Variational Principle for Nonhyperbolic Ergodic Measures: Skew Products and Elliptic Cocycles

For a large class of transitive non-hyperbolic systems, we construct nonhyperbolic ergodic measures with entropy arbitrarily close to its maximal possible value. The systems consider are partially hyperbolic one-dimensional central direction for which there positive whose Lyapunov exponent is negative, zero, or positive. We zero and the topological set points zero. This provides restricted variational principle (zero exponent) measures. result applied setting $$\mathrm {SL}(2,\mathbb {R})$$ matrix cocycles counterpart Furstenberg’s classical result: an open dense subset elliptic upper metric infinite products subexponential growth norm.