Invariant densities for random systems

Transcription

1 Invariant densities for random systems Marta Maggioni joint work with Charlene Kalle Universiteit Leiden Numeration208 May 23, 208 M. Maggioni / 22

2 Setting (Ω N, p) prob space T := {T j : X X, j Ω} family of maps T is a random system of the space X of probability p, if T (x) := T j (x) with probability p j M. Maggioni 2 / 22

3 Motivation Stochastic perturbations Particles systems Number expansions β, Lüroth, dyadic expansions, etc. M. Maggioni 3 / 22

4 Motivation Random β-transformations [DK03] T 0(x) = { βx, if x [0, ] β(β ) βx, if x (, ], T(x) = β(β ) β { βx, if x [0, β βx, if x [, β β β 2 β β β β 2 β β 0 β(β ) β 0 β β 0 β (a) T 0 (b) T (c) T M. Maggioni 4 / 22

5 ACIM: definition (X, B, µ p, T, p) random system ACIM: µ p (A) = h dλ = p j µ p (Tj A) for all A B A j Ω Perron-Frobenius operator A P T h dλ = T (A) h dλ P T h = j Ω P T j p j h P T h = h ACIM µ h M. Maggioni 5 / 22

6 Existing formulas Lasota-Yorke linear maps Same slopes Parry, Dajani, Kempton, Suzuki for the β-transformations (deterministic and random) Different slopes Kopf [Kop90], Góra (deterministic) Our approach (random) M. Maggioni 6 / 22

7 Results Setting T = {T j : [0, ] [0, ], j Ω} expanding on average wrt p sup x [0,] j Ω p j T j (x) < T j piecewise linear {I,..., I N } partition for the set {0 = z 0 < z <... < z N = } discontinuity points T i,j (x) = k i,j x + d i,j (Thm, [Ino2]) T admits an ACIM M. Maggioni 7 / 22

8 Results Assumptions. T (0), T () {0, } 2. for every i there exists n: p j j Ω k i,j d i,j p j j Ω k i,j j Ω p j k n,j d n,j j Ω p j 0 k n,j M. Maggioni 8 / 22

9 Results Definitions ω Ω N path y [0, ] point t N instant of time n {,..., N} interval τ ω (y, t) := pω t k i,ωt, if T ω t (y) I i δ ω (y, t) := t n=0 τ ω(y, n) KI n (y) := t ω Ω t δ ω(y, t) In (T ω t (y)) M. Maggioni 9 / 22

10 Results Results Thm. (Kalle, M., to appear) Under the previous assumptions, N h γ (x) = c m= is a T -invariant function. γ m l Ω [ pl L am,l (x) p ] l L bm,l (x) k m,l k m+,l a m,l = k m,l z m + d m,l, b m,l = k m+,l z m + d m+,l L y(x) = δ ω(y, t) [0,Tω(y))(x) t 0 ω Ω t M. Maggioni 0 / 22

11 Results Results Procedure: T M Mγ = 0 h γ for ( M = j Ω [ pj k i,j KI n (a i,j ) p ] ) j KI n (b i,j ) + q n,i k i+,j n,i M. Maggioni / 22

12 Results Results Not straightforward There always exists γ 0 (ass. 2.) h γ is T -invariant (ass..) {I,..., I N } arbitrary (endpoints and size of this set) M. Maggioni 2 / 22

13 Results Results Thm. (Kalle, M., to appear) For Ω finite and T expanding, the construction gives all possible T -invariant densities. Idea: M ˆM U γ ˆγ U M. Maggioni 3 / 22

14 Examples Example : random β-transformations [DdV07]: special β [Kem4]: h for all < β < 2, unbiased case h(x) = c n=0 ( ) (2β) n [0,R n ()] (x) + β,ω ωn [R n ( 2 β ω ω n {0,} n β,ω ωn β ), ](x) β [Suz7]: h for all < β < 2, biased cases M. Maggioni 4 / 22

15 Examples Example : random β-transformations h for all < β < 2 for p 0 = p = 2 β 2 β β 0 β h γ (x) = k t 0 ( ) (2β) t [0,Tω())(x) + [Tω( 2 β ω {0,} t β ), ](x) β M. Maggioni 5 / 22

16 Examples Example : random β-transformations KI n() ( KI 2 β ) n β ( KI n(0) KI ) n β c c 3 β 0 c 2 c c 3 c 0 β β + 2β (c β ) 2β c 3 ( ) β + 2β c 2 β 2β c 2 = 0 2β c 3 β 2β (c β ) M. Maggioni 6 / 22

17 Examples Example 2: random (α, β)-transformations T 0 (x) = { { βx, if x [0, /β) βx, if x [0, /β) α β (βx ), if x [/β, ] and T (x) = βx, if x [/β, ] β β β 3 0 β 3 β 2 β β β 2 = β +, α = /β h γ = c((β 2 p + β) A + (p + β) B + β C + D ) [DHK09] M. Maggioni 7 / 22

18 Examples Example 3: random Lüroth map with bounded digits x = n ( ) s n (r n + ω n ) n k= r k (r k ) T L (x) := n(n )x (n ) and T A (x) := T L (x) [Lür83, BBDK94, BI09, LY78, Pel84] M. Maggioni 8 / 22

19 Examples Example 3: random Lüroth map with bounded digits I 0 I (g) T 0 (h) T (i) T h γ (x) = 3/8(3 [ 3, 2 3 ](x) + 5 ( 2 3,](x)) digits frequency: 2 3/6, 3 3/6 M. Maggioni 9 / 22

20 Examples Thank you! M. Maggioni 20 / 22

21 Examples J. Barrionuevo, R. M. Burton, K. Dajani, and C. Kraaikamp. Ergodic properties of generalized Luroth series. TU Delft Report, 94-05: 6, 994. L. Barreira and G. Iommi. Frequency of digits in the Lüroth expansion. J. Number Theory, 29(6): , K. Dajani and M. de Vries. Invariant densities for random β-expansions. J. Eur. Math. Soc., 9():57 76, K. Dajani, Y. Hartono, and C. Kraaikamp. Mixing properties of (α, β)-expansions. Ergodic Theory Dynam. Systems, 29(4):9 40, K. Dajani and C. Kraaikamp. Random β-expansions. Ergodic Theory Dynam. Systems, 23(2):46 479, T. Inoue. Invariant measures for position dependent random maps with continuous random parameters. Studia Math., 208(): 29, 202. K. Kempton. On the invariant density of the random β-transformation. Acta Math. Hungar., 42(2):403 49, 204. C. Kopf. Invariant measures for piecewise linear transformations of the interval. Appl. Math. Comput., 39(2, part II):23 44, 990. M. Maggioni 2 / 22

22 Examples J. Lüroth. Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe. Math. Ann., 2(3):4 423, 883. T. Y. Li and J. A. Yorke. Ergodic transformations from an interval into itself. Trans. Amer. Math. Soc., 235:83 92, 978. S. Pelikan. Invariant densities for random maps of the interval. Trans. Amer. Math. Soc., 28(2):83 825, 984. S. Suzuki. Invariant density functions of random β-transformations. Ergodic Theory and Dynamical Systems, page 22, 207. M. Maggioni 22 / 22