Rectifiability of a class of invariant measures with one non-vanishing Lyapunov exponent

迭代函数系统所有李雅普诺夫指数的鲁棒消失

Robust vanishing of all Lyapunov exponents for iterated function systems

给定任何紧致连通流形$M$，我们描述$C^2$ -迭代函数系统（IFS）的开放集，允许全支持遍历措施的李雅普诺夫指数沿$M$都是零。此外，这些措施是近似的措施支持的周期轨道。我们还描述了$C^1$ -IFS的开放集，其正熵的遍历测度沿$M$的李雅普诺夫指数均为零。证明涉及到旗流形上诱导IFS的非双曲测度的构造。

极限模符号与李雅普诺夫谱

Limiting modular symbols and the Lyapunov spectrum

本文由限制模块符号的主题上的变化。涵盖的主题包括：作为连分数移位的李雅普诺夫指数的水平集上的Birkhoff平均的极限模符号的表达式，取决于广义Gauss-Kuzmin算子的谱性质的消灭定理，与模曲线上的非闭测地线相关联的某些非平凡同调类的构造，与移位不变集相关的某些Selberg zeta函数和C^* 代数。

Iterated function systems of affine expanding and contracting maps on the unit interval

We analyze the two-point motions of iterated function systems on the unit interval generated by expanding and contracting affine maps, where the expansion and contraction rates are determined by a pair $(M,N)$ of integers. This dynamics depends on the Lyapunov exponent. For a negative Lyapunov exponent we establish synchronization, meaning convergence of orbits with different initial points. For a vanishing Lyapunov exponent we establish intermittency, where orbits are close for a set of iterates of full density, but are intermittently apart. For a positive Lyapunov exponent we show the existe…

Randomly switched vector fields sharing a zero on a common invariant face

We consider a Piecewise Deterministic Markov Process given by random switching between finitely many vector fields vanishing at $0$. It has been shown recently that the behaviour of this process is mainly determined by the signs of Lyapunov exponents. However, results have only been given when all these exponents have the same sign. In this note, we consider the degenerate case where the process leaves invariant some face and results are stated when the Lyapunov exponents are of opposite signs. Applications are given to Lorenz vector fields with switching, and to SIRS model in random environme…

深度网络的最优初始化：深度Leaky ReLU网络的Lyapunov初始化与极限定理

Optimal Initialization in Depth: Lyapunov Initialization and Limit Theorems for Deep Leaky ReLU Networks

本文通过随机深度Leaky ReLU网络的严格概率分析，提出Lyapunov初始化方法，将Lyapunov指数设为零以确保激活稳定性，从而改善学习效果。

论非双曲遍历测度作为周期测度极限的存在性

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

本文利用跟踪引理在无扰动条件下统一重构 [GIKN] 与 [BBD1] 的构造，证明鲁棒传递非双曲及鲁棒环微分同胚的开稠子集上存在可由双曲周期测度逼近的非双曲遍历测度。