arXiv subjects

Wenmeng Zhang

Publications and source records attributed to Wenmeng Zhang.

9 recordsLinked to original sources

arXiv2026

Smooth linearization of nonautonomous dynamics under general dichotomic behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a very general form of dichotomy known as $μ$-dichotomy and that the associated $μ$-dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. We observe that our notion of $μ$-dichotomy encompasses the classical notions of exponential, polynomial and logarithmic dichotomies as very particular cases. In particular, our result is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between $μ$-dichotomy and exponential dichotomy via a suitable reparametrization of time.

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arXiv2026

Differentiable normal linearization of partially hyperbolic dynamical systems

A result on $C^0$ linearization which is differentiable at the hyperbolic fixed point is known. In this paper, we further investigate a partially hyperbolic diffeomorphism $F$ to find a local $C^0$ conjugacy, which is $C^1$ on the center manifold, to linearize the hyperbolic component (normal to the center direction) and obtain its Takens' normal form. Our result is optimal, as it needs no non-resonant condition usually required for smooth conjugacy (e.g., as in the Takens' theorem) and the $C^{1,α}$ $(α>0)$ smoothness condition is sharp. For the proof, the center direction obstructs the decoupling of $F$ as the stable and unstable foliations do not intersect. We overcome this difficulty via a semi-decoupling method only with the unstable foliation, where a modified Lyapunov-Perron equation needs to be established along the center direction. Subsequent issues of cocycle reduction and differentiable linearization for an expansive fiber-preserving mapping are then addressed by the Whitney's extension theory and a lifting technique, respectively. In the local context, our result improves the result of $C^0$ normal linearization by [C. Pugh and M. Shub, Invent. Math., 10 (1970): 187-198] to a differentiable one.

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arXiv2024

Takens Theorem for nonautonomous partially hyperbolic dynamical systems

Takens Theorem for a partially hyperbolic dynamics provides a normal linearization along the center manifold. In this paper, we give the nonautonomous version of Takens Theorem under non-resonance conditions formulated in terms of the dichotomy spectrum. In our proof, one difficulty is to solve homological equations for the normal form theory which involve a center variable, while another difficulty is to find the dichotomy spectrum of a certain matrix cocycle that is block lower triangular. In order to overcome those difficulties, in comparison with the autonomous case, we need an additional term in the (nonautonomous) non-resonance conditions to guarantee certain spectral gap conditions. This additional term disappears naturally in the autonomous case.

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arXiv2023

Smooth invariant foliations without a bunching condition and Belitskii's $C^{1}$ linearization for random dynamical systems

Smooth linearization is one of the central themes in the study of dynamical systems. The classical Belitskii's $C^1$ linearization theorem has been widely used in the investigation of dynamical behaviors such as bifurcations, mixing, and chaotic behaviors due to its minimal requirement of partial second order non-resonances and low regularity of systems. In this article, we revisit Belitskii's $C^1$ linearization theorem by taking an approach based on smooth invariant foliations and study this problem for a larger class of dynamical systems ({\it random dynamical systems}). We assumed that the linearized system satisfies the condition of Multiplicative Ergodic Theorem and the associated Lyapunov exponents satisfy Belitskii's partial second order non-resonant conditions. We first establish the existence of $C^{1,β}$ stable and unstable foliations without assuming the bunching condition for Lyapunov exponents, then prove a $C^{1,β}$ linearization theorem of Belitskii type for random dynamical systems. As a result, we show that the classical Belitskii's $C^1$ linearization theorem for a $C^{2}$ diffeomorphism $F$ indeed holds without assuming all eigenspaces of the linear system $DF(0)$ are invariant under the nonlinear system $F$, a requirement previously imposed by Belitskii in his proof.

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arXiv2022

Smooth linearization of nonautonomous dynamics under polynomial behaviour

The main purpose of this paper is to formulate new conditions for smooth linearization of nonautonomous systems with discrete and continuous time. Our results assume that the linear part admits a nonuniform polynomial dichotomy and that the associated polynomial dichotomy spectrum exhibits appropriate spectral gap and spectral band conditions. This is in sharp contrast to most of the previous results in the literature which assumed that the linear part admits an exponential dichotomy. Our techniques exploit the relationship between polynomial and exponential dichotomies via a suitable reparametrization of time.

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arXiv2019

Smooth Linearization of Nonautonomous Differential Equations with a Nonuniform Dichotomy

In this paper we give a smooth linearization theorem for nonautonomous differential equations with a nonuniform strong exponential dichotomy. In terms of discretized evolution operator with hyperbolic fixed point 0, we formulate its spectrum and then give a spectral bound condition for the linearization of such equations to be simultaneously differentiable at 0 and Hölder continuous near 0. Restricted in the autonomous case, our result is the first one that gives a rigorous proof for simultaneously differentiable and Hölder linearization of hyperbolic systems without any non-resonant conditions.

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arXiv2019

Smooth Linearization of Nonautonomous Difference Equations with a Nonuniform Dichotomy

In this paper we give a smooth linearization theorem for nonautonomous difference equations with a nonuniform strong exponential dichotomy. The linear part of such a nonautonomous difference equation is defined by a sequence of invertible linear operators on $\mathbb{R}^d$. Reducing the linear part to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in the case of uniform strong exponential dichotomies.

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arXiv2013

Sharp regularity of linearization for $C^{1,1}$ hyperbolic diffeomorphisms

$C^1$ linearization is of special significance because it preserves smooth dynamical behaviors and distinguishes qualitative properties in characteristic directions. However, $C^1$ smoothness is not enough to guarantee $C^1$ linearization. For $C^{1,1}$ hyperbolic diffeomorphisms on Banach spaces $C^1$ linearization was proved under a gap condition together with a band condition of the spectrum. In this paper, the result of $C^1$ linearization in Banach spaces is strengthened to $C^{1,β}$ linearization with a constant $β>0$ under a weaker band condition by a decomposition with invariant foliations. The weaker band condition allows the spectrum to be a union of more than two but finitely many bands but restricts those bands to be bounded by a number depending on the supremum of contractive spectrum and the infimum of expansive spectrum. Furthermore, we give an estimate for the exponent $β$ and prove that the estimate is sharp in the planar case.

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arXiv2013

Sharpness for $C^1$ linearization of planar hyperbolic diffeomorphisms

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 $α_0:=1-\log|λ_2|/\log|λ_1|<α\le 1$, where $λ_1$ and $λ_2$ are both eigenvalues such that $0<|λ_1|<|λ_2|<1$, admits $C^1$ linearization and the linearization is actually $C^{1,β}$. While a sharp Hölder exponent $β>0$ 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.

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