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This paper examines dimension of the graph of the famous Weierstrass non-differentiable function \[ W_{\lambda, b} (x) = \sum_{n=0}^{\infty}\lambda^n\cos(2\pi b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$. We prove that for every $b$ there exists (explicitly given) $\lambda_b \in (1/b, 1)$ such that the Hausdorff dimension of the graph of $W_{\lambda, b}$ is equal to $D = 2+\frac{\log\lambda}{\log b}$ for every $\lambda\in(\lambda_b,1)$. We also show that the dimension is equal to $D$ for almost every $\lambda$ on some larger interval. This partially solves a well-known thirty-year-old conjecture. Furthermore, we prove that the Hausdorff dimension of the graph of the function \[ f (x) = \sum_{n=0}^{\infty}\lambda^n\phi(b^n x) \] for an integer $b \ge 2$ and $1/b < \lambda < 1$ is equal to $D$ for a typical $\mathbb Z$-periodic $C^3$ function $\phi$.Comment: Final authors' version with a correction of an inexact statement in the introduction to the published version, concerning the box dimension of the graphs of functions of the form (1.1) and (1.2

Let f be an entire transcendental map of finite order, such that all the singularities of f −1 are contained in a compact subset of the immediate basin B of an attracting fixed point. It is proved that there exist geometric coding trees of preimages of points from B with all branches convergent to points from C. This implies that the Riemann map onto B has radial limits everywhere. Moreover, the Julia set of f consists of disjoint curves (hairs) tending to infinity, homeomorphic to a half-line, composed of points with a given symbolic itinerary and attached to the unique point accessible from B (endpoint of the hair). These facts generalize the corresponding results for exponential maps.