Ergodic measures with infinite entropy (Q6614531)

Let \( M \) be a compact \( C^1 \) manifold of finite dimension \( m \geq 1 \), endowed with a Riemannian metric \( \mathrm{dist} \). Define \( C_0(M) \) as the space of continuous functions \( f : M \to M \), equipped with the metric\N\[\N\|f - g\|_{C_0} := \max_{x \in M} \mathrm{dist}(f(x), g(x)) \qquad \text{for all } f, g \in C_0(M).\N\]\NIn this paper, the authors construct ergodic probability measures with infinite metric entropy for generic continuous maps and homeomorphisms on compact manifolds. They also construct sequences of such measures that converge to a measure with zero entropy. The central result of the paper is summarized in the following theorem: \N\NTheorem. For a generic map \( f \in C_0(M) \), there exists an ergodic Borel probability measure \( \mu \) such that \( h_\mu(f)= +\infty \). Moreover, there exists an integer \( p \geq 1 \) such that \( \mu \) is mixing for the map \( f^p \).\N\NLet \(\lim^{*}\) denote the limit in the space of probability measures endowed with the weak\(^{*}\) topology. The second result of the paper is as follows: \N\NTheorem. For a generic map \( f \in C_0(M) \), there exists a sequence of ergodic measures \( \mu_n \) such that for all \( n \geq 1 \), we have \( h_{\mu_n}(f) = +\infty \), and \( \lim^{*}_{n \to \infty} \mu_n = \mu\) with \(h_{\mu}(f) = 0.\)\N\NTo establish the main theorems, the authors provide a substantial number of supporting lemmas.