  This paper is an interval dynamics counterpart of three theories founded
earlier by the authors, S. Smirnov and others in the setting of the iteration
of rational maps on the Riemann sphere: the equivalence of several notions of
non-uniform hyperbolicity, Geometric Pressure, and Nice Inducing Schemes
methods leading to results in thermodynamical formalism. We work in a setting
of generalized multimodal maps, that is smooth maps f of a finite union of
compact intervals in R into R with non-flat critical points, such that on its
maximal forward invariant set K the map f is topologically transitive and has
positive topological entropy. We prove that several notions of non-uniform
hyperbolicity of f|_K are equivalent (including uniform hyperbolicity on
periodic orbits, TCE & all periodic orbits in K hyperbolic repelling, Lyapunov
hyperbolicity, and exponential shrinking of pull-backs). We prove that several
definitions of geometric pressure P(t), that is pressure for the map f|_K and
the potential - t log |f'|, give the same value (including pressure on periodic
orbits, "tree" pressure, variational pressures and conformal pressure). Finally
we prove that, provided all periodic orbits in K are hyperbolic repelling, the
function P(t) is real analytic for t between the "condensation" and "freezing"
parameters and that for each such t there exists unique equilibrium (and
conformal) measure satisfying strong statistical properties.
