Connectedness of the Set of Central Lyapunov Exponents

We show that there is a residual subset \(\varvec{\mathcal {R}}\) $\mathcal{R}$ of \(\varvec{{{\,\textrm{Diff}\,}}^1(M)}\) ${\text{Diff}}^{\mathbf{1}}\mathbf{(}\mathit{M}\mathbf{)}$ such that for any \(\varvec{f\in \mathcal {R}}\) $\mathit{f}\mathbf{\in }\mathcal{R}$ and any partially hyperbolic homoclinic class \(\varvec{H(p,f)}\) $\mathit{H}\mathbf{(}\mathit{p}\mathbf{,}\mathit{f}\mathbf{)}$ with one-dimensional central direction, the set of central Lyapunov exponents associated with the ergodic measure with either full support or positive entropy is an interval.