Symbolic extensions of smooth interval maps

In this course we will present the full proof of the fact that every smooth dynamical system on the interval or circle X, constituted by the forward iterates of a function f : X → X which is of class Cr with r > 1, admits a symbolic extension, i.e., there exists a bilateral subshift (Y, S) with Y a closed shift-invariant subset of Λ, where Λ is a finite alphabet, and a continuous surjection π : Y → X which intertwines the action of f (on X) with that of the shift map S (on Y ). Moreover, we give a precise estimate (from above) on the entropy of each invariant measure ν supported by Y in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure μ on X, the “Lyapunov exponent” of μ (the genuine Lyapunov exponent for ergodic μ, otherwise its analog), and the smoothness parameter r. This estimate agrees with a conjecture formulated in [15] around 2003 for smooth dynamical systems on manifolds. AMS 2000 subject classifications: Primary 37E05, 37C40; secondary 37A35.