Symbolic extensions of smooth interval maps

Abstract

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 C r 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 ℤ , 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.