The entropy spectra of chaotic maps of an interval

Abstract

For a continuous self-map f of a real interval L, let ent(f) denote the supremum of the topological entropies of the restrictions of f to its compact invariant sets; for a compact interval this is the usual topological entropy. For a class P of maps, the entropy spectrum of P is the set of values ent(f) taken by the maps f in P. We determine the entropy spectra of the three standard classes of chaotic maps of the compact interval [0,1]. Devaney chaotic maps have spectrum [(1/2) log 2, infinity], maps of positive entropy have spectrum (0, infinity], and Li-Yorke chaotic maps have spectrum [0, infinity]; thus every entropy compatible with the known lower bounds actually occurs, and the three spectra are strictly nested. Mixing maps have spectrum ((1/2) log 2, infinity], and Devaney chaotic maps with at least two fixed points have spectrum [log 2, infinity]. As a consequence, a Devaney chaotic map of the interval attains the minimal entropy (1/2) log 2 only if it has a unique fixed point and is not mixing. For an arbitrary interval, compact or not, maps with a point of least period n have spectrum [log sigma_n, infinity], where sigma_n are the Stefan constants, and Li-Yorke chaotic maps have spectrum [0, infinity]. The proofs combine a truncation theorem for maps of non-compact intervals, showing that clipping a map to a compact subinterval never creates entropy; a horseshoe argument showing that two fixed points force entropy log 2 on transitive maps; and explicit constructions, including an exact piecewise linear family, a mixing map of infinite entropy, and a square-root construction that halves the entropy of a transitive map.