Behavior of lacunary series at the natural boundary

We develop a local theory of lacunary Dirichlet series of the form , Re(z)>0 as z approaches the boundary iR, under the assumption g′→∞ and further assumptions on ck. These series occur in many applications in Fourier analysis, infinite order differential operators, number theory and holomorphic dynamics among others. For relatively general series with ck=1, the case we primarily focus on, we obtain blow up rates in measure along the imaginary line and asymptotic information at z=0.When sufficient analyticity information on g exists, we obtain Borel summable expansions at points on the boundary, giving exact local description. Borel summability of the expansions provides property-preserving extensions beyond the barrier.The singular behavior has remarkable universality and self-similarity features. If g(k)=kb, ck=1, b=n or b=(n+1)/n, n∈N, behavior near the boundary is roughly of the standard form Re(z)−b′Q(x) where Q(x)=1/q if x=p/q∈Q and zero otherwise.The Bötcher map at infinity of polynomial iterations of the form xn+1=λP(xn), |λ|<λ0(P), turns out to have uniformly convergent Fourier expansions in terms of simple lacunary series. For the quadratic map P(x)=x−x2, λ0=1, and the Julia set is the graph of this Fourier expansion in the main cardioid of the Mandelbrot set.