Thermodynamic formalism and multifractal analysis for general topological dynamical systems

We investigate to what degree results in dimension theory and multifractal formalism can be derived as a direct consequence of thermodynamic properties of a dynamical system. We show that under quite general conditions, various multifractal spectra (the entropy spectrum for Birkhoff averages and the dimension spectrum for pointwise dimensions, among others) may be obtained as Legendre transforms of functions T : R→R arising in the thermodynamic formalism. We impose minimal requirements on the maps we consider, and obtain partial results for any continuous map f on a compact metric space. In order to obtain complete results, the primary hypothesis we require is that the functions T be continuously differentiable. This makes rigorous the general paradigm of reducing questions regarding the multifractal formalism to questions regarding the thermodynamic formalism. These results hold for a broad class of measurable potentials, which includes (but is not limited to) continuous functions. We give applications that include most previously known results, as well as some new ones.;Along the way, we show that Bowen's equation, which characterises the Hausdorff dimension of certain sets in terms of the topological pressure of an expanding conformal map, applies in greater generality than has been heretofore established. In particular, we consider an arbitrary subset Z of a compact metric space and require only that the lower Lyapunov exponents be positive on Z, together with a tempered contraction condition. Among other things, this allows us to compute the dimension spectrum for Lyapunov exponents in terms of the entropy spectrum for Lyapunov exponents, and is also a crucial tool in the aforementioned results on the dimension spectrum for local dimensions.