Limit theorems for random intermittent maps Chris Bose University - PowerPoint PPT Presentation

Limit theorems for random intermittent maps Chris Bose University of Victoria, CANADA (joint with Wael Bahsoun, U. Loughborough, UK) Bielefeld, November 2015 U. Victoria: Bose CLT for random maps

Outline Random maps and skew product Warmup: Random piecewise expanding maps Intermittent maps – non-random limit theorems Random intermittent maps; annealed limit theorems U. Victoria: Bose CLT for random maps

Skew product, deterministic representation We consider random maps of the form { T 1 , T 2 , p 1 , p 2 } where the maps T i are chosen iid with probability p i . Classical setting: constant probabilities and skew product representation T ( x , ω ) = ( T ω 0 ( x ) , σ ( ω )) . We will consider (will need!) p i = p i ( x ) spatially dependent probabilities where the associated Markov process is P ( x , A ) = p 1 ( x ) 1 A ( T 1 ( x )) + p 2 ( x ) 1 A ( T 2 ( x )) . To realize this as a ‘skew product’ we use the following geometric idea U. Victoria: Bose CLT for random maps

Constant probabilities: X : ( x , ω ) ∈ [ 0 , 1 ] × [ 0 , 1 ] . p_2 p_1 x T_2(x) U. Victoria: Bose CLT for random maps

Spatially dependent probabilities p_4 p_3 p_2 p_1 x T_3(x) S ( x , ω ) = ( T ω 0 , ϕ ( x , ω )) U. Victoria: Bose CLT for random maps

Limit theorems for random expanding Assume T i ∈ expanding Lasota-Yorke maps. [ 0 , 1 ] = ∪ [ a j , a j + 1 ] = ∪ I ( i ) j T i : I ( i ) → [ 0 , 1 ] , C 2 and expanding j | T ′ i | ≥ λ i > 1 Assuming inf p i ( x ) > 0 the representation leads to a piecewise expanding, 2D-map of the unit square into itself. P_3 p_2 p_1 T_3 I_j I_j Works best if the p i are also locally smooth with respect to I j ; U. Victoria: Bose CLT for random maps

Correlation decay and Central Limit Theorem With S as above, BV = 2D bounded variation functions, the transfer operator P S is quasicompact. Then There is an ACIPM for S , d ν = h d ( m × m ) (m= Lebesgue). 1 There is a ρ < 1 such that f ∈ BV , g ∈ L ∞ and � f dx = 0 2 then � � � � f · g ◦ S n d ν � � ≤ C � f � BV � g � ∞ ρ n � � � � Assume S weakly mixing and f ∈ BV with fd ν = A . 3 There exists σ 2 ≥ 0 such that S n f − nA √ n → N ( 0 , σ ) Convergence is in distribution and σ 2 > 0 iff f is not a coboundary for S. U. Victoria: Bose CLT for random maps

A few remarks on CLT Quasicompactness and correlation decay: See Liverani (2011): Multidimensional . . . pedestrian approach. Spectral approach to CLT (and other limit theorems): See Gouëzel (2015?, expository) Why is f ∈ BV natural: Consider the perturbed transfer operator P t ( h ) = P S ( e itf h ) , t ∈ R and study spectral stability as t → 0. U. Victoria: Bose CLT for random maps

Embedding the Σ n f = � n − 1 k = 0 f ◦ S k � � P 2 P S e itf P S e itf ϕ · ψ dm t ϕ · ψ dm = � e itf P S e itf ϕ · ψ ◦ S dm = � e itf ◦ S e itf ϕ · ψ ◦ S 2 dm = � e it Σ 2 f ϕ · ψ ◦ S 2 dm = get � � e it Σ n f ϕ · ψ ◦ S n dm P n t ϕ · Ψ dm = Setting ϕ = ψ = 1 leads to characteristic function � � E ( e it Σ n f ) = P n P n t ϕ · Ψ dm = t 1 dm U. Victoria: Bose CLT for random maps

Variance and correlation decay In theorem above, we identify: � � σ 2 = ˜ ˜ f · ˜ f 2 dm + 2 � f ◦ S k dm , k where ˜ f = f − A . Key condition to obtain CLT via spectral argument is the summability of correlations: � � ˜ f · ˜ f ◦ S k dm < ∞ k as expected. Other decay rates like stretched exponential or even polynomial are known for maps with indifferent fixed points. These are the so-called intermittent maps. U. Victoria: Bose CLT for random maps

Intermittent maps of the interval An example. Fix 0 < α < ∞ . Set � x + 2 α x 1 + α x ∈ [ 0 , 1 / 2 ) T α ( x ) := 2 x − 1 x ∈ [ 1 / 2 , 1 ) U. Victoria: Bose CLT for random maps

Orbits are mostly spread chaotically throughout [ 0 , 1 ) interspersed with short periods getting ‘stuck’ near the neutral fixed point at x = 0. The periods of getting stuck are the intermittencies . U. Victoria: Bose CLT for random maps

An orbit histogram gives a picture of an invariant density for the map T α : It is known that the density has an order O ( x − α ) singularity near x = 0. U. Victoria: Bose CLT for random maps

History for single map 1 Liverani, Saussol, Vaienti (ETDS 1999) established regularity properties of the invariant density for T α and proved sub-exponential decay of correlation in the case of regular fixed point (i.e. 0 < α < 1 ) and finite ACIM: � � � ( g ◦ T n ) f d µ − Cor n ( g , f ) := g d µ f d µ α n 1 − 1 1 | Cor n ( g , f ) | ≤ C ( f ) || g || ∞ ( log n ) α for f ∈ C 1 and g ∈ L ∞ . µ is the ACIM The maps T α above are known as LSV-maps. Related: Pomeau-Manneville maps. U. Victoria: Bose CLT for random maps

History for single map 2 LS Young (Israel J. Math 1999) induced away from the fixed point and studied return time asymptotices on ∆ = [ 1 / 2 , 1 ] . Led to a systematic approach for many non-uniformly hyperbolic systems known as Young Towers or Markov extensions. Links invariant measures, mixing and correlation decay rates to a single intuitive estimate. U. Victoria: Bose CLT for random maps

Young Towers If R ( x ) = n + 1 then F ( x ) := T n + 1 ( x ) ∈ [ 1 / 2 , 1 ] α U. Victoria: Bose CLT for random maps

ACIM ν ∼ m ( m = Lebesgue) for T depends on � m (∆ k ) < ∞ k For f ∈ C β , g ∈ L ∞ � | Cor n ( g , f ) | ≤ C ( f ) || g || ∞ m (∆ k ) k > n For LSV, careful calculus estimate gives m (∆ k ) = 1 � n − 1 � 2 x k = O α Distortion control required: � DF ( x ) � � ≤ C θ d ( F ( x ) , F ( y )) � � DF ( y ) − 1 � � � U. Victoria: Bose CLT for random maps

Summary for LSV maps For T = T α , invariant ν = hdm , and Cor n ( f , g ) = O ( n 1 − 1 α ) H. Hu, 0. Sarig and S. Gouëzel (2002-2004) showed the correlation rate is sharp when 0 < α < 1. Central limit theorems hold when ν = 1 α − 1 > 1 ⇔ 0 < α < 1 2 When α ≥ 1 the ACIM is σ − finite. Melbourne &Terhesiu (Invent. 2012) established mixing and correlation decay estimates for suitably normalized correlation. U. Victoria: Bose CLT for random maps

Random intermittent maps Let 0 < α < β < ∞ and T α , T β two intermittent LSV maps and consider T := ( T α , T β , p 1 , p 2 ) the associated random dynamical system. We can represent T as a deterministic skew product on [ 0 , 1 ] × [ 0 , 1 ) by S ( x , y ) = ( T α ( ω ) ( x ) , σ ( ω )) Here � α if ω ∈ [ 0 , p 1 ) � ω p 1 if ω ∈ [ 0 , p 1 ) σ ( ω ) = if ω ∈ [ p 1 , 1 ) ; α ( ω ) = ω − p 1 β if ω ∈ [ p 1 , 1 ) p 2 This is just the independent ( p 1 , p 2 ) − shift U. Victoria: Bose CLT for random maps

In order to apply Young’s construction, we need analogues of the intervals I n and J n from the single map case. Since the position x n = x n ( ω ) (similarly x ′ n ( ω ) ), instead of intervals we see the following picture: ∆ 0 = [ 1 / 2 , 1 ) × [ 0 , 1 ) and the return sets I n and J n are unions of 2 n rectangles stacked ’vertically’ U. Victoria: Bose CLT for random maps

The key estimates are again � � � � E ω ( x ′ j ( ω ) − x ′ m × m ( J j ) = j + 1 ( ω )) k > n j ≤ 2 k k > n j = 1 � � E ω ( x j ( σω ) − x j + 1 ( σω )) 2 k > n j � � = E ω ( x j ( ω )) − E ω ( x j + 1 ( ω )) k > n j � = E ω ( x k ( ω )) k > n So we need to calculate the expected position of x k ( ω ) over the randomizing variable ω . U. Victoria: Bose CLT for random maps

The only completely obvious bounds are x n ( α ) ≤ x n ( ω ) ≤ x n ( β ) where x n ( α ) is the non-random point under parameter α and similar for x n ( β ) . With a little care, we can derive the following exact asymptotics: Proposition For 0 < α ≤ β < ∞ , for a.e. ω : 1 α x n ( ω ) n lim = 1 − 1 n 2 α − 1 1 α p α 1 − 1 So x n ( ω ) ∼ 1 / 2 α − 1 n − 1 α p α . We can see this is the ’right’ α 1 result by setting p 1 = 1 where we recover the same sharp estimate due to LS Young for a single map at parameter α . U. Victoria: Bose CLT for random maps

The heuristic: For large n , most strings ω n 0 see about p 1 · n occurrences of α , pushing the position strongly toward x n ( α ) . The fast escape process therefore dominates the asymtpotics. To make this precise, we need a large deviations result. Theorem (Hoeffding, 1963) Let X k be an independent sequence of RV, with 0 ≤ X k ≤ 1 ∀ k Set ¯ � n k = 1 X k and E n = E (¯ X n = 1 X n ) Then for each n 0 < t < 1 − p 1 P { ¯ X n − E n > t } ≤ exp ( − 2 nt 2 ) With exponentially decaying deviations, a simple Borel-Cantelli argument suffices to get pointwise convergence, almost surely. U. Victoria: Bose CLT for random maps