Multifractal analysis of some multiple ergodic averages in linear Cookie-Cutter dynamical systems

In this paper, we study the multiple ergodic averages of a locally constant real-valued function in linear Cookie-Cutter dynamical systems. The multifractal spectrum of these multiple ergodic averages is completely determined.


§1. Introduction and statement of results
Consider a piecewise linear map T on the unit interval with m branches (m ≥ 2).Let I 0 , • • • , I m−1 ⊂ [0, 1] be m closed intervals with disjoint interiors.Suppose that on each interval I i , the restriction T : I i → [0, 1] is bijective and linear with slop e λi (λ i > 0).Let J T := ∞ n=1 T −n [0, 1] be the corresponding Cookie-Cutter set.We will study the dynamical system (J T , T ).Let ℓ ≥ 2 be an integer and let ϕ be a real-valued function defined on [0, 1] ℓ .Let q ≥ 2 be an integer.We would like to calculate the Hausdorff dimensions of the following level sets If ϕ = g 1 ⊗ • • • ⊗ g ℓ is a tensor product of ℓ real-valued functions, the averages in (1) is nothing but the multiple ergodic averages which have been widely studied in ergodic theory.The research for the dimensions of the level sets (1), which was initiated by Fan, Liao and Ma [1], is concerned with multifractal analysis of multiple ergodic averages and has attracted much attention (see e.g.[6,7,3,4,9,5,10,8]).Most of the results concentrated on the symbolic space where the Lyapunov exponents for the shift transformation are constant.Liao and Rams [8] gave the multifractal analysis of a special multiple ergodic averages for systems with non-constant Lyapunov exponents.They considered the level set in (1) with m = ℓ = q = 2 and ϕ = 1 I1 ⊗1 I1 .Liao-Rams' argument was based on computations of Peres and Solomyak [9], which seems inconvenient to be adapted to general case.However, in this note we show how to adapt the arguments in [4] to arbitrary m, ℓ, q ≥ 2 and a class of functions ϕ.
We assume that ϕ is locally constant: for any (a With an abuse of notation, we write ϕ(a For simplicity of notation, we restrict ourselves to the case ℓ = 2 (the same arguments works for arbitrary ℓ ≥ 2 without any problem).For any s, r ∈ R, consider the non-linear transfer system e sϕ(i,j)+rλi t j (s, r), The difference between (2) and a similar system in [4] is that the Lyapunov exponents λ i 's are introduced in (2).It is easy to prove that the system admits a unique strictly positive solution (t 0 (s, r), • • • , t m−1 (s, r)) (see [4]).Define the pressure function by It will be shown (Lemma 1) that P is analytic and convex, and even strictly convex if ϕ is not constant and the λ i 's are not all the same.Let A and B be respectively the infimum and the supremum of the set It will be proved (Lemma 3) that for any α ∈ (A, B), there exists a unique solution (s(α), r(α)) ∈ R 2 to the system and that r(α) is analytic on (A, B), increasing on (A, ϕ 0 ) and decreasing on (ϕ 0 , B) where We sketch the proof.Let f i : [0, 1] → I i be the branches of Then we have π(Σ m ) = J T .Define the subset E ϕ (α) of Σ m which was studied in [3,4]: Then with a difference of a countable set, we have L ϕ (α) = π(E ϕ (α)).
In [3,4], a family of Gibbs-type measures called telescopic product measures were used to compute the Hausdorff dimension of E ϕ (α).Here we construct a similar class of measures in order to determine the Hausdorff dimension of L ϕ (α).In the following, we suppose that ϕ is not constant (otherwise the problem is trivial) and the λ i 's are not the same (otherwise the problem is reduced to the case considered in [3]).
Lemma 1.For any s, r ∈ R, the system (2) admits a unique solution (t 0 (s, r), • • • , t m−1 (s, r)) with strictly positive components, which are analytic functions of (s, r).The pressure function P is strictly convex in R 2 .
Proof It is essentially the same proof as in [4], with some small modifications.
The solution (t 0 (s, r), t 1 (s, r), • • • , t m−1 (s, r)) of the system (2) allows us to define a Markov measure µ s,r with initial law (π(i)) i∈S and probability transition matrix (p i,j ) S×S defined by , p i,j = e sϕ(i,j)+rλi t j (s, r) t i (s, r) 2 .
Then we decompose the set of positive integers N * into Λ i (q ∤ i) with Λ i = {iq k } k≥0 so that Σ m = i:q∤i S Λi .Take a copy µ s,r on each S Λi and define the product measure P s,r of these copies.Then P s,r is a probability measure on Σ m .Let ν s,r = π * P s,r = P s,r • π −1 and let D(ν s,r , x) be the lower local dimension of ν s,r at x. Lemma 2. Let α ∈ (A, B).For any (s, r) ∈ R 2 such that P (s, r) = α, for any x ∈ L ϕ (α) we have Proof It is the same proof as in [4].
It follows that dim H L ϕ (α) ≤ r q log m .Minimizing the right hand side gives an upper bound: where r(α) together with an s ∈ R constitute a (unique) solution of the system (3).
The following lemma says that such r(α) is really unique.
Proof Similar proof as in Proposition 4.13 of [2].
In order to get the lower bound, we only have to show that ν s(α),r(α) is supported on L ϕ (α) and the local dimension of ν s(α),r(α) is almost everywhere equal to r(α)/q log m.To this end, we need the following law of large numbers (LLN) (see Theorem 6 in [4]).
Applying the above lemma to F n (ω n , ω qn ) = ϕ(ω n , ω qn ) for all n and computing E Ps,r ϕ(ω n , ω qn ), we get Lemma 5.For P s,r -a.e.ω ∈ Σ m , we have Finally, it remains to show the measure ν s,r is exact in the sense that its local dimension is almost everywhere constant.Lemma 6.For ν s,r -a.e.x we have D(ν s,r , x) = r q log m + P (s, r) − s ∂P ∂s (s, r) λ(s, r) , where λ(s, r) is the expected limit with respect to P s,r of the Lyapunov exponent Thus we have finished the proof for α ∈ (A, B).If α = A (resp.B), as in the standard multifractal analysis, we use the measure ν s(α),r(α) and let α tend to A (resp.B).