Let $u=(u_n)$ be a sequence of real numbers and $\Gamma$ a curve in the complex plane which linearly joins consecutively the points $z_0$ to $z_1$, $z_1$ to $z_2$ etc, for $z_n=\sum_{k=0}^{n-1}\exp(2 \pi\imath u_k)$, $n=1,2,3$. Such patterns created in the complex plane by exponential sums have been given the name "curlicues". It turned out that curlicues are beautiful objects similar in their nature to fractals.

In the talk I will speak about some prominent results on curlicues proved by [Dekking and Mendes France, 1981] and [Sinai, 2008]. I will also present my results, joint with A. Tahzibi and M. Ponce, referring to the case when $u_n=f^n(x_0)$ with $f$ being a lift of a circle homeomorphism. We will see how dynamical properties of circle homeomorphisms/diffeomorphisms affect geometric properties of such curves. This is a work in progress.