Dynamical Systems Method for solving nonlinear equations with locally Holder continuous monotone operators

Continuity of Lyapunov exponents in all Hölder topologies for irreducible cocycles

We prove that a locally constant $SL_{2}(\mathbb{R})$-valued cocycle over the shift generated by an irreducible collection of matrices is a continuity point for Lyapunov exponents in the $α$-Hölder topology for every $α> 0$. This gives negative answers to conjectures of Viana and the author; we pose a new conjecture to replace these conjectures. We show that an analogous continuity result also holds for $GL_{2}(\mathbb{R})$-valued cocycles that admit canonical holonomies.

平坦 Wieler 螺线管的横向 Hölder 正则性

On transversal Hölder regularity for flat Wieler solenoids

本文围绕平坦 Wieler 螺线管，考察其上同调、Ruelle 谱刻画的混合速率及本原替换子移位的上同调方程解，强调 Cantor 集上 α-Hölder 正则性的统一作用。

Non-degenerate locally connected models for plane continua and Julia sets

Suppose that a $X$ is an \emph{unshielded} plane continuum (i.e., $X$ coincides with the boundary of the unbounded complementary component of $X$). Then there exists a \emph{finest monotone} map $m:X\to L$, where $L$ is a locally connected continuum (i.e., $m^{-1}(y)$ is connected for each $y\in L$, and any monotone map $φ:X\to L'$ onto a locally connected continuum is a composition $φ=φ'\circ m$ where $φ':L\to L'$ is monotone). Such finest locally connected model $L$ of $X$ is easier to understand because $L$ is locally connected (in particular it can be described by a picture) and represents…

Hölder continuity of the Lyapunov exponents of linear cocycles over hyperbolic maps

Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a centra…

On fair entropy of the tent family

The notions of fair measure and fair entropy were introduced by Misiurewicz and Rodrigues recently, and discussed in detail for piecewise monotone interval maps. In particular, they showed that the fair entropy $h(a)$ of the tent map $f_a$, as a function of the parameter $a=\exp(h_{top}(f_a))$, is continuous and strictly increasing on $[\sqrt{2},2]$. In this short note, we extend the last result and characterize regularity of the function $h$ precisely. We prove that $h$ is $\frac{1}{2}$-Hölder continuous on $[\sqrt{2},2]$ and identify its best Hölder exponent on each subinterval of $[\sqrt{2}…

Discontinuity of Lyapunov Exponents Near Fiber Bunched Cocycles

We give examples of locally constant $SL(2,\mathbb{R})$-cocycles over a Bernoulli shift which are discontinuity points for Lyapunov exponents in the Hölder topology and are arbitrarily close to satisfying the fiber bunching inequality. Backes, Brown, and the author have shown that the Lyapunov exponents vary continuously when restricted to the space of fiber bunched Hölder continuous cocycles. Our examples give evidence that this theorem is optimal within certain families of Hölder cocycles.

度量Anosov流的Hölder正则性

Hölder regularity for metric Anosov flows

本文定义$C^\alpha$度量Anosov流，给出其正则性判据，证明这类流的局部乘积结构、拓扑共轭、马尔可夫编码的Hölder正则性，并将结果应用于负曲率流形的测地流。