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 $f : X \rightarrow X$ 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 $\Lambda^{\mathbb{Z}}$, where $\Lambda$ is a finite alphabet, and a continuous surjection $\pi : Y \rightarrow X$ 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 $\upsilon$ supported by $Y$ in an optimized symbolic extension. This estimate depends on the entropy of the underlying measure $\mu$ on $X$, the “Lyapunov exponent” of $\mu$ (the genuine Lyapunov exponent for ergodic $\mu$, 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.