[Paper Review] Sharpness for $C^1$ linearization of planar hyperbolic diffeomorphisms

Wenmeng Zhang, Weinian Zhang|arXiv (Cornell University)|May 17, 2013

Mathematical Dynamics and Fractals30 references3 citations

TL;DR

This paper establishes sharp Hölder regularity thresholds for $C^1$ linearization of planar hyperbolic diffeomorphisms in both Poincaré and Siegel domains. By analyzing invariant manifolds and functional equations via the Whitney extension theorem, it proves that the previously known $C^{1,eta}$ linearization exponent $\beta$ is optimal when the diffeomorphism is $C^{1,\alpha}$ with $\alpha = \alpha_0$ in the Poincaré domain and $\alpha \in (0,1]$ in the Siegel domain, resolving sharpness of regularity bounds.

ABSTRACT

Planar hyperbolic diffeomorphisms can be referred to two cases: Poincaré domain (both eigenvalues lie inside the unit circle $S^1$) and Siegel domain (one eigenvalue inside $S^1$ but the other outside $S^1$). In Poincaré domain it was proved that $C^{1,α}$ smoothness with $α_0:=1-\log|λ_2|/\log|λ_1|0$ is given, an interesting problem is: Is the exponent $α_0$ also sharp? On the other hand, in Siegel domain we only know that $C^{1,α}$ smoothness with $α\in (0,1]$ admits $C^1$ linearization. In this paper we further study the sharpness for $C^1$ linearization in both cases.

Motivation & Objective

• To determine the sharp Hölder exponent $\alpha_0$ for $C^1$ linearization of planar hyperbolic diffeomorphisms in the Poincaré domain, where $0 < |\lambda_1| < |\lambda_2| < 1$.
• To establish sharp regularity bounds for $C^1$ linearization in the Siegel domain, where $0 < |\lambda_1| < 1 < |\lambda_2|$.
• To resolve whether the exponent $\alpha_0 = 1 - \log|\lambda_2|/\log|\lambda_1|$ is sharp for $C^1$ linearization in the Poincaré domain.
• To derive sharp estimates for the Hölder exponent $\beta$ of the linearizing conjugacy $\Phi$ in terms of $\alpha$ in both domains.

Proposed method

• Analyzing the convergence of the sequence $\Lambda^{-n}F^n$ in the $C^1$ norm using precise estimates on the invariant manifold tangent to the stronger contractive direction.
• Employing functional equations derived from the conjugacy $\Phi \circ F = \Lambda \circ \Phi$ to study the regularity of the linearizing map $\Phi$.
• Applying the Whitney extension theorem to extend $C^{1,\alpha}$ data on a closed set to a global $C^{1,\beta}$ diffeomorphism, ensuring smoothness of the conjugacy.
• Using a duality argument with the inverse map $G = F^{-1}$ to extend results to the case $|\lambda_1\lambda_2| > 1$ in the Siegel domain.
• Deriving sharp bounds on $\beta$ by contradiction, assuming $\beta > -\sigma(1 - \sigma)^{-1}\alpha$ and showing inconsistency with the asymptotic behavior of iterates.
• Introducing a modified diffeomorphism $F_*$ with controlled nonlinear terms to isolate the regularity of the conjugacy and analyze its $C^{1,\beta}$ behavior.

Experimental results

Research questions

• RQ1 Is the exponent $\alpha_0 = 1 - \log|\lambda_2|/\log|\lambda_1|$ sharp for $C^1$ linearization of planar hyperbolic diffeomorphisms in the Poincaré domain when $0 < |\lambda_1| < |\lambda_2| < 1$?
• RQ2 What is the optimal Hölder exponent $\beta > 0$ for the linearizing conjugacy $\Phi$ when the diffeomorphism is $C^{1,\alpha}$ with $\alpha = \alpha_0$ in the Poincaré domain?
• RQ3 Can $C^1$ linearization be achieved for $C^{1,\alpha}$ diffeomorphisms in the Siegel domain with $\alpha \in (0,1]$, and is the resulting conjugacy $\Phi$ necessarily $C^{1,\beta}$ for some $\beta > 0$?
• RQ4 What are the sharp bounds on $\beta$ in terms of $\alpha$ when $|\lambda_1\lambda_2| \leq 1$ and $|\lambda_1\lambda_2| > 1$ in the Siegel domain?
• RQ5 Is the bound $\beta \leq -\sigma(1 - \sigma)^{-1}\alpha$ sharp for $C^{1,\beta}$ linearization when $|\lambda_1\lambda_2| \leq 1$ in the Siegel domain?

Key findings

• The exponent $\alpha_0 = 1 - \log|\lambda_2|/\log|\lambda_1|$ is sharp for $C^1$ linearization in the Poincaré domain, meaning that $C^1$ linearization fails for $\alpha < \alpha_0$.
• For $\alpha = \alpha_0$ in the Poincaré domain, the linearizing conjugacy $\Phi$ is $C^{1,\beta}$ with $\beta > 0$, and the bound $\beta \leq -\sigma(1 - \sigma)^{-1}\alpha$ is sharp.
• In the Siegel domain with $|\lambda_1\lambda_2| \leq 1$, the bound $\beta \leq -\sigma(1 - \sigma)^{-1}\alpha$ is sharp, and $\Phi$ is $C^{1,\beta}$ for $\alpha \in (0,1]$.
• For $|\lambda_1\lambda_2| > 1$ in the Siegel domain, the sharp bound is $\beta \leq (1 - \sigma)^{-1}\alpha$, which is optimal.
• The results confirm that the previously known $C^{1,\beta}$ linearization with $\beta > 0$ is optimal when $\alpha = \alpha_0$ in the Poincaré domain and $\alpha \in (0,1]$ in the Siegel domain.
• The sharpness of the bounds is established via contradiction by analyzing the asymptotic behavior of iterates and the regularity of the conjugacy on invariant manifolds.