Given a piecewise $C^{1+\beta }$ map of the interval, possibly with critical points and discontinuities, we construct a symbolic model for invariant probability measures with nonuniform expansion that do not approach the critical points and discontinuities exponentially fast almost surely. More specifically, for each $\chi >0$ we construct a finite-to-one Hölder continuous map from a countable topological Markov shift to the natural extension of the interval map, that codes the lifts of all invariant probability measures as above with Lyapunov exponent greater than χ almost everywhere.

中文翻译：

---

具有非均匀扩展的一维地图的符号动力学

给定分段 $C^{\mathrm{1个}+\beta }$区间的地图，可能带有临界点和不连续点，我们为具有不均匀扩展的不变概率测度构建了一个符号模型，该模型几乎可以肯定地快地没有以指数方式快速接近临界点和不连续点。更具体地说，对于每个$\chi >0$我们从可数的拓扑马尔可夫位移到间隔图的自然扩展，构造了一个有限对一的Hölder连续图，该图对上述所有不变概率测度的升力进行编码，几乎所有地方的Lyapunov指数都大于χ。