Entropy for Three-dimensional Vector Fields

In this paper, we show that for any \(C^1\) $C^1$ three-dimensional vector fields with positive topological entropy, the topological entropy can be approximated by horseshoes. Precisely, for any \(C^1\) $C^1$ three-dimensional vector field X with positive topological entropy, there exists a vector field Y arbitrarily close (in the \(C^1\) $C^1$ topology) to X exhibiting a horseshoe \(\Lambda \) $\mathrm{\Lambda }$ such that the topological entropy of Y restricted on \(\Lambda \) $\mathrm{\Lambda }$ can arbitrarily approximate the topological entropy of X. This extends a classical result (Katok in Inst Hautes Études Sci Publ Math 51:137–173, 1980) of Katok for \(C^{1+\alpha }(\alpha >0)\) $C^{1+\alpha }(\alpha >0)$ surface diffeomorphisms and a result (Wu and Liu in Proc Am Math Soc 148(1):223–233, 2020) for \(C^1\) $C^1$ surface diffeomorphisms.