| This thesis examines the thermodynamic formalism of nonuniformly hyperbolic dynamical systems in two cases.;In the first part, we study the nonadditive thermodynamic formalism for the class of almost-additive sequences of potentials. We define the topological pressure PZ(phi) of an almost-additive sequence phi, on a compact f-invariant set Z. We give conditions which allow us to establish a variational principle for the topological pressure. We state conditions for the existence and uniqueness of equilibrium measures. In the special case of subshifts of finite type we state conditions for the existence and uniqueness of Gibbs measures. We compare our results for almost-additive sequences to the thermodynamic formalism for additive sequences [Rue78, Sin72, Bow70], nonadditive sequences [Bar96], subadditive sequences [Fal88], and the almost-additive sequence studied by Feng and Lau [FL02, Fen04].;Second, we study the thermodynamic formalism for discontinuous potentials. We give conditions under which the topological pressure of a discontinuous potential can be defined. A corresponding variational principle is established, no additional conditions are required. This thermodynamic formalism is applied to nonuniformly hyperbolic maps f and the corresponding potentials ϕ t(x) = -t log Jac( df∣Eux ). Several specific examples are considered, namely countable Markov shifts [Sar99], one-dimensional maps [PS05], and diffeomorphisms admitting a Young tower [You98]. |
