Thermodynamic formalism. In this section we introduce the tools from thermodynamic formalism which will be used extensively in the proofs of many results throughout this thesis. Thermodynamic formalism is a branch of ergodic theory which originated in statistical mechanics. While the ergodic theorem provides us with a useful tool for studying the orbits of typical points, it does not provide us with a ‘natural’ invariant measure to equip the space with. Here ‘natural’ is used loosely and depends on the characteristic of the dynamical system that one is interested in, such as for instance an invariant measure which maximises dimension or entropy. Xxxxx measures were translated from statistical mechanics to the setting of dynamical systems by Xxxxxx and Sinai beginning with [S], providing a class of invariant measures whose properties were closely connected with the properties of the Xxxxx potential. The subsequent body of work that followed connecting Xxxxx measures with other analogues of notions from statistical mechanics such as pressure, equilibrium states and entropy all in one beautiful and interwoven theory is now called thermodynamic formalism. The connections established by this theory have proved to be powerful tools in many areas of dynamical systems including its dimension theory, rates of mixing and statistical properties of dynamical systems. The monographs of Xxxxx and Xxxxxx [Bo; Ru] provide classical expositions of thermodynamic formalism in the original settings in which it was developed. Of course this body of work has since grown and indeed, thermodynamic formalism will appear in a number of different settings in this thesis. Therefore this section will be split into three parts, each of which summarises the relevant results that will be used from each setting. Firstly, we briefly touch upon the thermodynamic formalism of H¨older continuous potentials for the subshift of finite type. Only very basic results from this setting will be used. Secondly, the most detailed account of results from the thermodynamic formalism will be given for the setting of the countable shift, which will be used throughout Chapters 3-5. Finally, we will give an overview that contains the analogue of these ideas in the sub-additive setting, that is, for sub-additive potentials on a subshift of finite type. Results from this setting will be used in Chapter 6.
