MPEJ Volume 6, No.1, 18 pp. Received June 21 1999, Revised Jan 9 2000, Accepted Jan 9 2000 M. Salmhofer, Chr. Wieczerkowski Construction of the renormalized $GN_{2-\epsilon}$ trajectory ABSTRACT: We construct the renormalized Gross-Neveu trajectory in $2-\epsilon$ dimensions. Our construction uses a contraction mapping for an extended renormalization group. The extension is a running coupling constant with linear step $\beta$ function. The contraction mapping relies on norm estimates for a fermionic momentum space renormalization group.