The Myers-Steenrod theorem for Finsler manifolds of low regularity

We prove a version of Myers-Steenrod’s theorem for Finsler manifolds under minimal regularity hypothesis. In particular we show that an isometry between $C^{k,\alpha }$-smooth (or partially smooth) Finsler metrics, with $k+\alpha >0$, $k\in \mathbb{N}\cup \{0\}$, and $0\le \alpha \le 1$ is necessary a diffeomorphism of class $C^{k+1,\alpha }$. A generalisation of this result to the case of Finsler 1-quasiconformal mapping is given. The proofs are based on the reduction of the Finlserian problems to Riemannian ones with the help of the the Binet-Legendre metric.