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

Abstract

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