OAI.StandardMapEntropy.main_entropy_complete

The theorem states that there is a threshold k₀>0 such that, for every real k≥k₀, three things hold for the standard map on the torus (ℝ/ℤ)², given by (x,y) ↦ (x+y+k sin(2πx), y+k sin(2πx)) with coordinates taken mod 1, and with area the product of Lebesgue measures on the two circles. First, the metric entropy of the standard map with respect to area is strictly positive, where this entropy (valued in [0,∞]) is the supremum over finite measurable partitions p of the infimum over n≥1 of the block entropy of the partition refined along the first n iterates, divided by n. Second, PositiveLargestLyapunov(k) holds: there is a measurable set E of positive area such that every point z in E has some l>0 for which the limit of (1/n) log of the operator norm of the n-step derivative product at z equals l, and LyapunovSpectrumAt(k,z,l) holds, meaning that either vectors all grow at exponential rate 0, or l>0 and there is a nonzero direction s such that every nonzero vector v has growth rate −l if s and v are parallel (wedge zero) and +l otherwise. The derivative product is built from a coordinate-conjugated 2×2 transfer matrix with potential 2+2πk cos(2πx). Third, the entropy is positive if and only if PositiveLargestLyapunov(k) holds; since both hold, this equivalence is true for large k.