Sinai's conjecture: positive metric entropy of the standard map for a positive-measure set of parameters

Theorem 1.1: there is $k_0>0$ with $h_m(f_k)>0$ for every $k\ge k_0$; equivalently, by Pesin's formula, the top Lyapunov exponent is positive on a set of positive area. Corollary 1.2: a positive-area ergodic component with nonzero exponents that splits into finitely many cyclic Bernoulli pieces. It does NOT claim ergodicity of area, positive exponents almost everywhere, a uniform lower bound on the entropy, or any explicit $k_0$, and says nothing about small or moderate $k$.

Theorem 1.1: there is $k_0>0$ with $h_m(f_k)>0$ for every $k\ge k_0$; equivalently, by Pesin's formula, the top Lyapunov exponent is positive on a set of positive area. Corollary 1.2: a positive-area ergodic component with nonzero exponents that splits into finitely many cyclic Bernoulli pieces. It does NOT claim ergodicity of area, positive exponents almost everywhere, a uniform lower bound on the entropy, or any explicit $k_0$, and says nothing about small or moderate $k$.