happen. Rather it can happen for interval translation maps, an example is given in [26], and a whole class of examples is studied in [27], where it is shown that the appropriate Hausdorff measure restricted to the attractor is invariant (but they do not know if it is 0, finite or infinite). Lemma 3.3. If A ⊆ M is compact, f ∈ PWI(M), μA ∈ M(M, f ), and for all measurable B with A ⊂ B ⊆ M, (B \ A) > 0, μB ∈ / M(M, f ), then A is the global attractor for f . Proof. We only need to show that for any A1 with A ⊆ A1 ⊂ M, the local maximal invariant set of f over A1 is almost equal to A, i.e., A =0 Mf (A1 ) B. ∈ M(M, f ), f −1 (A) =
From μA 0 A, so A is almost invariant, therefore A ⊆0 B. Suppose that A =0 B, without loss of the generality, we may suppose A ⊂ B and (B \ A) > 0. From [20], B is invariant, and by Lemma 2.1 μB ∈ M(M, f ), a contradiction. This confirms that A =0 Mf (A1 ). 2 From Lemmas 3.2 and 3.3 we have the following theorem: Theorem 3.4. (Characterizations of global attractors by invariant measures.) If A ⊆ M is compact and f ∈ PWI(M), then A is the global attractor for f if and only if μA ∈ M(M, f ), and / M(M, f ) for all measurable B with A ⊂ B ⊆ M, (B \ μB ∈ A) > 0. If A is the global attractor for f ∈ PWI(M); then f : (M, B(M), μA ) → (M, B(M), μA ) is invertible mod 0 ⇔ f˜−1 B(M) = B(M) [28]. Denote by C + (M) all the positively valued continuous functions ϕ : M → R+ , i.e., ϕ(x) > 0, ∀x ∈ M.
Lemma 3.5. Let f ∈ PWI(M), and A ⊆ M be compact, then μA ∈ M(M, f ) if and only if (ϕ − ϕ ◦ f ) dμA = 0, ∀ϕ ∈ C + (M). M
Proof. Let m be the measure on M such that m(S) = μA f −1 (S) , ∀S ∈ B(M). We have χS dm = χS ◦ f dμA ,
∀S ∈ B(M).
So if s is positive-valued, and is a linear combination of some characteristic functions, i.e., s is a simple function with positive coefficients, we have s dm = s ◦ f dμA . Therefore we have ϕ dm = ϕ ◦ f dμA
(3.1)
for all ϕ ∈ C + (M), as this can be done by choosing an increasing sequence of simple functions with positive coefficients which converges pointwise to ϕ. If μA ∈ M(M, f ), then μA (f −1 (S)) = μA (S), ∀S ∈ B(M), that is, m(S) = μA (S), therefore from (3.1), ϕ ◦ f dμA = ϕ dm = ϕ dμA . This implies (ϕ − ϕ ◦ f ) dμA = 0,
∀ϕ ∈ C + (M).
Conversely, if ∀ϕ ∈ C + (M), (ϕ − ϕ ◦ f ) dμA = 0,
from (3.1), ϕ dm = ϕ ◦ f dμA = ϕ dμA ,
∀ϕ ∈ C + (M).
The proof is done if we can show m = μA , and it suffices to show that m(B) = μA (B) for all bounded closed sets B ⊆ M. Let ε > 0. There exists an open set O with B ⊂ O and μA (O \ B) < ε. Define ϕ : M → R+ by ε x ∈ M \ O, 1+ε , ϕ(x) = d(x, M\O) 1 1+ε ( d(x,M\O)+d(x,B) + ε), x ∈ O, ε then ϕ is continuous, ϕ = 1+ε on M \ O, ϕ = 1 on B, and ε 0 < 1+ε ϕ(x) 1, ∀x ∈ M. So ∀ε > 0 we have ε m(B) ϕ dm = ϕ dμA μA (M \ O) + μA (O) 1+ε ε < (A) + μA (B) + ε, 1+ε
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hence m(B) μA (B); similarly, we can obtain that μA (B) m(B) by using symmetry. Therefore μA = m. That is to say, μA ∈ M(M, f ). 2 Therefore from Theorem 3.4 and Lemma 3.5 we have the theorem below: Theorem 3.6. (Characterizations of global attractors by positive continuous functions.) If A ⊆ M is compact and f ∈ PWI(M), then A is the global attractor for f if and only if (ϕ − ϕ ◦ f ) dμA = 0 and (ϕ − ϕ ◦ f ) dμB > 0 M
M
for all ϕ ∈ C + (M), and all measurable B with A ⊂ B ⊆ M, (B \ A) > 0. 4. Discussions
289
any measurable set B with A ⊂ B ⊆ M, (B \ A) > 0, we have / M(R2 , f ), due to the phase space reduction effect. While μB ∈ from Fig. 2, the set A cannot be further reduced to a smaller set, as it consists of infinitely many invariant disks [15]. Therefore μA ∈ M(R2 , f ). Thus by Theorem 3.4, the set A is the global attractor for the sigma–delta map. More generally, the dynamics of piecewise continuous or piecewise smooth maps is not Markov in general. This makes the dynamics of such maps much richer than their globally continuous or globally smooth counterparts. This Letter shows that it is important to understand the structure of invariant measures of piecewise continuous maps. Note that the Lebesgue measure restricted to a maximal invariant set is invariant under such a map, but in general it is not necessarily true that it is ergodic. In particular, it may be the case that this restriction is trivial in which case the empirical measures can still be defined. Moreover, by analogy with the interval translation maps [26,29], we think that there may be cases where a Hausdorff measure restricted to a maximal invariant set is invariant.
The main result (Theorem 3.4) of this Letter shows that invariant measures may be used to characterize global attractors of non-invertible planar piecewise isometric maps. Here we give a typical example to geometrically and numerically illustrate the characterization. We take the model for a bandpass sigma–delta modulator introduced by Feely and Fitzgerald [10] as an example. By an appropriate linear shear the model can be transformed into a piecewise isometric map defined on the whole plane. We call this piecewise isometry the sigma–delta map f [17]. Indeed f is non-invertible. It was conjectured that the sigma–delta map f has a bounded global attractor [17]. By using very complicated mathematical analysis, and assisted with numerical simulations, Deane [18], and Yu et al. [19] proved the conjecture. By using the main result Theorem 3.4 of this Letter, however, it is rather easy to show that the irreducible invariant set in Fig. 2 is the global attractor for the sigma–delta map. Since from Fig. 1, under the iteration of f , the set consisting of 8 trapezia reduces to the set consisting of 4 trapezia, and the latter further reduces to the set consisting of 2 trapezia, which we denote as set A. Then it can be shown geometrically that for
Fig. 1. The reduction of the phase space.
Fig. 2. The irreducible invariant set.
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Acknowledgements The research was supported jointly by NSFC Grant 10471087 and the Illinois Institute of Technology via a Visiting Professorship for X. Fu. This work is also partially supported by the NSF Grant DMS-0542450 and NSFC Grant 10672146. A part of this work was done while J. Duan was visiting the American Institute of Mathematics, Palo Alto, California, USA. The authors would like to thank Prof. Peter Ashwin and Dr. Miguel Mendes for their helpful discussions. The authors also thank the anonymous referees for their invaluable comments. References [1] A. Goetz, Illinois J. Math. 44 (2000) 465. [2] J. Buzzi, Ergodic Theory Dynam. Systems 21 (2001) 1371. [3] R.L. Adler, B. Kitchens, C. Tresser, Ergodic Theory Dynam. Systems 21 (2001) 959. [4] P. Ashwin, X.C. Fu, On the dynamics of some non-hyperbolic areapreserving piecewise linear maps, in: J.G. McWhirter, I.K. Proudler (Eds.), Proceedings of the Fifth IMA Conference on Maths in Signal Processing, in: Mathematics in Signal Processing, vol. V, Oxford Univ. Press, 2002, pp. 137–145. [5] S. Vaienti, J. Stat. Phys. 67 (1992) 251. [6] L.O. Chua, T. Lin, IEEE Trans. CAS 35 (1988) 648. [7] A.C. Davies, Philos. Trans. R. Soc. London A 353 (1995) 85. [8] L. Kocarev, C.W. Wu, L.O. Chua, IEEE Trans. CAS-II 43 (1996) 234. [9] P. Ashwin, W. Chambers, G. Petkov, Int. J. Bifur. Chaos 7 (1997) 2603. [10] O. Feely, D. Fitzgerald, Non-ideal and chaotic behaviour in bandpass sigma–delta modulators, in: Proceedings of Workshop on Nonlinear Dynamics of Electronic Systems, Sevilla, 1996.
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