The first main topic of this thesis is the thorough analysis of two families of piecewise linear maps on the unit interval, the α-Lüroth and α-Farey maps. Here, α denotes a countably infinite partition of the unit interval whose atoms only accumulate at the origin. The basic properties of these maps will be developed, including that each α-Lüroth map (denoted Lα) gives rise to a series expansion of real numbers in [0,1], a certain type of Generalised Lüroth Series. The first example of such an expansion was given by Lüroth. The map Lα is the jump transformation of the corresponding α-Farey map Fα. The maps Lα and Fα share the same relationship as the classical Farey and Gauss maps which give rise to the continued fraction expansion of a real number. We also consider the topological properties of Fα and some Diophantine-type sets of numbers expressed in terms of the α-Lüroth expansion. Next we investigate certain ergodic-theoretic properties of the maps Lα and Fα. It will turn out that the Lebesgue measure λ is invariant for every map Lα and that there exists a unique Lebesgue-absolutely continuous invariant measure for Fα. We will give a precise expression for the density of this measure. Our main result is that both Lα and Fα are exact, and thus ergodic. The interest in the invariant measure for Fα lies in the fact that under a particular condition on the underlying partition α, the invariant measure associated to the map Fα is infinite. Then we proceed to introduce and examine the sequence of α-sum-level sets arising from the α-Lüroth map, for an arbitrary given partition α. These sets can be written dynamically in terms of Fα. The main result concerning the α-sum-level sets is to establish weak and strong renewal laws. Note that for the Farey map and the Gauss map, the analogue of this result has been obtained by Kesseböhmer and Stratmann. There the results were derived by using advanced infinite ergodic theory, rather than the strong renewal theorems employed here. This underlines the fact that one of the main ingredients of infinite ergodic theory is provided by some delicate estimates in renewal theory. Our final main result concerning the α-Lüroth and α-Farey systems is to provide a fractal-geometric description of the Lyapunov spectra associated with each of the maps Lα and Fα. The Lyapunov spectra for the Farey map and the Gauss map have been investigated in detail by Kesseböhmer and Stratmann. The Farey map and the Gauss map are non-linear, whereas the systems we consider are always piecewise linear. However, since our analysis is based on a large family of different partitions of U , the class of maps which we consider in this paper allows us to detect a variety of interesting new phenomena, including that of phase transitions. Finally, we come to the conformal systems of the title. These are the limit sets of discrete subgroups of the group of isometries of the hyperbolic plane. For these so-called Fuchsian groups, our first main result is to establish the Hausdorff dimension of some Diophantine-type sets contained in the limit set that are similar to those considered for the maps Lα. These sets are then used in our second main result to analyse the more geometrically defined strict-Jarník limit set of a Fuchsian group. Finally, we obtain a “weak multifractal spectrum” for the Patterson measure associated to the Fuchsian group.

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In this thesis, we explore the robustness of heteroclinic cycles which can appear as solutions of dynamical systems subject to certain constraints. We develop a method, using topological notions, to inspect the dynamics of simple heteroclinic cycles; in particular, in the first part of the thesis we develop a “restricted Conley index”. This tool is defined by restricting the general Conley index to specific invariant subspaces associated with constraints on the vector field. The resulting restricted Conley index allows us to find connections that are robust to perturbations that respect these constraints. In the second part of this thesis we study the dynamical problem of designing a system of globally coupled oscillator systems with a specific structure. We extend some known results on conditions for stability of cluster states in these systems and, as an example, we give sufficient stability conditions for cluster states with three non-trivial clusters, (2, 2, 2)−cluster states, in a system of six globally coupled oscillators. We show that robust heteroclinic cycles connecting these states can appear as a result of our investigation on dynamics of a systems of six globally coupled oscillators, and we use the restricted Conley index to investigate the robustness of heteroclinic cycles between three nontrivial clusters.

The interaction of topology and dynamics has attracted a great deal of attention from numerous mathematicians. This thesis is devoted to the study of dynamical systems on low-dimensional manifolds. In the order of dimensions, we first look at the case of two-manifolds (surfaces) and derive explicit differential equations for dynamical systems defined on generic surfaces by applying elliptic and automorphic function theory to uniformise the surfaces in the upper half of the complex plane with the hyperbolic metric. By modifying the definition of the standard theta series, we will determine general meromorphic systems on a fundamental domain in the upper half plane, the solution trajectories of which 'roll up' onto an appropriate surface of any given genus. Meanwhile, we will show that a periodic nonlinear, time-varying dissipative system that is defined on a genus-p surface contains one or more invariant sets which act as attractors. Moreover, we shall generalize a result in [Martins, 2004] and give conditions under which these invariant sets are not homeomorphic to a circle individually, which implies the existence of chaotic behaviour. This is achieved by analyzing the topology of inversely unstable solutions contained within each invariant set. Then the thesis concerns a study of three-dimensional systems. We give an explicit construction of dynamical systems (defined within a solid torus) containing any knot (or link) and arbitrarily knotted chaos. The first is achieved by expressing the knots in terms of braids, defining a system containing the braids and extending periodically to obtain a system naturally defined on a torus and which contains the given knotted trajectories. To get explicit differential equations for dynamical systems containing the braids, we will use a certain function to define a tubular neighbourhood of the braid. The second one, generating chaotic systems, is realized by modelling the Smale horseshoe. Moreover, we shall consider the analytical and topological structure of systems on 2- and 3- manifolds. By considering surgery operations, such as Dehn surgery, Heegaard splittings and connected sums, we shall show that it is possible to obtain systems with 'arbitrarily strange' behaviour, Le., arbitrary numbers of chaotic regimes which are knotted and linked in arbitrary ways. We will also consider diffeomorphisms which are defined on closed 3-manifolds and contain generalized Smale solenoids as the non-wandering sets. Motivated by the result in [Jiang, Ni and Wang, 2004], we will investigate the possibility of generating dynamical systems containing an arbitrary number of solenoids on any closed, orientable 3-manifold. This shall also include the study of branched coverings and Reeb foliations. Based on the intense development from four-manifold theory recently, we shall consider four-dimensional dynamical systems at the end. However, this part of the thesis will be mainly speculative.

Molecular dynamics trajectories that sample from a Gibbs, or canonical, distribution can be generated by introducing a modified Hamiltonian with additional degrees of freedom as described by Nose [46]. Although this method has found widespread use in its time re-parameterized Nose-Hoover form, the lack of a Hamiltonian, and the need to 'tune' thermostatting parameters has limited, its use compared to stochastic methods. In addition, since the proof of the correct sampling is based on an ergodic assumption, thermostatting small of stiff systems often does not given the correct distributions unless the Nose-Hoover chains [43] method is used, which inherits the Nose-Hoover deficiencies noted above. More recently the introduction of the Hamiltonian Nose-Poincare method [11], where symplectic integrators can be used for improved long term stability, has renewed interest in the possibility of Hamiltonian methods which can improve dynamical sampling. This class of methods, although applicable to small systems, has applications in large scale systems with complex chemical structure, such as protein-bath and quantum-classical models.;For Nose dynamics, it is often stated that the system is driven to equilibrium through a resonant interaction between the self-oscillation frequency of the thermostat variable and a natural frequency of the underlying system. By the introduction of multiple thermostat Hamiltonian formulations, which are not restricted to chains, it has been possible to clarify this perspective, using harmonic models, and exhibit practical deficiencies of the standard Nose-chain approach. This has led to the introduction of two Hamiltonian schemes, the Nose-Poincare chains method and the Recursive Multiple Thermostat (RMT) method. The RMT method obtains canonical sampling without the stability problems encountered with chains with the advantage that the choice of Nose mass is independent of the underlying system.

The universal theme connecting the topics of this thesis is how microscopic details can help us understand the macroscopic behaviour of Hamiltonian systems. We will focus on three aspects of Hamiltonian dynamics which we now describe, this is not intended to be a comprehensive description, we provide a comprehensive introduction to each aspect in the corresponding chapters.