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 Λℤ, 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.
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