- Email: [email protected]
Stability and approximation of invariant measures for a class of nonexpanding transformations
Nonlinear
Analysis,
Theory,
Methods
& Applications,
Pergamon
Vol. 23, No. 8, pp. 1013-1025, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7X0+ .@I
0362-546X(93)E0031-W
STABILITY AND APPROXIMATION OF INVARIANT MEASURES FOR A CLASS OF NONEXPANDING TRANSFORMATIONS WALTER Department (Received
23 March
of Mathematics,
Howard
M. MILLER University,
1993; received in revised form 4 October
Key words and phruses: Frobenius-Perron convex transformation, irreducible.
operator,
Washington,
DC 20059, U.S.A.
1993; received for publication
invariant
measures,
Ulam’s
20 December
method,
1993)
piecewise
1. INTRODUCTION
Invariant measures for a dynamical system give a statistical description of its long term behaviour-information which is especially useful when the system is considered “chaotic”. As such the determination of these measures is of practical as well as theoretical significance; especially those measures which enjoy some character of stability and are thus in some sense “physical”; see [l]. finite approximation method-the In 1960, Ulam conjectured that a certain “natural” approximants of which are fixed eigenvectors of stochastic matrices-should converge to an invariant measure of a self-transformation of an interval; see [2]. Later, using a key inequality of Lasota and Yorke relating the variation of any initial density to that of its transformation, Li, in [3], gave sufficient conditions guaranteeing just such convergence. He showed the following: if a piecewise C2 transformation T of an interval Z satisfies J := infl T’( > 2 then Ulam’s method, applied to T, converges strongly in L,(Z) to the density of a T-invariant measure. When T is just expanding (J > l), his method, as well as the results of Keller [4] and Blank [5], imply the convergence of Ulam’s method for sufficiently large iterates, Tk, of T. An advantage of Ulam’s approach is the robustness of its numerical implementation, see [3]. Arguably, however, as noted by Gora and Boyarsky in [6], there are two drawbacks of the method. One regards the difficulty, in general, of computing the approximating Ulam matrices themselves. This issue is addressed by Hunt in [7]. Another regards the restriction J > 2. Several authors, notably Keller in [8], conjecture that this restriction is not necessary. This is borne out by the numerical evidence of Ding and Li in [9], which in fact suggests a far wider applicability of the method, even for nonexpanding maps. We address this latter issue here, and in Section 3 we show the convergence of Ulam’s method for a class of piecewise monotone maps of the interval that are not necessarily expanding, i.e. for which J < 1. These are the piecewise convex (PCVX) maps of an interval Z introduced by Lasota and Yorke in [lo]. For these maps we are able to circumvent the “obstruction” J < 1 by taking advantage of a crucial property-their corresponding Frobenius-Perron operators, as transformations on the subspace of densities in L,(Z, m), leave invariant the positive cone of decreasing functions. This cone will, for us, serve a similar purpose as the subspace BI’ of functions of bounded variation, does for Li in [3], Kowalski in [ll], and Keller in [4]. Our strategy is as follows: first using the cone-invariance property alluded to above, we demonstrate the existence of a convergent family of fixed points f,, of P,,(T). Then to show 1013
1014
W. M. MILLER
that any family of nonnegative normalized fixed points of P,(T) are convergent, we show that there is in fact only one such family, i.e. we show uniqueness. It is important to note that we are unable to invoke the results as in [3,4] to show existence, or the techniques used in [12], for example, to show uniqueness, because Tis not necessarily expanding, even though by [13] some sufficiently large iterate, Tk, is. Instead we use the more general “ergodic” notion of irreducibility as introduced by Hunt and Miller in [14]; see lemma 2.1 and corollary 2.2 in Section 2. In this regard our methods and those in [12] may be viewed as complementary. Let us also note that as a corollary of our method we are able to deduce a result of some computational significance-the approximating densities, f, , may be numerically computed by repeated iteration of the approximating matrices P,(T) themselves. Finally, as Keller notes in [4] the convergence of Ulam’s method may be viewed as a type of stability of the invariant measure-a stability that allows it to be approximable by this particular method when applied to a particular map. Now by the results in [4, 11, 151 one sees that this and other “stochastic stability” properties hold for piecewise expanding maps in general, and, therefore, by [13], for sufficiently large iterates of PCVX maps, in particular. Is one, as shown in Section 3 for Ulam’s method, also able to approximate the invariant measure of a PCVX map by these other methods when applied to the map itself as opposed to a sufficiently large iterate of the map? In Section 4 we give a positive answer to this question and following [4] we show invariant measures of the class of PCVX maps are stable with respect to three types of perturbations: deterministic perturbations of piecewise convex transformations, perturbations, occurring deterministic stochastic and randomly perturbations-again making crucial use of the cone-invariance property mentioned above. Note that these cases have special physical significance, as they serve to model the “real world” effects of “noise” or “error” on the determination of “observables” of the system by the viewer.
2. PRELIMINARIES
(A) Frobenius-Perron
operators,
invariant
AND
measures
LEMMAS
and Ulam ‘s method
An extremely useful tool for investigating invariant measures of a nonsingular transformation T on a measure space (X, m) is the Frobenius-Perron (FP) operator P = PT corresponding to T. The action of Ton points of X induces the action of P on density functions on X, which may be characterized as follows. For any measurable set A and f E L,(X, m) we have
(1) P is a positive linear Markov operator on L,(X, m), mapping densities to densities, and any positive normalized fixed point of P corresponds to an absolutely continuous (with respect to m) T-invariant measure on X. By the Birkhoff individual ergodic theorem we know that invariant measures encode statistical properties of the asymptotics of the dynamical system (T, X, m); thus, the keen interest in the determination of fixed points of P; see [16]. In 1960 Ulam, in [2], proposed the following method for approximating fixed points of P. For a given (equi)partition II, of the space X into 1X,, X, , . . . , X,J define a row stochastic
Invariant
matrix
P, , the (i, j)th
entry of which gives the fraction
[PJij:=
1015
measures
m(Xi fl
of Xi that is mapped
by T to Xj
T-'Xj)
m(xi)
.
P,, is in a sense a finite approximation to the transformation T, but more precisely it is a finite Markov approximation to the corresponding Frobenius-Perron operator, P, on the space of densities ‘Z) = a>(X) in ,5,(X, m). It is not hard to show, by the Brouwer fixed point theorem, for example, that the Ulam matrix, P,, , corresponding to each equipartition, II,, has at least one left fixed eigenvector f,, which corresponds to a piecewise constant (with respect to the given partition) density. Ulam conjectured that the sequence of fixed points, f,, , of the matrices P, corresponding to densities on finer and finer partitions of X, converge to a fixed point of the Frobenius-Perron operator itself. Markov Given a partition II, = (Xj Ij = 1, . . . , n) of X let Q, denote the corresponding projection operator on L,(X) where for any f in L,(X), Q,f is the piecewise constant function whose value at x equals the average value off on that pixel of the partition containing x
Q,f := i i=l
Ci*
Ix,,
Ci:=&*Sx,fdm-
We have the following fundamental relation between partition and the Frobenius-Perron operator P
PnQnf = Q, Pf
the Ulam
(3) matrices
P,, defined
by a
(4)
for any f in L,(X); see [3, 141. Now let f *be a P-invariant density. We will say that P is ergodic (w.r.t. f *)if for any density f the sequence {Pkf 1k L 0) is weakly Cesaro convergent, in L,(x, m), to f *; this implies that f * is the unique density such that Pf * = f *; see [ 16, Section 41. We will say T is ergodic (w.r.t. f *) if PT is likewise ergodic. We have the following rather general approximation result which as realized in the subsequent corollary relates the support of the Ulam eigenvectors f,, to that of an ergodic P-invariant density f *. Note that the stated set equalities and inclusions hold mod m.
LEMMA 2.1. Let P = Pr: L,(X, m) -+ L,(X, m) denote the Frobenius-Perron operator, with fixed density f *, of a nonsingular ergodic transformation Ton a measure space (X, m). Given an (equi)partition II, = 1Xj: 1 5 j 5 n) of X let Q, denote the corresponding piecewise constant averaging operator; P,, = QnPT the corresponding Markov finite approximation to PT; and f, any corresponding nonnegative normalized fixed point, f,, = P,, f, . Let S denote the support off * and S, the support off,, . Then: (a) S c T-‘S; (b) S,, E T-‘S,,. If A is any set of positive measure such that A E T-‘A then (c) SEA. If A is any union of partition boxes such that A & T-IA then (d) S, EA.
1016
W. M. MILLER
Proof of (a). By definition
1 = Isf’dm and 1 = ST-IS f *dm implies Proof of(b).
= iilsf*drn
= isPrf*dm n
S E T-‘S.
Here by the P,,-invariance
off,
1 = isnfndm But since S,, is a union
off *, we have
of S and the P-invariance
and the definition
= jsnQ.PTfndm.
of pixels of II, we have, by definition
i’sn
of S,, we have
Q,,gdm
=
of Q,,
for any g
gdm i S”
(5)
so we have I= and 1 = ST-as, f,dm Proof of(c).
Let
P,f, QnPTfndm = 53, .i sn
fA := (l/m(A))
f, dm T_‘S,
n
S, C T-IS,,.
implies
dm =
. IA, then since A C T-IA
we have
~=ilf~dm~i’T_,*f~dm=ilPf,dmr---r%Pkf~dm by induction,
for any integer
k I 0. However,
1 I jA P”f, dm implies PkfA 1, dm.
Ergodicity
implies _ :j,
jX
f*l,dm
PkfA 1, dm +
=
iX
as n + 00. So 1 I jA f * dm, which implies
f*dm i A
S E A.
H
Proof of(d). Assume A E T-‘A. This implies for R := XL4 that T-‘R E R which implies S, fI T-‘R G S, fl R. Letting B := S,, fl R, it suffices to show that m(B) = 0. We have f,dm the last equality
holding
f,dm
=
=
P,f,dm
since R is a union
= sR
sR
s T-‘R
c &I-IT-‘R
of partition
QnPTfndm
boxes of n,.
However,
Q,PTf,, =
Ptzfn = f* 9 so f,dm s S,i’T-‘R
= i,fi,dm
= isnnRfndm.
(6)
Invariant
This equality (6) combined with S, (l T-‘R
G S, fl R implies
S,,fl T-'R = S,nR, since f, > 0 on S, . Therefore, B = so B E T-‘B. S&B. W
1017
measures
(mod m)
(7)
by the above equality (7) and the inequality in part (b),
s, n R = s, n T-'R E ~-ls,n T-1~ c T-$9, no) = T-'B
However, by part (c) above this implies m(B) = 0 since B E x‘\S precludes
COROLLARY 2.2. If the hypotheses on Tare as above then (e) S E S,. Moreover, if T is such that S E int(T-‘S) then for sufficiently large n, i.e. for any sufficiently fine partition II,, (f) s E s, E [S], where [S] denotes the partition neighbourhood of S,
Proof. Since S 5 int(T-‘S), if II,, is sufficiently fine then [S] G int(T-‘S) c T-'S c T-'[S],so [S]E T-'[S].Therefore, lemma 2.1, part (d) implies S, E [S], since [S] is a union of partition boxes. That S E S, holds is implied by parts (b) and (c) of the same lemma. n (B) Piecewise convex transformations
and their Frobenius-Perron
operators
We will be interested in the following class of maps. A transformation T: [0, l] + [0, l] is said to be piecewise convex with strong repeller (PCVX) if T satisfies the following conditions: (i) there is a partition 0 = a, < a, < ..a < a, = 1 of [0, l] such that for each integer i= 1, ..-, r the restriction of T to [ai_l, ai) is a C2-function; (ii) T'(x)> 0 and T"(x)2 0 for all x in [0, l), [T’(aJ and T”(aJ are right derivatives]; (iii) for each integer i = 1, . . . , r we have T(ai_ J = 0; (iv) I := T'(0)> 1. This class of transformation was introduced by Lasota and Yorke in [lo] where they showed the following theorem. THEOREM2.3. Let a transformation T satisfy (i)-(iv) above. Then there exists a unique normalized absolutely continuous (w.r.t. m) invariant measure pg that is invariant under T. Its density g is bounded, decreasing and ergodic; in fact P is asymptotically stable, i.e. lim P”f = g, as n + 00, for any initial density f, where m denotes standard Bore1 measure on [0, l] and P = PT is the Frobenius-Perron operator corresponding to T. Next we consider the following abstract set-up, the relevance of which will be made clear in the lemma immediately following. Let 9% denote the subset of &(I, m) consisting of bounded decreasing densities. We will say that a Markov operator P: L,(Z, m) -+ L,(Z, m) is of class S if: (Sl) P: a>zD--t L-&D,; (S2) for some I > 1 and A4 > 0, we have for any f in ED IPf L
5 wn)lf less+ A4
(9)
1018
W.M.MlLLER
where If I,,, denotes the (essential) supremum off (with respect to m). For a particular A and A4 we will say that P is of class s(A, M), and a sequence (P,) of Markov operators will be said to be S-bounded if for some I and M, independent of n, each P, is of class S(A, M). It turns out that the FP-operators of the piecewise convex transformations above are of class S. More precisely we have the following lemma. LEMMA 2.4 (Lasota and Yorke). where A := T’(0) and
If T satisfies (i)-(iv) then its FP-operator
M :=
c
PT is of class S(A, M)
(10)
(g;(O)/aj_,).
i=2
Proof.
See [lo; 16, Section
For an abstract
operator
6.31.
W
of class S one is able to show the following
lemma.
2.5. Let P be of class S(I, M). If Q is of class S(1, 0), i.e. (i) Q: 99 -+ DD; and (ii) ID],,, 5 IfI,,,, then QP and PQ are of class S(I, M).
LEMMA
Proof.
IQWI,,,
This is straightforward,
5 Ipfi,,, 5 WVlfIess
since lPQfless I (I/A)/Qfl,,,
+ M.
+ A4 5
(l/l)ljl,,,
LEMMA 2.6. Let P be any operator of class S(A, M), L > 1. Then P has a bounded fixed density g and any such g satisfies lg],,, I AM/(1 - 1).
Proof.
By induction
+ M, and
n decreasing
we are able to show that (S2) implies
(11)
IP”fles,< (Wklfless + AM/U - 1) for any positive integer k, whenever sequence (A,(f)), given by
f is a bounded
A(f)
decreasing
density.
This implies
that the
:= $iOPkf
(12)
is uniformly supremum bounded and so weakly precompact in Li(Z, m). Therefore, by theorem 5.2.1 in [16], the sequence (12) converges strongly to some fixed point g of P such that
ILL 5 nMl(n - 1) as n + co. By lemma 2.4 the g so constructed is actually a decreasing function since each Cesaro sum A,(f) is a decreasing density because each of its summands Pkf is decreasing, since f is decreasing and P: DD -+ 339. n 2.7. If (f,) is a sequence of densities such that: (i) each f, is decreasing; (ii) 1f,I,,, I K, for some K independent of n; then (f,) is strongly precompact in L,(Z, m) and any limit point f of (f,) is also decreasing satisfies 1f less 5 K. LEMMA
and
1019
Invariant measures
Proof. We have Var(f,) I K, for each n; i.e. the set [f,) is of uniformly bounded variation, so Helly’s theorem applies, see [17]. n LEMMA 2.8. Let (PJ be a sequence of Markov operators on L,(m) with corresponding fixed densities (f,). If f is any density such that f, + f, and P any operator such that we have IlP,f - Pf (1+ 0, then f is a fixed point of P, i.e. Pf = f.
Proof. We have Ilf - Pf II 5 Ilf - f,ll + Ilf, - Pf II, andIlf, - Pf II = IlP,f, - Pf II 5 lkfn - P,f II + IPnf - Pf II 5 IlP,(fn- f)ll + IPnf - Pf II 5 Ilfn- fll + IPnf - Pf II. so Ilf - Pf II 5 2llfn-f II + IPnf - Pf II9for all n > 0. The hypotheses imply that Ilf - Pf 1)is
arbitrarily small, thus equal to zero.
n
Using the above lemmas we are able to show the following proposition. PROPOSITION 2.9. Let (P,) denote a sequence of S-bounded operators pointwise convergent, on %D, to an operator P. Then: (a) there is a sequence 5 := If,) of bounded decreasing normalized fixed points of P, ; (b) any such sequence 5 is strongly precompact in L, ; (c) any limit point of 5 is a bounded decreasing fixed point of P; (d) if P has a unique invariant density g the sequence 5 = (f,,) is convergent and f,, + g.
Proof. Lemma 2.6 implies the existence of such an 5, each member of which is uniformly bounded. So lemma 2.7 implies (b), and with lemma 2.8, implies (c), which in turn implies (d). n 3. CONVERGENCE
Heretofore
OF ULAM’S
APPROXIMATION
METHOD
in the literature the convergence of Ulam’s method has only been demonstrated
analytically for piecewise expanding maps; and even for these maps in general, and PCVX
maps in particular, these results would at best imply convergence of the method when applied to sufficiently large iterates of the respective maps; see [13]. We will show here, however, that Ulam’s method converges, in fact, for any piecewise convex transformation T with strong repellor even if it is not expanding-a result which supports the numerical evidence of a much wider applicability of the method; see [9]. We have the following theorem. Let Z = [0, 11. If T:I + I is piecewise convex with strong repellor, then any sequence of nonnegative normalized fixed points of P,, converges in L,(Z, m) to the density g of the unique absolutely continuous T-invariant measure; i.e. Ulam’s method converges. THEOREM 3.1.
Proof. PT is of class S(A, M) for d and A4 given by lemma 2.4; and it is clear that Q, maps to DaD, and IQ,f less5 If less,so by lemma 2.5, each P,, is also of class S(A, M). Now Ilf - Q,f II + 0 f or each f in L,(m), see [3; 18, theorem 8.181; so P, = QnPT + PT pointwise on all of L,(m). By theorem 2.3, PT has a unique invariant density g, therefore, by proposition 2.9 there is a sequence (f,) of normalized fixed points of P,, such that f, + g.
99
1020
W. M. MILLER
Now we show that for sufficiently large n, P,, , like P itself, has only one fixed nonnegative normalized eigenvector. Recall, by theorem 2.3, that P has a unique bounded fixed density g that is decreasing. As such its support, supp(g) := clos(x 1g(x) > 0], is an interval of the form [0,6], for some b > 0. Let S = [0, b] denote the support of g. We consider the following two cases.
Case 1 (b = 1). If g > 0 a.e. then by lemma 2.1 Tis irreducible, in the sense that if A c T-‘(A) then m(A) = 0 or m(A) = 1. This implies each P,, is irreducible (in the sense of matrix theory), and, therefore, has a unique fixed eigenvector f,; see [14, Section 11. So in the case of g > 0 we have convergence of Ulam’s method. Case 2 (0 < b < 1). We will show that S is contained in the interior of T-‘S. Now lemma 2.1, part (a), implies S c T-‘(S). However, if we have equality then S would be invariant and, thus, R := AS would be an invariant set of m-measure 1 - b > 0 which contradicts the asymptotic stability of Pralluded to in theorem 2.3 since iterates of densities supported on R would remain supported on R and, thus, never converge to g-a density supported on S. So S is properly contained in T-‘(S). Recall S = [0, b] is a closed interval, so U := T-‘(S) = T-‘[O, b] is just a union of Ui := z-‘[O, b], 1 I i 5 r. However, each q is a smooth monotone mapping of the interval [ai_l, ai) on to [0, bi) z [0, 11, (if i = r the interval from a,_, to a, is closed and we might have [0, b,] = [0, 1]), see [16, Section 6.31; so each Ui is an interval of the form [ai_ , pi], where pi < ai, if b < bi; or of the form [ai_,,/3J, pi = ai, if b > 6,; SO
LO,bl C
6 [ai-1 Pi1 9
i=l
whereai_,
fixed point f,, of P,, the restriction
off,
on [S] is uniquely
Proof. The part of f, on S is uniquely determined since the [S] x [S] block of P, is an irreducible stochastic sub-matrix. This is because S E T-‘S and any reducible sub-block of pixels in [S] corresponds to a nontrivial reducible proper subset of S, under T, which by part (c) of lemma 2.1 is not possible. See [14, Section I].
Invariant measures
1021
Therefore, for sufficiently large n, each P,, has a unique fixed nonnegative normalized eigenvector which is the& constructed above (see Section 2B), any sequence of which converges to the unique P,-invariant density g. n Recall that the Frobenius-Perron operator P of any PCVX map T has the following striking property: for any initial density f, Pkf -+ g, as k 4 00, i.e. P is asymptotically stable. Now from a computational point of view it would be fortuitous to know that the approximating operators, P,(T), inherited this property, because then we would have the option of numerically approximating their fixed points f, by iteration, since P,“f + f, , as k + 00, for any initial nonnegative normalized vector f; say f = 1, for example. We have the following positive result. PROPOSITION 3.3. The irreducible [S] x [S] Ulam sub-matrices of P, = P,(T) described above with unique fixed points f, are such that P,“f -+ f,, , as k --t 03. Proof. It suffices to show that the [S] x [S] sub-matrices of P,(T) are primitive. Since these sub-matrices are nonnegative and irreducible by a standard result in matrix theory we just have to show that their trace is positive; see [19, Section 8.51, for example. However, ([S] x [SJ),, = (P,Jll ,because S = [0, b]; and
[Pnll, :=
m(X, rl T,-IX,)
m(XJ
(13)
gives the fraction of the partition box X, = [0, xi] mapped by T into X1 . Therefore, (P,,), 1 > 0, since Tl : [0, al) 4 [0, b,) is a C2 increasing map satisfying T,(O) = 0. n 4. STABILITY OF INVARIANT MEASURES OF PCVX MAPS Various results in the literature imply other “stochastic stability” properties for PCVX maps-properties that imply approximability of their invariant measures by “methods” applied to sufficiently large iterates of these maps. The results here show that these other “methods” also work when applied to just the map itself, as opposed to some iterates. (A) Deterministic perturbations
of piecewise convex transformations
By theorem 2.3 we know that each member, T, of PCVX possesses a unique bounded decreasing density g*(T) corresponding to a unique absolutely continuous T-invariant measure. Here we consider the stability of invariant densities of piecewise convex transformations with respect to perturbations in PCVX. For example, as in [4], introduce the following metric on PCVX: for Tl and T2 in PCVX, let dist(T, , T,) := infle > 0 1for some A E I, cr: Z + I, such that m(A) > 1 - E, (T a diffeomorphism, T,lA = T, 0 oIA, and for all x in A: 1CT(X)- XI < E and 1l/a’(x) - 1I < E). We then have the following result due to Keller.
(14)
1022
W. M.
MILLER
LEMMA 4.1. If P, and Pz are the Frobenius-Perron operators of two transformations in PCVX, then for any function f of bounded variation, we have
T, and T2
lipIf- Sf 11,I: 12*dist(T,,G) . IifIIy, whereIlfIIv := Ilflll + Var(f) Proof. See [4, Section 31.
denotes
the variation
(BV) norm
(15)
off.
n
We are now able to show that the map T + g*(T) from the metric space PCVX to L,(I) has the following continuity property. THEOREM
4.2. Let T, (T,] be piecewise convex transformations Frobenius-Perron operators; and g*, (g,] their corresponding unique is S-bounded and dist(T,, T) --t 0 then g, 4 g* in L,(I).
in PCVX; P, (P,) their invariant densities. If {P,)
Proof. By lemma 4. I the hypothesis that dist(T,, T) -+ 0 implies P,, + P pointwise on Da>, since a>% is a subset of BV. By hypothesis (P,) is S-bounded and theorem 2.3 implies P has a unique stationary density; so proposition 2.9 implies the conclusion of the theorem as stated. n (B) Stochastic stability under random perturbations Here we show stochastic stability under random perturbations of piecewise convex transformations. These perturbations model the presence of error or the addition of noise to the unperturbed dynamical system governed by the transformation T. The perturbed situation may be described as a particle jumping from x to y = TX and then dispersing randomly near y with some localized distribution given by a density K, = &(y, z) defined in an E neighbourhood of y; see [15]. More precisely we let K,(y, z) denote a given bistochastic kernel, see [4]; this induces a Markov process on I which induces a corresponding evolution of densities given by
Q,fW := IfW *Ke@, xl du. Now the time evolution the Frobenius-Perron stochastically perturbed
of densities for the unperturbed deterministic process T is governed by the time evolution of densities for the operator P. Therefore, process described here is given by P,f := Q,Pf, so
P,f(x) := 1I
Pf(u) * &@, 4 du
see [4, 5, I 51. It turns out that we are able to exploit the special property if the Q8 operators satisfy the following conditions.
Condition 1. Q, is of class S(1, 0), i.e. (a)
Q,: zm + 99 and@I IQ,fless5 if less.
Condition 2. ~~ K,(u,x)
(16)
= 6,(u), i.e. Fz Q,f = f.
(17) of our transformations
1023
Invariant measures
Remark 1. It is not hard to show that any Markov satisfies
condition
process
defined
by a bistochastic
kernel K
1 (a).
Remark 2. Conditions
1 (b) and (2) are satisfied, for example, by the following bistochastic kernels: (a) the kernel K,(x, y) describing Ulam’s method described above, see [4, Section 41; (b) the kernel KE(u, x) = (l/m@,(u))) - lBecuj(x)which models a uniform random dispersion in an c-neighbourhood of u, (see [5]); where to accommodate for the boundary of [0, l] we have
m@,(u))
=
U+&
ifOlu
2E
ife52451--e
1 l--U+& (c) the kernel K,(x, y) describing see [9, 121. We have the following stability first part of theorem 3.1.
ifl-s
a continuous
result,
(18)
piecewise
linear version
of Ulam’s
the proof of which is exactly analogous
method;
to that of the
THEOREM 4.3. Let T: [0, l] + [0, l] be a piecewise convex transformation with strong repellor and T, denote a family of random perturbations of T with transition probability densities p,(x,y) = K,(Tx,y). Assume the bistochastic kernels K, satisfy the assumptions outlined above. Then for each E > 0 there exists a bounded, decreasing c-invariant density f, and f, -+ g* in L,([O, 11, m), as E + 0, where g* is the density of the unique T-invariant measure.
(C) Randomly
occurring deterministic perturbations
The model of “noise” or “error” here involves the application of different piecewise convex transformations chosen at random, as in [20], which may be interpreted as random perturbations of the viewer of the unperturbed dynamical system (T, Z, m). As above this random process on points induces a deterministic dynamical system on the space of densities and we consider the stability of the density of the unperturbed T-invariant measure with respect to the indicated perturbations. We describe the situation more precisely. Let T,, o an element of a measure space of parameters (a, 5), denote a family of nonsingular transformations on a measure space (I, m). For each o in Q let P, denote the Frobenius-Perron operator corresponding to the transformation T, on I. Given a probability measure p on Q let P,: 9(Z, m) + a>(Z,m) be defined by
P,f(x)
:= 5n
P,f(x)
* &(a).
(19)
T, from the The Markov process, x~+~ = T,(x,,), on Z, of randomly selecting a transformation family of transformations (T, : o in Q) to act on the space Z at time n + 1, with probability of selection given by the measure p on Q, induces a time evolution on D(Z, m) which is given by Pfi; see [4, 201. We have the following
straightforward
lemma.
W. M. MILLER
1024
LEMMA 4.4. If for each o in CJ, P, is of class S(A, M), then for any probability Q P, is also of class S(A, M). Proof. For any bounded decreasing family of decreasing densities, P,f bounded. H LEMMA 4.5. If v is the point iIP,f
Proof.
We have P,f
measure
,D on
density f, P,f being an average, with respect to p, of a I (l/A)1 f less + M, is itself decreasing, and likewise
mass at o,, E Q, then - P,f
- P,f
12llfiv
iI1 5
disUk7T,,,)*ddw).
= Sn (P, f - Pa0 f) . d&),
ii&f - Pvfl/~:=
I
iP,f(x)
- Pvf(x)I
IP,f(x)
5
a
IlP,f
SO * dm(x)
- P,,fW
- P,,f
(20)
* W-G
* d/do>
111* Mw)
12llf11 y * disttir,, T,J - @to),
I cl
by lemma
4.1.
n
Now for measures
p and v as above,
define dist(T, , T,J * dp(o).
6(& v) := n Then we have the stability
result contained
THEOREM 4.6. Let ,u,, be a sequence
in the following
of measures sequence of Markov operators as defined above If (P, I w E ~2) is S-bounded and a(,~,, v) + 0 fn + g*, where g* is the density of the unique
(21)
theorem.
on (C& 5) and P,, := Pp. be the corresponding in (19). Let v denote the point mass at o,, E a. then each P,, has a fixed density f., such that TUO-invariant measure.
Proof. By lemma 4.4, P,, and each P,, are of class S(I, M), for some A > 1 and M > 0, and PO, has a unique invariant density, by theorem 2.3; so to invoke proposition 2.9 it suffices that W p, + p,,, pointwise on a>%,, which is the case by lemma 4.5. Acknowledgement-This
work was partially supported by a University of Maryland post-doctoral grant.
Invariant
measures
1025
REFERENCES 1. ECKMANN J. P. & RUELLE D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57,617-656 (1985). 2. ULAM S., Problems in Modern Mathematics. Interscience, New York (1960). 3. LI T.-Y., Finite approximation for the Frobenius-Perron operator. A solution to Ulam’s conjecture, J. Approx. Theory 17, 177-186 (1976). 4. KELLER G., Stochastic stability in some chaotic dynamical systems, A4h. Math. 94, 313-333 (1982). 5. BLANK M. L., Stochastic properties of deterministic dynamical systems, Sov. Sci. Rev. C. Math./Phys. 6, 243-271 (1987). 6. GORA P. & BOYARSKY A., Compactness of invariant densities for families of expanding, piecewise monotonic .’ : 7*I transformations (to appear). 7. HUNT F. Y., A Monte-Carlo approach to the approximation of invariant measures, ./National’ Institute of Standards and Technology Internal Report, NISTIR 4980’(1993). 8. KELLER G., Stochastic perturbations of some strange attractors, Lecture Notes in Physics, Vol. 79, pp. 192-193. Springer, New York (1982). 9. DING J. & LI T.-Y., Markov finite approximation of the Frobenius-Perron operator, Nonlinear Analysis 17, 759-772 (1991). 10. LASOTA A. & YORKE J., Exact dynamical systems and the Frobenius-Perron operator, Trans. Am. math. Sot. 273, 375-384 (1982). 11. KOWALSKI Z. S., Invariant measures for piecewise monotonic transformations, Lecture Notes in Mathematics, Vol. 472, pp. 77-94. Springer, New York (1975). 12. CHIU C., DU Q. & LI T.-Y., Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Analysis 19, 291-308 (1992). 13. JABLONSKI M. & MALCZAK J., The rate of convergence of iterates of the Frobenius-Perron operator for piecewise convex transformations, Bull. Acad. Polon. Sci. 31, 250-253 (1983). 14. HUNT F. Y. &MILLER W. M., On the approximation of invariant measures, J. Stat. Phys. 66, 535-548 (1992). 15. KIFER Y., Random Perturbations of Dynamical Systems. Birkhauser, Boston (1988). 16. LASOTA A. & MACKEY M., Probabilistic Properties of Deterministic Systems. Cambridge University Press, New York (1985). 17. NATANSON I. P., Theory of Functions of a Real Variable, Vol. 1, revised edition. Ungar, New York (1961). 18. DUNFORD N. & SCHWARTZ J., Linear Operators, Part 1. Wiley, New York (1988). 19. HORN R. & JOHNSON C., Matrix Analysis. Cambridge University Press, New York (1985). 20. KIFER Y., Ergodic Theory of Random Transformations. Birkhauser, Boston (1986).
Analysis,
Theory,
Methods
& Applications,
Pergamon
Vol. 23, No. 8, pp. 1013-1025, 1994 Copyright 0 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0362-546X/94 $7X0+ .@I
0362-546X(93)E0031-W
STABILITY AND APPROXIMATION OF INVARIANT MEASURES FOR A CLASS OF NONEXPANDING TRANSFORMATIONS WALTER Department (Received
23 March
of Mathematics,
Howard
M. MILLER University,
1993; received in revised form 4 October
Key words and phruses: Frobenius-Perron convex transformation, irreducible.
operator,
Washington,
DC 20059, U.S.A.
1993; received for publication
invariant
measures,
Ulam’s
20 December
method,
1993)
piecewise
1. INTRODUCTION
Invariant measures for a dynamical system give a statistical description of its long term behaviour-information which is especially useful when the system is considered “chaotic”. As such the determination of these measures is of practical as well as theoretical significance; especially those measures which enjoy some character of stability and are thus in some sense “physical”; see [l]. finite approximation method-the In 1960, Ulam conjectured that a certain “natural” approximants of which are fixed eigenvectors of stochastic matrices-should converge to an invariant measure of a self-transformation of an interval; see [2]. Later, using a key inequality of Lasota and Yorke relating the variation of any initial density to that of its transformation, Li, in [3], gave sufficient conditions guaranteeing just such convergence. He showed the following: if a piecewise C2 transformation T of an interval Z satisfies J := infl T’( > 2 then Ulam’s method, applied to T, converges strongly in L,(Z) to the density of a T-invariant measure. When T is just expanding (J > l), his method, as well as the results of Keller [4] and Blank [5], imply the convergence of Ulam’s method for sufficiently large iterates, Tk, of T. An advantage of Ulam’s approach is the robustness of its numerical implementation, see [3]. Arguably, however, as noted by Gora and Boyarsky in [6], there are two drawbacks of the method. One regards the difficulty, in general, of computing the approximating Ulam matrices themselves. This issue is addressed by Hunt in [7]. Another regards the restriction J > 2. Several authors, notably Keller in [8], conjecture that this restriction is not necessary. This is borne out by the numerical evidence of Ding and Li in [9], which in fact suggests a far wider applicability of the method, even for nonexpanding maps. We address this latter issue here, and in Section 3 we show the convergence of Ulam’s method for a class of piecewise monotone maps of the interval that are not necessarily expanding, i.e. for which J < 1. These are the piecewise convex (PCVX) maps of an interval Z introduced by Lasota and Yorke in [lo]. For these maps we are able to circumvent the “obstruction” J < 1 by taking advantage of a crucial property-their corresponding Frobenius-Perron operators, as transformations on the subspace of densities in L,(Z, m), leave invariant the positive cone of decreasing functions. This cone will, for us, serve a similar purpose as the subspace BI’ of functions of bounded variation, does for Li in [3], Kowalski in [ll], and Keller in [4]. Our strategy is as follows: first using the cone-invariance property alluded to above, we demonstrate the existence of a convergent family of fixed points f,, of P,,(T). Then to show 1013
1014
W. M. MILLER
that any family of nonnegative normalized fixed points of P,(T) are convergent, we show that there is in fact only one such family, i.e. we show uniqueness. It is important to note that we are unable to invoke the results as in [3,4] to show existence, or the techniques used in [12], for example, to show uniqueness, because Tis not necessarily expanding, even though by [13] some sufficiently large iterate, Tk, is. Instead we use the more general “ergodic” notion of irreducibility as introduced by Hunt and Miller in [14]; see lemma 2.1 and corollary 2.2 in Section 2. In this regard our methods and those in [12] may be viewed as complementary. Let us also note that as a corollary of our method we are able to deduce a result of some computational significance-the approximating densities, f, , may be numerically computed by repeated iteration of the approximating matrices P,(T) themselves. Finally, as Keller notes in [4] the convergence of Ulam’s method may be viewed as a type of stability of the invariant measure-a stability that allows it to be approximable by this particular method when applied to a particular map. Now by the results in [4, 11, 151 one sees that this and other “stochastic stability” properties hold for piecewise expanding maps in general, and, therefore, by [13], for sufficiently large iterates of PCVX maps, in particular. Is one, as shown in Section 3 for Ulam’s method, also able to approximate the invariant measure of a PCVX map by these other methods when applied to the map itself as opposed to a sufficiently large iterate of the map? In Section 4 we give a positive answer to this question and following [4] we show invariant measures of the class of PCVX maps are stable with respect to three types of perturbations: deterministic perturbations of piecewise convex transformations, perturbations, occurring deterministic stochastic and randomly perturbations-again making crucial use of the cone-invariance property mentioned above. Note that these cases have special physical significance, as they serve to model the “real world” effects of “noise” or “error” on the determination of “observables” of the system by the viewer.
2. PRELIMINARIES
(A) Frobenius-Perron
operators,
invariant
AND
measures
LEMMAS
and Ulam ‘s method
An extremely useful tool for investigating invariant measures of a nonsingular transformation T on a measure space (X, m) is the Frobenius-Perron (FP) operator P = PT corresponding to T. The action of Ton points of X induces the action of P on density functions on X, which may be characterized as follows. For any measurable set A and f E L,(X, m) we have
(1) P is a positive linear Markov operator on L,(X, m), mapping densities to densities, and any positive normalized fixed point of P corresponds to an absolutely continuous (with respect to m) T-invariant measure on X. By the Birkhoff individual ergodic theorem we know that invariant measures encode statistical properties of the asymptotics of the dynamical system (T, X, m); thus, the keen interest in the determination of fixed points of P; see [16]. In 1960 Ulam, in [2], proposed the following method for approximating fixed points of P. For a given (equi)partition II, of the space X into 1X,, X, , . . . , X,J define a row stochastic
Invariant
matrix
P, , the (i, j)th
entry of which gives the fraction
[PJij:=
1015
measures
m(Xi fl
of Xi that is mapped
by T to Xj
T-'Xj)
m(xi)
.
P,, is in a sense a finite approximation to the transformation T, but more precisely it is a finite Markov approximation to the corresponding Frobenius-Perron operator, P, on the space of densities ‘Z) = a>(X) in ,5,(X, m). It is not hard to show, by the Brouwer fixed point theorem, for example, that the Ulam matrix, P,, , corresponding to each equipartition, II,, has at least one left fixed eigenvector f,, which corresponds to a piecewise constant (with respect to the given partition) density. Ulam conjectured that the sequence of fixed points, f,, , of the matrices P, corresponding to densities on finer and finer partitions of X, converge to a fixed point of the Frobenius-Perron operator itself. Markov Given a partition II, = (Xj Ij = 1, . . . , n) of X let Q, denote the corresponding projection operator on L,(X) where for any f in L,(X), Q,f is the piecewise constant function whose value at x equals the average value off on that pixel of the partition containing x
Q,f := i i=l
Ci*
Ix,,
Ci:=&*Sx,fdm-
We have the following fundamental relation between partition and the Frobenius-Perron operator P
PnQnf = Q, Pf
the Ulam
(3) matrices
P,, defined
by a
(4)
for any f in L,(X); see [3, 141. Now let f *be a P-invariant density. We will say that P is ergodic (w.r.t. f *)if for any density f the sequence {Pkf 1k L 0) is weakly Cesaro convergent, in L,(x, m), to f *; this implies that f * is the unique density such that Pf * = f *; see [ 16, Section 41. We will say T is ergodic (w.r.t. f *) if PT is likewise ergodic. We have the following rather general approximation result which as realized in the subsequent corollary relates the support of the Ulam eigenvectors f,, to that of an ergodic P-invariant density f *. Note that the stated set equalities and inclusions hold mod m.
LEMMA 2.1. Let P = Pr: L,(X, m) -+ L,(X, m) denote the Frobenius-Perron operator, with fixed density f *, of a nonsingular ergodic transformation Ton a measure space (X, m). Given an (equi)partition II, = 1Xj: 1 5 j 5 n) of X let Q, denote the corresponding piecewise constant averaging operator; P,, = QnPT the corresponding Markov finite approximation to PT; and f, any corresponding nonnegative normalized fixed point, f,, = P,, f, . Let S denote the support off * and S, the support off,, . Then: (a) S c T-‘S; (b) S,, E T-‘S,,. If A is any set of positive measure such that A E T-‘A then (c) SEA. If A is any union of partition boxes such that A & T-IA then (d) S, EA.
1016
W. M. MILLER
Proof of (a). By definition
1 = Isf’dm and 1 = ST-IS f *dm implies Proof of(b).
= iilsf*drn
= isPrf*dm n
S E T-‘S.
Here by the P,,-invariance
off,
1 = isnfndm But since S,, is a union
off *, we have
of S and the P-invariance
and the definition
= jsnQ.PTfndm.
of pixels of II, we have, by definition
i’sn
of S,, we have
Q,,gdm
=
of Q,,
for any g
gdm i S”
(5)
so we have I= and 1 = ST-as, f,dm Proof of(c).
Let
P,f, QnPTfndm = 53, .i sn
fA := (l/m(A))
f, dm T_‘S,
n
S, C T-IS,,.
implies
dm =
. IA, then since A C T-IA
we have
~=ilf~dm~i’T_,*f~dm=ilPf,dmr---r%Pkf~dm by induction,
for any integer
k I 0. However,
1 I jA P”f, dm implies PkfA 1, dm.
Ergodicity
implies _ :j,
jX
f*l,dm
PkfA 1, dm +
=
iX
as n + 00. So 1 I jA f * dm, which implies
f*dm i A
S E A.
H
Proof of(d). Assume A E T-‘A. This implies for R := XL4 that T-‘R E R which implies S, fI T-‘R G S, fl R. Letting B := S,, fl R, it suffices to show that m(B) = 0. We have f,dm the last equality
holding
f,dm
=
=
P,f,dm
since R is a union
= sR
sR
s T-‘R
c &I-IT-‘R
of partition
QnPTfndm
boxes of n,.
However,
Q,PTf,, =
Ptzfn = f* 9 so f,dm s S,i’T-‘R
= i,fi,dm
= isnnRfndm.
(6)
Invariant
This equality (6) combined with S, (l T-‘R
G S, fl R implies
S,,fl T-'R = S,nR, since f, > 0 on S, . Therefore, B = so B E T-‘B. S&B. W
1017
measures
(mod m)
(7)
by the above equality (7) and the inequality in part (b),
s, n R = s, n T-'R E ~-ls,n T-1~ c T-$9, no) = T-'B
However, by part (c) above this implies m(B) = 0 since B E x‘\S precludes
COROLLARY 2.2. If the hypotheses on Tare as above then (e) S E S,. Moreover, if T is such that S E int(T-‘S) then for sufficiently large n, i.e. for any sufficiently fine partition II,, (f) s E s, E [S], where [S] denotes the partition neighbourhood of S,
Proof. Since S 5 int(T-‘S), if II,, is sufficiently fine then [S] G int(T-‘S) c T-'S c T-'[S],so [S]E T-'[S].Therefore, lemma 2.1, part (d) implies S, E [S], since [S] is a union of partition boxes. That S E S, holds is implied by parts (b) and (c) of the same lemma. n (B) Piecewise convex transformations
and their Frobenius-Perron
operators
We will be interested in the following class of maps. A transformation T: [0, l] + [0, l] is said to be piecewise convex with strong repeller (PCVX) if T satisfies the following conditions: (i) there is a partition 0 = a, < a, < ..a < a, = 1 of [0, l] such that for each integer i= 1, ..-, r the restriction of T to [ai_l, ai) is a C2-function; (ii) T'(x)> 0 and T"(x)2 0 for all x in [0, l), [T’(aJ and T”(aJ are right derivatives]; (iii) for each integer i = 1, . . . , r we have T(ai_ J = 0; (iv) I := T'(0)> 1. This class of transformation was introduced by Lasota and Yorke in [lo] where they showed the following theorem. THEOREM2.3. Let a transformation T satisfy (i)-(iv) above. Then there exists a unique normalized absolutely continuous (w.r.t. m) invariant measure pg that is invariant under T. Its density g is bounded, decreasing and ergodic; in fact P is asymptotically stable, i.e. lim P”f = g, as n + 00, for any initial density f, where m denotes standard Bore1 measure on [0, l] and P = PT is the Frobenius-Perron operator corresponding to T. Next we consider the following abstract set-up, the relevance of which will be made clear in the lemma immediately following. Let 9% denote the subset of &(I, m) consisting of bounded decreasing densities. We will say that a Markov operator P: L,(Z, m) -+ L,(Z, m) is of class S if: (Sl) P: a>zD--t L-&D,; (S2) for some I > 1 and A4 > 0, we have for any f in ED IPf L
5 wn)lf less+ A4
(9)
1018
W.M.MlLLER
where If I,,, denotes the (essential) supremum off (with respect to m). For a particular A and A4 we will say that P is of class s(A, M), and a sequence (P,) of Markov operators will be said to be S-bounded if for some I and M, independent of n, each P, is of class S(A, M). It turns out that the FP-operators of the piecewise convex transformations above are of class S. More precisely we have the following lemma. LEMMA 2.4 (Lasota and Yorke). where A := T’(0) and
If T satisfies (i)-(iv) then its FP-operator
M :=
c
PT is of class S(A, M)
(10)
(g;(O)/aj_,).
i=2
Proof.
See [lo; 16, Section
For an abstract
operator
6.31.
W
of class S one is able to show the following
lemma.
2.5. Let P be of class S(I, M). If Q is of class S(1, 0), i.e. (i) Q: 99 -+ DD; and (ii) ID],,, 5 IfI,,,, then QP and PQ are of class S(I, M).
LEMMA
Proof.
IQWI,,,
This is straightforward,
5 Ipfi,,, 5 WVlfIess
since lPQfless I (I/A)/Qfl,,,
+ M.
+ A4 5
(l/l)ljl,,,
LEMMA 2.6. Let P be any operator of class S(A, M), L > 1. Then P has a bounded fixed density g and any such g satisfies lg],,, I AM/(1 - 1).
Proof.
By induction
+ M, and
n decreasing
we are able to show that (S2) implies
(11)
IP”fles,< (Wklfless + AM/U - 1) for any positive integer k, whenever sequence (A,(f)), given by
f is a bounded
A(f)
decreasing
density.
This implies
that the
:= $iOPkf
(12)
is uniformly supremum bounded and so weakly precompact in Li(Z, m). Therefore, by theorem 5.2.1 in [16], the sequence (12) converges strongly to some fixed point g of P such that
ILL 5 nMl(n - 1) as n + co. By lemma 2.4 the g so constructed is actually a decreasing function since each Cesaro sum A,(f) is a decreasing density because each of its summands Pkf is decreasing, since f is decreasing and P: DD -+ 339. n 2.7. If (f,) is a sequence of densities such that: (i) each f, is decreasing; (ii) 1f,I,,, I K, for some K independent of n; then (f,) is strongly precompact in L,(Z, m) and any limit point f of (f,) is also decreasing satisfies 1f less 5 K. LEMMA
and
1019
Invariant measures
Proof. We have Var(f,) I K, for each n; i.e. the set [f,) is of uniformly bounded variation, so Helly’s theorem applies, see [17]. n LEMMA 2.8. Let (PJ be a sequence of Markov operators on L,(m) with corresponding fixed densities (f,). If f is any density such that f, + f, and P any operator such that we have IlP,f - Pf (1+ 0, then f is a fixed point of P, i.e. Pf = f.
Proof. We have Ilf - Pf II 5 Ilf - f,ll + Ilf, - Pf II, andIlf, - Pf II = IlP,f, - Pf II 5 lkfn - P,f II + IPnf - Pf II 5 IlP,(fn- f)ll + IPnf - Pf II 5 Ilfn- fll + IPnf - Pf II. so Ilf - Pf II 5 2llfn-f II + IPnf - Pf II9for all n > 0. The hypotheses imply that Ilf - Pf 1)is
arbitrarily small, thus equal to zero.
n
Using the above lemmas we are able to show the following proposition. PROPOSITION 2.9. Let (P,) denote a sequence of S-bounded operators pointwise convergent, on %D, to an operator P. Then: (a) there is a sequence 5 := If,) of bounded decreasing normalized fixed points of P, ; (b) any such sequence 5 is strongly precompact in L, ; (c) any limit point of 5 is a bounded decreasing fixed point of P; (d) if P has a unique invariant density g the sequence 5 = (f,,) is convergent and f,, + g.
Proof. Lemma 2.6 implies the existence of such an 5, each member of which is uniformly bounded. So lemma 2.7 implies (b), and with lemma 2.8, implies (c), which in turn implies (d). n 3. CONVERGENCE
Heretofore
OF ULAM’S
APPROXIMATION
METHOD
in the literature the convergence of Ulam’s method has only been demonstrated
analytically for piecewise expanding maps; and even for these maps in general, and PCVX
maps in particular, these results would at best imply convergence of the method when applied to sufficiently large iterates of the respective maps; see [13]. We will show here, however, that Ulam’s method converges, in fact, for any piecewise convex transformation T with strong repellor even if it is not expanding-a result which supports the numerical evidence of a much wider applicability of the method; see [9]. We have the following theorem. Let Z = [0, 11. If T:I + I is piecewise convex with strong repellor, then any sequence of nonnegative normalized fixed points of P,, converges in L,(Z, m) to the density g of the unique absolutely continuous T-invariant measure; i.e. Ulam’s method converges. THEOREM 3.1.
Proof. PT is of class S(A, M) for d and A4 given by lemma 2.4; and it is clear that Q, maps to DaD, and IQ,f less5 If less,so by lemma 2.5, each P,, is also of class S(A, M). Now Ilf - Q,f II + 0 f or each f in L,(m), see [3; 18, theorem 8.181; so P, = QnPT + PT pointwise on all of L,(m). By theorem 2.3, PT has a unique invariant density g, therefore, by proposition 2.9 there is a sequence (f,) of normalized fixed points of P,, such that f, + g.
99
1020
W. M. MILLER
Now we show that for sufficiently large n, P,, , like P itself, has only one fixed nonnegative normalized eigenvector. Recall, by theorem 2.3, that P has a unique bounded fixed density g that is decreasing. As such its support, supp(g) := clos(x 1g(x) > 0], is an interval of the form [0,6], for some b > 0. Let S = [0, b] denote the support of g. We consider the following two cases.
Case 1 (b = 1). If g > 0 a.e. then by lemma 2.1 Tis irreducible, in the sense that if A c T-‘(A) then m(A) = 0 or m(A) = 1. This implies each P,, is irreducible (in the sense of matrix theory), and, therefore, has a unique fixed eigenvector f,; see [14, Section 11. So in the case of g > 0 we have convergence of Ulam’s method. Case 2 (0 < b < 1). We will show that S is contained in the interior of T-‘S. Now lemma 2.1, part (a), implies S c T-‘(S). However, if we have equality then S would be invariant and, thus, R := AS would be an invariant set of m-measure 1 - b > 0 which contradicts the asymptotic stability of Pralluded to in theorem 2.3 since iterates of densities supported on R would remain supported on R and, thus, never converge to g-a density supported on S. So S is properly contained in T-‘(S). Recall S = [0, b] is a closed interval, so U := T-‘(S) = T-‘[O, b] is just a union of Ui := z-‘[O, b], 1 I i 5 r. However, each q is a smooth monotone mapping of the interval [ai_l, ai) on to [0, bi) z [0, 11, (if i = r the interval from a,_, to a, is closed and we might have [0, b,] = [0, 1]), see [16, Section 6.31; so each Ui is an interval of the form [ai_ , pi], where pi < ai, if b < bi; or of the form [ai_,,/3J, pi = ai, if b > 6,; SO
LO,bl C
6 [ai-1 Pi1 9
i=l
whereai_,
fixed point f,, of P,, the restriction
off,
on [S] is uniquely
Proof. The part of f, on S is uniquely determined since the [S] x [S] block of P, is an irreducible stochastic sub-matrix. This is because S E T-‘S and any reducible sub-block of pixels in [S] corresponds to a nontrivial reducible proper subset of S, under T, which by part (c) of lemma 2.1 is not possible. See [14, Section I].
Invariant measures
1021
Therefore, for sufficiently large n, each P,, has a unique fixed nonnegative normalized eigenvector which is the& constructed above (see Section 2B), any sequence of which converges to the unique P,-invariant density g. n Recall that the Frobenius-Perron operator P of any PCVX map T has the following striking property: for any initial density f, Pkf -+ g, as k 4 00, i.e. P is asymptotically stable. Now from a computational point of view it would be fortuitous to know that the approximating operators, P,(T), inherited this property, because then we would have the option of numerically approximating their fixed points f, by iteration, since P,“f + f, , as k + 00, for any initial nonnegative normalized vector f; say f = 1, for example. We have the following positive result. PROPOSITION 3.3. The irreducible [S] x [S] Ulam sub-matrices of P, = P,(T) described above with unique fixed points f, are such that P,“f -+ f,, , as k --t 03. Proof. It suffices to show that the [S] x [S] sub-matrices of P,(T) are primitive. Since these sub-matrices are nonnegative and irreducible by a standard result in matrix theory we just have to show that their trace is positive; see [19, Section 8.51, for example. However, ([S] x [SJ),, = (P,Jll ,because S = [0, b]; and
[Pnll, :=
m(X, rl T,-IX,)
m(XJ
(13)
gives the fraction of the partition box X, = [0, xi] mapped by T into X1 . Therefore, (P,,), 1 > 0, since Tl : [0, al) 4 [0, b,) is a C2 increasing map satisfying T,(O) = 0. n 4. STABILITY OF INVARIANT MEASURES OF PCVX MAPS Various results in the literature imply other “stochastic stability” properties for PCVX maps-properties that imply approximability of their invariant measures by “methods” applied to sufficiently large iterates of these maps. The results here show that these other “methods” also work when applied to just the map itself, as opposed to some iterates. (A) Deterministic perturbations
of piecewise convex transformations
By theorem 2.3 we know that each member, T, of PCVX possesses a unique bounded decreasing density g*(T) corresponding to a unique absolutely continuous T-invariant measure. Here we consider the stability of invariant densities of piecewise convex transformations with respect to perturbations in PCVX. For example, as in [4], introduce the following metric on PCVX: for Tl and T2 in PCVX, let dist(T, , T,) := infle > 0 1for some A E I, cr: Z + I, such that m(A) > 1 - E, (T a diffeomorphism, T,lA = T, 0 oIA, and for all x in A: 1CT(X)- XI < E and 1l/a’(x) - 1I < E). We then have the following result due to Keller.
(14)
1022
W. M.
MILLER
LEMMA 4.1. If P, and Pz are the Frobenius-Perron operators of two transformations in PCVX, then for any function f of bounded variation, we have
T, and T2
lipIf- Sf 11,I: 12*dist(T,,G) . IifIIy, whereIlfIIv := Ilflll + Var(f) Proof. See [4, Section 31.
denotes
the variation
(BV) norm
(15)
off.
n
We are now able to show that the map T + g*(T) from the metric space PCVX to L,(I) has the following continuity property. THEOREM
4.2. Let T, (T,] be piecewise convex transformations Frobenius-Perron operators; and g*, (g,] their corresponding unique is S-bounded and dist(T,, T) --t 0 then g, 4 g* in L,(I).
in PCVX; P, (P,) their invariant densities. If {P,)
Proof. By lemma 4. I the hypothesis that dist(T,, T) -+ 0 implies P,, + P pointwise on Da>, since a>% is a subset of BV. By hypothesis (P,) is S-bounded and theorem 2.3 implies P has a unique stationary density; so proposition 2.9 implies the conclusion of the theorem as stated. n (B) Stochastic stability under random perturbations Here we show stochastic stability under random perturbations of piecewise convex transformations. These perturbations model the presence of error or the addition of noise to the unperturbed dynamical system governed by the transformation T. The perturbed situation may be described as a particle jumping from x to y = TX and then dispersing randomly near y with some localized distribution given by a density K, = &(y, z) defined in an E neighbourhood of y; see [15]. More precisely we let K,(y, z) denote a given bistochastic kernel, see [4]; this induces a Markov process on I which induces a corresponding evolution of densities given by
Q,fW := IfW *Ke@, xl du. Now the time evolution the Frobenius-Perron stochastically perturbed
of densities for the unperturbed deterministic process T is governed by the time evolution of densities for the operator P. Therefore, process described here is given by P,f := Q,Pf, so
P,f(x) := 1I
Pf(u) * &@, 4 du
see [4, 5, I 51. It turns out that we are able to exploit the special property if the Q8 operators satisfy the following conditions.
Condition 1. Q, is of class S(1, 0), i.e. (a)
Q,: zm + 99 and@I IQ,fless5 if less.
Condition 2. ~~ K,(u,x)
(16)
= 6,(u), i.e. Fz Q,f = f.
(17) of our transformations
1023
Invariant measures
Remark 1. It is not hard to show that any Markov satisfies
condition
process
defined
by a bistochastic
kernel K
1 (a).
Remark 2. Conditions
1 (b) and (2) are satisfied, for example, by the following bistochastic kernels: (a) the kernel K,(x, y) describing Ulam’s method described above, see [4, Section 41; (b) the kernel KE(u, x) = (l/m@,(u))) - lBecuj(x)which models a uniform random dispersion in an c-neighbourhood of u, (see [5]); where to accommodate for the boundary of [0, l] we have
m@,(u))
=
U+&
ifOlu
2E
ife52451--e
1 l--U+& (c) the kernel K,(x, y) describing see [9, 121. We have the following stability first part of theorem 3.1.
ifl-s
a continuous
result,
(18)
piecewise
linear version
of Ulam’s
the proof of which is exactly analogous
method;
to that of the
THEOREM 4.3. Let T: [0, l] + [0, l] be a piecewise convex transformation with strong repellor and T, denote a family of random perturbations of T with transition probability densities p,(x,y) = K,(Tx,y). Assume the bistochastic kernels K, satisfy the assumptions outlined above. Then for each E > 0 there exists a bounded, decreasing c-invariant density f, and f, -+ g* in L,([O, 11, m), as E + 0, where g* is the density of the unique T-invariant measure.
(C) Randomly
occurring deterministic perturbations
The model of “noise” or “error” here involves the application of different piecewise convex transformations chosen at random, as in [20], which may be interpreted as random perturbations of the viewer of the unperturbed dynamical system (T, Z, m). As above this random process on points induces a deterministic dynamical system on the space of densities and we consider the stability of the density of the unperturbed T-invariant measure with respect to the indicated perturbations. We describe the situation more precisely. Let T,, o an element of a measure space of parameters (a, 5), denote a family of nonsingular transformations on a measure space (I, m). For each o in Q let P, denote the Frobenius-Perron operator corresponding to the transformation T, on I. Given a probability measure p on Q let P,: 9(Z, m) + a>(Z,m) be defined by
P,f(x)
:= 5n
P,f(x)
* &(a).
(19)
T, from the The Markov process, x~+~ = T,(x,,), on Z, of randomly selecting a transformation family of transformations (T, : o in Q) to act on the space Z at time n + 1, with probability of selection given by the measure p on Q, induces a time evolution on D(Z, m) which is given by Pfi; see [4, 201. We have the following
straightforward
lemma.
W. M. MILLER
1024
LEMMA 4.4. If for each o in CJ, P, is of class S(A, M), then for any probability Q P, is also of class S(A, M). Proof. For any bounded decreasing family of decreasing densities, P,f bounded. H LEMMA 4.5. If v is the point iIP,f
Proof.
We have P,f
measure
,D on
density f, P,f being an average, with respect to p, of a I (l/A)1 f less + M, is itself decreasing, and likewise
mass at o,, E Q, then - P,f
- P,f
12llfiv
iI1 5
disUk7T,,,)*ddw).
= Sn (P, f - Pa0 f) . d&),
ii&f - Pvfl/~:=
I
iP,f(x)
- Pvf(x)I
IP,f(x)
5
a
IlP,f
SO * dm(x)
- P,,fW
- P,,f
(20)
* W-G
* d/do>
111* Mw)
12llf11 y * disttir,, T,J - @to),
I cl
by lemma
4.1.
n
Now for measures
p and v as above,
define dist(T, , T,J * dp(o).
6(& v) := n Then we have the stability
result contained
THEOREM 4.6. Let ,u,, be a sequence
in the following
of measures sequence of Markov operators as defined above If (P, I w E ~2) is S-bounded and a(,~,, v) + 0 fn + g*, where g* is the density of the unique
(21)
theorem.
on (C& 5) and P,, := Pp. be the corresponding in (19). Let v denote the point mass at o,, E a. then each P,, has a fixed density f., such that TUO-invariant measure.
Proof. By lemma 4.4, P,, and each P,, are of class S(I, M), for some A > 1 and M > 0, and PO, has a unique invariant density, by theorem 2.3; so to invoke proposition 2.9 it suffices that W p, + p,,, pointwise on a>%,, which is the case by lemma 4.5. Acknowledgement-This
work was partially supported by a University of Maryland post-doctoral grant.
Invariant
measures
1025
REFERENCES 1. ECKMANN J. P. & RUELLE D., Ergodic theory of chaos and strange attractors, Rev. Mod. Phys. 57,617-656 (1985). 2. ULAM S., Problems in Modern Mathematics. Interscience, New York (1960). 3. LI T.-Y., Finite approximation for the Frobenius-Perron operator. A solution to Ulam’s conjecture, J. Approx. Theory 17, 177-186 (1976). 4. KELLER G., Stochastic stability in some chaotic dynamical systems, A4h. Math. 94, 313-333 (1982). 5. BLANK M. L., Stochastic properties of deterministic dynamical systems, Sov. Sci. Rev. C. Math./Phys. 6, 243-271 (1987). 6. GORA P. & BOYARSKY A., Compactness of invariant densities for families of expanding, piecewise monotonic .’ : 7*I transformations (to appear). 7. HUNT F. Y., A Monte-Carlo approach to the approximation of invariant measures, ./National’ Institute of Standards and Technology Internal Report, NISTIR 4980’(1993). 8. KELLER G., Stochastic perturbations of some strange attractors, Lecture Notes in Physics, Vol. 79, pp. 192-193. Springer, New York (1982). 9. DING J. & LI T.-Y., Markov finite approximation of the Frobenius-Perron operator, Nonlinear Analysis 17, 759-772 (1991). 10. LASOTA A. & YORKE J., Exact dynamical systems and the Frobenius-Perron operator, Trans. Am. math. Sot. 273, 375-384 (1982). 11. KOWALSKI Z. S., Invariant measures for piecewise monotonic transformations, Lecture Notes in Mathematics, Vol. 472, pp. 77-94. Springer, New York (1975). 12. CHIU C., DU Q. & LI T.-Y., Error estimates of the Markov finite approximation of the Frobenius-Perron operator, Nonlinear Analysis 19, 291-308 (1992). 13. JABLONSKI M. & MALCZAK J., The rate of convergence of iterates of the Frobenius-Perron operator for piecewise convex transformations, Bull. Acad. Polon. Sci. 31, 250-253 (1983). 14. HUNT F. Y. &MILLER W. M., On the approximation of invariant measures, J. Stat. Phys. 66, 535-548 (1992). 15. KIFER Y., Random Perturbations of Dynamical Systems. Birkhauser, Boston (1988). 16. LASOTA A. & MACKEY M., Probabilistic Properties of Deterministic Systems. Cambridge University Press, New York (1985). 17. NATANSON I. P., Theory of Functions of a Real Variable, Vol. 1, revised edition. Ungar, New York (1961). 18. DUNFORD N. & SCHWARTZ J., Linear Operators, Part 1. Wiley, New York (1988). 19. HORN R. & JOHNSON C., Matrix Analysis. Cambridge University Press, New York (1985). 20. KIFER Y., Ergodic Theory of Random Transformations. Birkhauser, Boston (1986).