Why do many people link entropy to chaos?

I would say the connection between chaos and entropy is through ergodic theory, and the fundamental assumption of statistical mechanics that a system with a given energy is equally likely to be found in any 'microstate' with that energy.

Although chaos is a very general aspect of dynamical systems, Hamiltonian chaos (encountered in classical mechanics) is characterized by a paucity of conserved quantities, such as energy, and total linear/angular momentum. The crucial fact is not that these conserved quantities are merely difficult to find, but that they do not exist. Because of this, the trajectories of a chaotic dynamical system will trace out a high-dimensional submanifold of phase space, rather than a simple 1 dimensional curve. Each trajectory is locally 1 dimensional, but if you looked at the set of all points in phase space traced out over all time, you would find a higher-dimensional space, with dimension $2D-N_C$, where $N_C$ is the number of globally conserved quantities.

In most many body systems, $N_C$ is finite, or at least subextensive (i.e. any 'hidden' conservation laws are insignificant in the scaling limit as volume and particle number go to infinity, while keeping intrinsic parameters such as density and temperature constant). One takes total energy as the single most important conserved quantity, and the rest is textbook statistical mechanics.

Now, the precise type of non-linear behavior that would introduce ergodicity to the system is usually ignored in physics, because everything absorbs and emits radiation, which almost always causes systems to reach equilibrium. However, going a step deeper to consider self-thermalization of non-linear wave equations historically lead to the important discovery of the Fermi-Pasta-Ulam problem. Essentially, the discovery was that many nonlinear differential equations have hidden conservation laws that cause the heuristic argument described above to break down.

I'll have to disagree that those notions of entropy are disjoint. I'll try to explain my view.

In Statistical Mechanics entropy is defined in terms of accessible regions in phase space. It is the logarithm of this volume times a constant. In the process of deriving this formula starting from the number of accessible configurations it is postulated that all configurations must be equally accessible. This postulate is called the Ergodic Hypothesis. Since you're a mathematician I think you're probably familiarized with the term ergodic: it is a system whose evolution preserves measure (in our case, Liouville measure, which is Lebesgue's measure on phase space). Now, not every system is ergodic. Even though, estimates can be carried out and point that in a general gas, which has a huge number of particles, non-ergodicity would result in an extremely small error in Physics measurements (Laudau does that in his first volume on Thermodynamics). Even though, systems like spin glasses are canonical examples of non-ergodic systems where usual Statistical Mechanics is not applicable.

You see that the ergodic hypothesis is a key assumption in Statistical Mechanics. But what does chaos mean in Classical Mechanics? Well, it means your trajectories will cover your whole phase space. If you take a chaotic system (which is not only ergodic but also strongly mixing), the particle's trajectory will cover each and every bit of phase space accessible to it, bounded by energy conservation laws.

The conclusion is that if you assume Statistical Mechanics as being applicable, this is the same as assuming you cannot predict trajectories in phase space, either because you have too many initial conditions or because you can't track each and every trajectory, and after an infinite time they'll also cover the whole phase space. This is intrinsically connected to the notion of chaos in Classical Mechanics.

In Thermodynamics I think no one really understood what entropy meant, so I can't elaborate on that. It only gets clear in Statistical Mechanics.