Math Conjectures

Positive metric entropy for the standard map

OPENLandmarkOpen problemProposed c. 1980 · Canonical special case

Canonical statement

There exists K∈R∖{0}K\in\mathbb R\setminus\{0\} such that the area-preserving standard map
SK(x,y)=(x+y+Ksin⁡(2πx), y+Ksin⁡(2πx))(modZ2) S_K(x,y)=\bigl(x+y+K\sin(2\pi x),\, y+K\sin(2\pi x)\bigr)\pmod{\mathbb Z^2}
of T2\mathbb T^2 has positive Kolmogorov--Sinai entropy with respect to normalized Lebesgue measure.
View source LaTeX
There exists \(K\in\mathbb R\setminus\{0\}\) such that the area-preserving standard map \[ S_K(x,y)=\bigl(x+y+K\sin(2\pi x),\, y+K\sin(2\pi x)\bigr)\pmod{\mathbb Z^2} \] of \(\mathbb T^2\) has positive Kolmogorov--Sinai entropy with respect to normalized Lebesgue measure.
DYN-008:[Chirikov1979UniversalInstability][Sinai1994TopicsErgodicTheory][Knill2005StandardMapEntropy][Blumenthal2020CompositionsStandardMaps]
Dynamical systemsMathematical physics

Notes

The standard map SKS_K is the basic area-preserving twist map of the two-torus, introduced by Chirikov as a model of instability in kicked oscillator systems [Chirikov1979UniversalInstability]. The problem asks whether for even a single nonzero parameter KK the map has positive Kolmogorov–Sinai entropy with respect to Lebesgue measure — that is, whether the numerically visible chaotic sea genuinely carries positive-measure chaos. Sinai promoted the metric-entropy question in the early 1980s [Sinai1994TopicsErgodicTheory].

What is proved falls short in a characteristic way. Horseshoes give positive topological entropy for broad parameter sets, and numerical evidence predicts a chaotic region of positive area for large KK; but horseshoes carry zero Lebesgue measure, and the possible presence of elliptic islands obstructs the known approaches to positive metric entropy [Knill2005StandardMapEntropy]. Positive results exist in modified regimes, such as statistical properties for compositions of standard maps with increasing coefficient [Blumenthal2020CompositionsStandardMaps].

No single parameter value is currently known to yield positive Lebesgue metric entropy, and the problem — emblematic of the coexistence question for conservative dynamics — remains open.

References (4)

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