Ergod. Th. & Dynam. Sys. (,, Printed in the United Kingdom c Cambridge University Press A Family of Piecewise Expanding Maps having Singular Measure as a it of ACIM s Zhenyang Li,Pawe l Góra, Abraham Boyarsky, Harald Proppe and Peyman Eslami Department of Mathematics and Statistics, Concordia University, 455 de Maisonneuve Blvd. West, Montreal, Quebec H3G M8, Canada (e-mail: zhenyangemail@gmail.com, (e-mail: pgora@mathstat.concordia.ca, (e-mail: boyar@alcor.concordia.ca, (e-mail: proppe@alcor.concordia.ca, (e-mail: peyman eslami@yahoo.com (Received September 0, 0 Abstract. Keller [9] introduced families of W shaped maps that can have a great variety of behavior. As a family approaches a it W map, he observed behavior that was either described by a probabilty density function (pdf or by a singular point measure. Based on this, Keller conjectured that instability of the absolutely continuous invariant measure (acim can result only from the existence of small invariant neighbourhoods of the fixed critical point of the it map. In this note we show that the conjecture is not true. We construct a very simple family of W maps with acim s supported on the whole interval, whose iting dynamical behavior is captured by a singular measure. Key to the analysis is the use of a general formula for invariant densities of piecewise linear and expanding maps [6].. Introduction Usually, the absolutely continuous invariant measure of a piecewise expanding map of an interval is stable under deterministic or even random perturbations. This means that if we consider a family of piecewise expanding maps τ a, a > 0 with acims µ a, converging to a piecewise expanding map τ 0 with acim µ 0, then under general assumptions µ a s converge to µ 0. One such assumption is that for some positive ε, τ a > + ε for all a 0. Actually, much more is proven in [0] and []. This stability property is often useful, for example, in establishing results about metastable systems [4]. The research of the authors was supported by NSERC grants. [Version: 999/07/ v.0]
Zh. Li, P. Góra, A. Boyarsky, H. Proppe,P. Eslami Keller [9] introduced the family of W-maps that are piecewise expanding and exhibit a wide variety of behaviour. This was done to understand whether in dimension one the expanding constant ensuring stability is really rather than as for zero-dimensional systems [7]. This regularity was later confirmed in [8] by showing that this constant for a piecewise expanding n-dimensional system with rectangular partition is n +. Key to the complexity of Keller s families is the fact that, as the parameter approaches 0, say, the behavior near a folded critical point plays a crucial role. This critical point has slope on one side and on the other. Thus, the entire family is uniformly piecewise expanding and each member has a unique absolutely continuous invariant measure. However, the stability of probability density functions that one might expect in families of uniformly piecewise expanding maps does not occur. Keller provided an example of a family for which the it of acims is a singular measure. This occurred because of the existence of diminishing invariant neighbourhoods of the critical point. Keller conjectured that this is the only mechanism which can cause such iting behaviour. In this paper we construct a family of simple W-maps which disproves Keller s conjecture. All our maps are piecewise expanding with slopes strictly greater than in magnitude and are exact with their acim s supported on all of [0, ]. Standard bounded variation methods cannot be applied in this setting as the slopes of maps in our family are not uniformly bounded away from. The Lasota- Yorke BV inequality [] cannot be invoked for families of maps whose slopes are not bounded away from in magnitude. Other research related to the objectives of this note can be found in [3], [3] and []. However, none of the methods used in these papers apply to the family of maps we consider. In a recent paper [5], a 3-parameter countable family of transitive maps was constructed which converges to the W-map for which the acim s approach any given convex combination of the point measure at / and the W-map acim. All maps in this family are Markov, which makes the analysis easier. In this note we shall utilize the main result of [6] which proves that the invariant pdf for any piecewise linear map which is eventually expanding has a convenient infinite series expansion. The estimates on the family of pdfs derived from this representation allows us to prove our main result, that the acim s of the family of W-maps approach a combination of an absolutely continuous and a singular measure rather than the acim of the it map. In Section we introduce our family and state the main theorem, which is proved in Section 3. In Section 4, we show computational results for some pdfs of the W a maps when a is small.
A Family of Piecewise Expanding Maps 3. Family of W a maps and the main result We consider the family {W a : 0 a} of maps of [0, ] onto itself defined by 4x, for 0 x < /4 ; ( + a(x /4, for /4 x < / ; W a (x = / + a/4 ( + a(x /, for / x < 3/4 ; 4(x 3/4, for 3/4 x. ( Figure. a map W 0, b map W a, a > 0, with a few first points of the trajectory of /. The map W 0 is Keller s W-map [9]. We consider only small a > 0 as we are interested in the iting behaviour of the W a s as a 0. Fig. shows the graphs of W a for a = 0 and a > 0. Every W a is a piecewise linear, piecewise expanding map with minimal modulus of the slope equal to +a. Every W a has a unique acim µ a supported on [0, ] and is exact with respect to this measure. The transitivity of such maps is proven in [5], and the uniqueness of acim and exactness follow directly from the Li-Yorke paper []. Let h a denote the normalized density of µ a, a 0. It is easy to check that for W 0, µ 0 has density { 3, for 0 x < / ; h 0 =, for / x. ( Our goal is to prove the following theorem: Theorem.. As a 0 the measures µ a converge -weakly to the measure 3 µ 0 + 3 δ (, where δ ( is the Dirac measure at point /.
4 Zh. Li, P. Góra, A. Boyarsky, H. Proppe,P. Eslami The proof relies on the general formula for invariant densities of piecewise linear maps [6] and direct calculations. The calculations depend on the parameter a, but we suppress it whenever there is no confusion. 3. Proofs This section contains the proof of Theorem., divided into a number of steps. 3.. Formula for non-normalized invariant density of W a We adapt the general formulas of [6] to our case and obtain the following formula for f a : Lemma 3.. For small a > 0 there exists A < such that ( χ s (β(/, n, Wa n (/ f a = + A. (3 β(/, n n= is a W a -invariant non-normalized density. Here, χ s (t, y = { χ[0,y] for t > 0 ; χ [y,] for t < 0, and β(/, n is the cumulative slope along the n steps of the trajectory of / defined by: β(/, = + a, and β(/, n = ( + a W a(w a (/ W a(w a (/ W a(w n a (/, for n. The detailed justification of formula (3 is in Subsection 3.. For small positive a, the first image of / is W a (/ = / + a/4 and the next image lands just below the fixed point slightly less than /. The following forward images of / form a decreasing sequence until they go below /4. Let k be the first iterate j when W j a(/ is less than /4. That is, k = min{j : W j a(/ /4}. Then, the consecutive cumulative slopes of /, namely β(/, j, j k, are ( + a, ( + a, ( + a 3,..., ( + a k, and f a = + A χ [0,W a(/] ( + a + k j= χ [W j a (/,] ( + a j +.... (4 3.. Justification of the formula for f a Using the notation of [6], we have the following lemma: Lemma 3.. (a N=4, K=, L=0; (b α = (, / + a/4, / + a/4,, β = ( 4, + a, ( + a, 4, γ = (0, 0, 0, 0; (c The digits A = (a, a, a 3, a 4, where a =, a = / + a/4, a 3 = 3/ 3a/4, a 4 = 3;
A Family of Piecewise Expanding Maps 5 (d There are two c i s, which are c = (/, and c = (/, 3, and j(c =, j(c = 3. Then, W u = {c, c },W l =, U l = {c },U r = {c }. (e β(c, = + a since j(c =, then β(c, = ( + a and β(c, k = ( + a k up to k defined in Subsection 3., k = min{j : Wa(/ j /4}; (f β(c, = ( + a since j(c = 3, then β(c, = ( + a and β(c, k = ( + a k up to the same k in part (e, Wa n (c = Wa n (c for all n; (g Based on (f, we have the following for the matrix S = S i,j,i, j =, : For c U r δ(β((c, n > 0δ(Wa n (c > / + δ(β((c, n < 0δ(Wa n (c < / S, =, β(c, n S, = n= n= δ(β((c, n > 0δ(Wa n(c > / + δ(β((c, n < 0δ(Wa n(c < /. β(c, n For c U l δ(β((c, n < 0δ(Wa n (c > / + δ(β((c, n > 0δ(Wa n (c < / S, =, β(c, n S, = n= n= δ(β((c, n < 0δ(Wa n (c > / + δ(β((c, n > 0δ(Wa n (c < /, β(c, n where δ( condition is equal to if the condition holds and to 0 if it does not. Remark: It follows from (e, f of Lemma 3. that S i,j are equal for i, j =,. Let Id be the identity matrix, V = [, ]. Then, for the solution, D = [D, D ], of the following system : ( S T + Id D T = V T, (5 we have D = D. Let us denote them by A. Let I, I, I 3, I 4 be the partition of I = [0, ], where I = [0, /4, I = (/4, /, I 3 = (/, 3/4 and I 4 = (3/4, ]. Let β = 4, β = +a, β 3 = (+a, and β 4 = 4.We define the following index: and and j(x = j for x I j, j =,, 3, 4, j(c =, j(c = 3. Already defined for Lemma 3. we have cumulative slopes for iterates of points: β(x, = β j(x, and β(x, n = β(x, n β j(w n a (x, n, χ s (t, y = { χ[0,y] for t > 0 ; χ [y,] for t < 0. Using Theorem in [6] directly, we obtain Lemma 3.. Now, we can prove Lemma 3.:
6 Zh. Li, P. Góra, A. Boyarsky, H. Proppe,P. Eslami Proof: First, by part (g of Lemma 3., since the first and fourth branches of W a have slope of modulus 4 > + a, S i,j ( + a = n + a <. On the other hand, for small a n= S i,j + a + ( + a > /. Now, the solution of the system (5 will be D = D = S, <. By Theorem in [6], it follows from (d, e, f of Lemma 3. that: f a = + D = + A = + A n= n= ( χ s (β(c, n, W n a (c β(c, n χ s (β(c, n, W n a (/ β(c, n n= which completes the proof. + D + A χ s (β(/, n, W n a (/ β(/, n n= n=, χ s ( β(c, n, W n a (c β(c, n χ s ( β(c, n, W n a (/ β(c, n 3.3. Estimates on f a Recall that k = min{j : Wa j (/ /4}. Clearly, k >. Furthermore, we have the following lemma: Lemma 3.3. (I for m k, Wa m (/ = 4 a a(+am + +a + ; (II ak = 0; (III = 0. a(+a k Moreover, if we let k = [ 3k] (integer part of k/3, we have (IV a(+a = 0; k (V a ( + a k = 0; (V I Wa k( =. Proof: Suppose (I is true. By the definition of k, 0 Wa k (/ /4. That is, The first inequality of (6 implies and the second implies and so 0 4 aa( + ak + + a + 4. (6 a ( + a k, (7 a ( + a k, (8 inf a ( + a k supa ( + a k 4.
A Family of Piecewise Expanding Maps 7 Assertions (II, (III and (V follow immediately. To prove (IV we have from (7 and so using also (8: a + a ( + a k/, ( + ak k = k a( + a a( + a k a( + ak k + a + a ( + a k/ + a ( + = ak k ( + a k k/ 0. Finally (VI follows from (I and (V. Now, let us prove (I. For m =, it is easy to check that Wa (/ = a a 4 which is the same as 4 a a(+a+ +. Suppose (I holds for m = i < k, that is +a Then for m = i +, This completes the proof. W i a (/ = 4 aa( + ai + + a +. Wa i+ (/ = ( + a( 4 aa( + ai + + + a 4 = 4 aa( + ai + + a + + a + a 4 = 4 aa( + ai + + + a. Let δ( condition be equal to if the condition holds and to 0 if it does not. Lemma 3. implies that S, = n= δ(β((/, n > 0δ(W n a (/ > / + δ(β((/, n < 0δ(W n a (/ < / β(/, n Also, it was shown there that A = and we have S, A l = k n= S, S,. Since ( + a n = n= + a a + (+a k +a (+a k +, +a ( + a = n + a, A + a a = A h. (9 Note that, for small a, both estimates A l and A h are smaller that..
8 Zh. Li, P. Góra, A. Boyarsky, H. Proppe,P. Eslami Let us define, g l = χ [0,W a(/] ( + a k + j= χ [W j a(/,] ( + a j, and g h = g l + j=k + ( + a = g j l +. ( + a( + a k Let us further define f l = + A l g h and f h = + A h g l. It follows from (4 and (9 that f l f a f h. (0 Let χ = χ [0,/+a/4], χ j = χ [W j a(/,/+a/4], j =, 3,..., k, χ c = χ (/+a/4,]. Now we will represent the functions f l and f h as combinations of functions χ j, j =,..., k and χ c. After some calculations, we obtain: f l = k χ + A l ( + a A l + +a + (A (+a k l + a + n= χ n ( + a n + χ c + A l ( + a( + a k ; f h = [ ( A h ]χ + a + k χ n + A h ( + a n + n= +a (+a k a + χ c. Note that (9 implies that both A l, A h are smaller than ( + a. Using this we can show that all coefficients in the representation of f l and f h are negative for sufficiently small a. 3.4. Normalization Let us define J = [0, W k a (/], J = (W k a (/, /+a/4], J 3 = (/ + a/4, ]. We will calculate integrals of f h over each of these intervals and use them to normalize f h. We have C = f h dλ = J = J [ [ ( + a a + a ( + a a + a + ] + χ dλ ] W k a ( = a + 3a (a ( + a W k a (. C Using Lemma 3.3, we have a = 3 4. In the same way we can see that for any 0 < α < /, we obtain a α 0 f h dλ = 3 α. (
A Family of Piecewise Expanding Maps 9 On the interval J, the integral of f h is: [ ( + a C = f h dλ = J a + a = J a + 3a (a ( + a [ + + a a + ] χ dλ + + a a ( + a 4 W k a ( (k a 4( + a( + a + a 4( + a k j= J ] (+a k + a χ j ( + a j dλ Using Lemma 3.3, we have C a =. ( On the interval J 3, the integral of f h is: ( +a C 3 = f h dλ = (+a k + χ c dλ J 3 J 3 a ( Using Lemma 3.3, we have = +a (+a k a + ( a 4. C 3 a = 4. In the same way we can see that for any 0 < α < /, we obtain a /+αf h dλ = ( α. (3 If we define B = C + C + C 3, then fh B is a normalized density. We see that B a = 3. 3.5. Conclusion of the proof Now, we will use our foregoing calculations to show that the normalized measures (f h /B λ converge -weakly to the measure 3 µ 0 + 3 δ (, as a 0. For any interval [0, α], 0 < α < / as a 0, formula ( implies α 0 f h B dλ = 3 α 3 = α. (4 For J, which converges to the point /, formula ( implies f h J B dλ = 3 = 3. (5 For any interval [/ + α, ], 0 < α < /, formula (3 implies ( f h B dλ = α = ( 3 α. (6 /+α 3.
0 Zh. Li, P. Góra, A. Boyarsky, H. Proppe,P. Eslami Formulas (4, (5 and (6 together show that measures (f h /B λ converge - weakly to the sum of the measure with density χ [0,/] + 3 χ [/,] and 3 of a unit point mass at /, i.e., to the measure 3 µ 0 + 3 δ (. Now, we will show the same for the normalized measure defined using f l. To this end, let us note that f h f l = A h g l A l g h = (A h A l g l A l ( + a( + a k = + a /( + a k g l A l, a a + /( + a k ( + a( + a k where g l and A l =. Using Lemma 3.3 once again, we can show that, for any subinterval J [0, ], we have (f h f l dλ = 0. a J For J = [0, ] this means that the normalizations of f l and f h are asymptotically the same. Thus, the it for a general J implies that the -weak it of normalized measures defined using f l is the same as for those defined using f h. Together with inequality (0 this proves Theorem.. 4. Computational Results We present in Fig. graphs of W a normalized invariant densities for a: a = 0., b: a = 0.05 and c: a = 0.0. They were obtained using Maple 3. Note that the vertical scales of the graphs are very different. Figure. W a-invariant pdf s for a: a = 0., b: a = 0.05 and c: a = 0.0. Acknowledgment: The authors are grateful to an anonymous reviewer for comments.
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