A Conjectura de entropia para dinâmicas infinitamente diferenciáveis

Abstract

This work lies in the theory of dynamical systems and has as its central theme Shub's Entropy Conjecture, which compares the topological entropy h(f) of a system (N, f) with the homology induced dynamics via the spectral radii sp(fi∗ ). The main objective is to present and prove in detail Yomdin's Theorem, which establishes the validity of the conjecture for C∞ maps on compact manifolds. The strategy follows the introduction of an intermediate invariant of volume growth, v(f), which bridges homological invariants and entropy: one shows S(f) ≤ v(f), and obtains an upper bound v(f) ≤ h(f) under regularity assumptions, concluding S(f) ≤ h(f) in the C∞ case. The technical core involves local volume control via reparametrizations and tools from Semialgebraic Geometry, from which one obtains decompositions with controlled derivatives and uniform bounds for the volume of the relevant pieces in the iterates of f.