TY - BOOK ID - 46358325 TI - Geometric pressure for multimodal maps of the interval AU - Przytycki, Feliks AU - Rivera-Letelier, Juan PY - 2019 SN - 9781470435677 1470435675 PB - Providence, RI : American Mathematical Society, DB - UniCat KW - Lyapunov functions. KW - Riemann surfaces. KW - Differential equations. KW - Riemann, Surfaces de. KW - Liapounov, Fonctions de. KW - Équations différentielles. KW - Conformal geometry. KW - Mappings (Mathematics) KW - Géométrie conforme KW - Applications (Mathématiques) KW - Conformal geometry KW - Riemann surfaces KW - Maps (Mathematics) KW - Functions KW - Functions, Continuous KW - Topology KW - Transformations (Mathematics) KW - Surfaces, Riemann KW - Circular geometry KW - Geometry of inverse radii KW - Inverse radii, Geometry of KW - Inversion geometry KW - Möbius geometry KW - Geometry UR - https://www.unicat.be/uniCat?func=search&query=sysid:46358325 AB - "This memoir 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 I 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 pullbacks). 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"-- ER -