預估自由半群動力系統之局部熵潛能量將有上半連續(upper semicontinuity)的性質, 共同擁有交換性架構, 亦期望發現 Birkhoff Ergodic Theorem 相似成果. 當系統的自由度無限增大時, 遍歷的可能性也就越來越增大,乘幕法則將影響半群系統不確定性, 想理下合理預測 Shannon-McMillan-Breimann 定理在動力系統中, 共同擁有不變測度. 是否有相似成果, 將進一步審查 power rule, product rule, affinity, generator是否保持. 因為混沌半群系統的發展正處於實質性應用開發的研究階段, 符號動力學(symbolic dynamics)已是非線性混沌系統研究的核心部分. 在不變系統架構的假設之下, 預估把多函數半群動力系統乘幕法則的探討, 提起到符號動力學上研究它的遞移性(transitivity)、遞迴性(recurrence) 及周期性質.
Predict this semigroup system keeps the property of upper semicontinuity. The community of the measure-thoeretic entropy will be correct. We also check similar Shannon-McMillan-Breimann under this system. Moreover, under the symbolic dynamics, we also can discuss the transitivity, recurrence and periodic properties in the semigroup case.
