Almost automorphy of minimal sets for $C^1$-smooth strongly monotone skew-product semiflows on Banach spaces

TL;DR

This paper proves that in certain smooth, strongly monotone dynamical systems on Banach spaces, linearly stable minimal sets are necessarily almost automorphic, extending previous results to broader regularity conditions.

Contribution

It extends the known results on almost automorphy of minimal sets to $C^1$-smooth systems, broadening the applicability to less regular differential equations.

Findings

Linearly stable minimal sets are almost automorphic in $C^1$-smooth systems.

Extension of Shen and Yi's results to broader regularity conditions.

Broader applicability to almost periodically forced differential equations.

Abstract

We focus on the presence of almost automorphy in strongly monotone skew-product semiflows on Banach spaces. Under the $C^1$-smoothness assumption, it is shown that any linearly stable minimal set must be almost automorphic. This extends the celebrated result of Shen and Yi [Mem. Amer. Math. Soc. 136(1998), No. 647] for the classical $C^{1,\alpha}$-smooth systems. Based on this, one can reduce the regularity of the almost periodically forced differential equations and obtain the almost automorphic phenomena in a wider range.