We develop a local theory of lacunary Dirichlet series of the form
$\sum\limits_{k=1}^{\infty}c_k\exp(-zg(k)), \Re(z)>0$ as $z$ approaches the
boundary i\RR, under the assumption $g'\to\infty$ and further assumptions on
$c_k$. 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 $c_k=1$, 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 $g(k)=k^b$, $c_k=1$, $b=n$ or
$b=(n+1)/n$, n\in\NN, behavior near the boundary is roughly of the standard
form $\Re(z)^{-b'}Q(x)$ where $Q(x)=1/q$ if x=p/q\in\QQ and zero otherwise.
The B\"otcher map at infinity of polynomial iterations of the form
$x_{n+1}=\lambda P(x_n)$, $|\lambda|<\lambda_0(P)$, turns out to have uniformly
convergent Fourier expansions in terms of simple lacunary series. For the
quadratic map $P(x) =x-x^2$, $\lambda_0=1$, and the Julia set is the graph of
this Fourier expansion in the main cardioid of the Mandelbrot set