The notion of singular hyperbolicity for vector fields has been introduced by Morales, Pacifico and Pujals in order to extend the classical uniform hyperbolicity and include the presence of singularities. This covers the Lorenz attractor. I will present a joint work with Dawei Yang which proves a dichotomy in the space of three-dimensional $C^{1}$-vector fields, conjectured by J. Palis: every three-dimensional vector field can be $C^{1}$-approximated by one which is singular hyperbolic or by one which exhibits a homoclinic tangency.[-]

Fatou noticed in 1926 that certain transcendental entire functions have Julia sets in which there are curves of points that escape to infinity under iteration and he wondered whether this might hold for a more general class of functions. In 1989, Eremenko carried out an investigation of the escaping set of a transcendental entire function f, $I(f)=\left \{ z\in\mathbb{C}:\left | f^{n}\left ( z \right ) \right | \rightarrow \infty \right \}$ and produced a conjecture with a weak and a strong form. The strong form asks if every point in the escaping set of an arbitrary transcendental entire function can be joined to infinity by a curve in the escaping set.
This was answered in the negative by the 2011 paper of Rottenfusser, Rückert, Rempe, and Schleicher (RRRS) by constructing a tract that produces a function that cannot contain such a curve. In the same paper, it was also shown that if the function was of finite order, that is, log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )$ as $\left | z \right |\rightarrow \infty$, then every point in the escaping set can indeed be connected to infinity by a curve in the escaping set.
The counterexample $f$ used in the RRRS paper has growth such that log log $\left | f\left ( z \right ) \right |= \mathcal{O}\left ( log\left | z \right | \right )^{k}$ where $K > 12$ is an arbitrary constant. The question is, can this exponent, K, be decreased and can explicit calculations and counterexamples be performed and constructed that improve on this?[-]