标题： 非双曲步偏移积：遍历逼近 显示英文标题

标题： Nonhyperbolic step skew-products: Ergodic approximation

摘要： 我们研究在完全移位$k$, $k\ge2$的基础上建立的传递步态乘积映射，其纤维映射定义在圆上，并具有交织的收缩和扩张区域。 这些动力系统是真正非双曲的，并同时表现出具有正、负和零指数的遍历测度。 我们为纤维映射引入了一组公理，并研究了由此产生的乘积的动力学。 这些公理被证明捕捉了具有紧致中心叶的非双曲鲁棒传递映射的动力学的关键机制。 专注于这些系统的非双曲遍历测度（具有零纤维指数），我们证明这样的测度可以在弱$\ast$拓扑和熵的意义下被双曲测度逼近。 我们还证明它们位于具有正纤维指数的测度和具有负纤维指数的测度的凸包的交集中。 我们的方法也允许扰动双曲测度。 我们可以直接将一个具有负指数的测度扰动为一个具有正指数的测度（反之亦然），然而在这个过程中我们会损失一些熵。 熵的损失由测度的李雅普诺夫指数之间的差异决定。 显示英文摘要

摘要： We study transitive step skew-product maps modeled over a complete shift of $k$, $k\ge2$, symbols whose fiber maps are defined on the circle and have intermingled contracting and expanding regions. These dynamics are genuinely nonhyperbolic and exhibit simultaneously ergodic measures with positive, negative, and zero exponents. We introduce a set of axioms for the fiber maps and study the dynamics of the resulting skew-product. These axioms turn out to capture the key mechanisms of the dynamics of nonhyperbolic robustly transitive maps with compact central leaves. Focusing on the nonhyperbolic ergodic measures (with zero fiber exponent) of these systems, we prove that such measures are approximated in the weak$\ast$ topology and in entropy by hyperbolic ones. We also prove that they are in the intersection of the convex hulls of the measures with positive fiber exponent and with negative fiber exponent. Our methods also allow us to perturb hyperbolic measures. We can perturb a measure with negative exponent directly to a measure with positive exponent (and vice-versa), however we lose some amount of entropy in this process. The loss of entropy is determined by the difference between the Lyapunov exponents of the measures.