Large entropy implies existence of a maximal entropy measure for interval maps

Abstract

We give a new type of sufficient condition for the existence of measures with maximal entropy for an interval map f , using some non-uniform hyperbolicity to compensate for a lack of smoothness of f . More precisely, if the topological entropy of a C 1 interval map is greater than the sum of the local entropy and the entropy of the critical points, then there exists at least one measure with maximal entropy. As a corollary, we obtain that any C r interval map f such that h top (f ) > 2 log f ∞ /r possesses measures with maximal entropy. * Let us notice that in this definition the supremum over all points x can be moved outside of the limits on , δ and n (see [3] or [11]).

surface diffeomorphisms with no maximal entropy measure

“…Indeed, the examples built here satisfy h top (f ) ≤ log Lip(f )/r. This is known in the setting of interval maps [7,3].…”

“…Division of the orbit. For x ∈]−1/2, 1/2| 2 \Q and v ∈ R 2 \{0}, we are going to show (7) with χ = log A for some A > 1 and "large" parameters K, L, n 0 (how large will be specified).…”

“…The variational principle states that continuity is enough to ensure the existence of measures with entropy arbitrarily close to h top (f ). Sufficient conditions for achieving this value include uniform hyperbolicity and, by a classical result of Newhouse [11], C ∞ smoothness (or a combination [7]). Newhouse theorem is sharp: for every r < ∞, there exist C r smooth dynamical systems on compact manifolds with no maximal measures.…”

Existence, uniqueness and stability of equilibrium states for non-uniformly expanding maps

“…This list is certainly not complete. Some results, including [17] and [38] are specific for measures of maximal entropy. An important notion of entropy-expansiveness was introduced by Buzzi [12], which influenced [14,38] among other papers.…”

Measures of maximal entropy for surface diffeomorphisms

“…However their topological entropy is smaller than (but arbitrarily close to) λ min (f )/r. It has been asked [31] whether this is optimal as it is for interval maps (see [37,22]):…”