The theory of Fatou and Julia is extended to include the dynamics of functions f which are meromorphic in \widehat{\mathbb{C}} outside a totally disconnected compact set E(f) at whose points the cluster set of f is \widehat{\mathbb{C}}. The Julia set is defined not only by the standard approach but is also characterized in terms of the set of points whose orbits approach a point of E(f). For the subclass where E(f) has a complement of class OAD and the inverse of f has a finite set of singular points it is shown that neither wandering components nor Baker domains occur in F(f)>. As an application, functions of a certain general class are shown to have a totally disconnected Julia set.

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This work explores the potential application of naturally-occurring minerals as inexpensive catalysts in heterogeneous Fenton, namely catalytic wet peroxide oxidation (CWPO). The availability, low cost and environmentally friendly character of those materials make them interesting candidates for such application. The performance of magnetite, hematite and ilmenite as CWPO catalysts has been tested under different working conditions, which include temperature (25–90 °C), H2O2 dose (250–1000 mg L−1) and catalyst concentration (1–4 g L−1). The operating temperature plays a key role on the rate of H2O2 decomposition so that with magnetite H2O2 conversion after 4 h increased from 8 to 99% by increasing the temperature from 25 to 90 °C. Based on the reaction mechanism proposed, a kinetic model was developed which successfully described the experimental results on H2O2 decomposition. The catalytic performance of the minerals tested at temperatures above the ambient was demonstrated using phenol (100 mg L−1) as target pollutant. Unprecedented efficiencies of H2O2 consumption, higher than 80% were achieved, allowing high oxidation and mineralization, i.e. complete phenol conversion and almost 80% TOC reduction at 75 °C with a catalyst loading of 2 g L−1 and the theoretical stoichiometric amount of H2O2 for complete mineralization of phenol (500 mg L−1). Magnetite is particularly attractive, since it showed the highest activity and can be easily separated from the liquid phase given its magnetic properties. All the minerals tested suffered low iron leaching and magnetite and hematite showed a good reusability upon three consecutive runs. However, in this case long-term durability is not a crucial issue, given the availability and low cost of these mineralsThis research has been supported by the Spanish MINECO through the project CTQ2013-4196-R and by the CM through the project S2013/MAE-2716. M. Muñoz thanks the Spanish MINECO for a Juan de la Cierva-Incorporación postdoctoral contract (IJCI-2014-19427

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The residual Julia set, denoted by J r (f ), is defined to be the subset of those points of the Julia set which do not belong to the boundary of any component of the Fatou set. The points of J r (f ) are called buried points of J(f ) and a component of J(f ) which is contained in J r (f ) is called a buried component. In this paper we survey the most important results related with the residual Julia set for several classes of functions. We also give a new criterium to deduce the existence of buried points and, in some cases, of unbounded curves in the residual Julia set (the so called Devaney hairs). Some examples are the sine family, certain meromorphic maps constructed by surgery and the exponential family.