2011/11/10 by De‐Jun Feng, Feng, De-Jun
Mathematics · Physics and Astronomy · #11K16 (Secondary) #28A78 (Primary) 28A80 #Chaos control and synchronization #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Quantum chaos and dynamical systems
paper · pdf · doi:10.48550/arxiv.1111.2414
openalex publication_date 2011/11/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the multifractal structure of the Bernoulli convolution νλ, where λ-1 is a Salem number in (1,2). Let τ(q) denote the Lq spectrum of νλ. We show that if α∈ [τ'(+∞), τ'(0+)], then the level set E(α):=x∈ \R: limr→ 0(log νλ([x-r, x+r]))/(log r)=α is non-empty and dimHE(α)=τ^*(α), where τ^* denotes the Legendre transform of τ. This result extends to all self-conformal measures satisfying the asymptotically weak separation condition. We point out that the interval [τ'(+∞), τ'(0+)] is not a singleton when λ-1 is the largest real root of the polynomial xn-xn-1-... -x+1, n≥ 4. An example is constructed to show that absolutely continuous self-similar measures may also have rich multifractal structures.