Networks are intrinsic to a broad spectrum of complex phenomena in the world around us: thoughts and memory emerge from the interconnection of neurons in the brain, nutrients and waste are transported through the cardiovascular system, and social and business networks link people. River networks stand as an archetypal example of branching networks, an important sub-class of all network structures. Of significant physical interest in and of themselves, river networks thus also provide an opportunity to develop results which are extendable to branching networks in general. To this end, this thesis carries out a thorough examination of river network geometry. The work combines analytic results, numerical simulations of simple models and measurements of real river networks. We focus on scaling laws which are central to the description of river networks. Starting from a few simple assumptions about network architecture, we derive all known scaling laws showing that only two scaling exponents are independent. Having thus simplified the description of networks we pursue the precise measurement of real network structure and the further refining of our descriptive tools. We address the key issue of universality, the possibility that scaling exponents of river networks take on specific values independent of region. We find that deviations from scaling are significant enough to preclude exact, definitive measurements. Importantly, geology matters as the externality of basin shape is shown to be part of the reason for these deviations. This implies that theories that do not incorporate boundary conditions are unable to produce realistic river network structures. We also extend a number of scaling laws to incorporate fluctuations about simple scaling. Going further than this, we find we are able to identify joint probability distributions that underlie these scaling laws. We generalize a well-known description of the size and number of network components as well as a description of network architecture, how these network components fit together. Both of these generalizations demonstrate that the spatial distribution of network components is random and, in this sense, we obtain the most basic level of network description.

The chaotic phenomenon of intermittency is modeled by a simple map of the unit interval, the Farey map. The long term dynamical behaviour of a point under iteration of the map is translated into a spin system via symbolic dynamics. Methods from dynamical systems theory and statistical mechanics may then be used to analyse the map, respectively the zeta function and the transfer operator. Intermittency is seen to be problematic to analyze due to the presence of an `indifferent fixed point'. Points under iteration of the map move away from this point extremely slowly creating pathological convergence times for calculations. This difficulty is removed by going to an appropriate induced subsystem, which also leads to an induced zeta function and an induced transfer operator. Results obtained there can be transferred back to the original system. The main work is then divided into two sections. The first demonstrates a connection between the induced versions of the zeta function and the transfer operator providing useful results regarding the analyticity of the zeta function. The second section contains a detailed analysis of the pressure function for the induced system and hence the original by considering bounds on the radius of convergence of the induced zeta function. In particular, the asymptotic behaviour of the pressure function in the limit β, the inverse of `temperature', tends to negative infinity is determined and the existence and nature of a phase transition at β=1 is also discussed.