F. E. Brochero Martinez, L. Lopez-Hernanz
Let $F$ be an analytic diffeomorphism in $(\C^m,0)$ tangent to the identity of order $n$. The infinitesimal generator of $F$ is the formal vector field $X$ such that $\Exp X=F$. In this paper we provide an elementary proof of the fact that $X$ belongs to the Gevrey class of order $1/n$.
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We characterize infinitesimal generators of semigroups of holomorphic self-maps of strongly convex domains using the pluricomplex Green function and the pluricomplex Poisson kernel. Moreover, we study boundary regular fixed points of semigroups. Among other things, we characterize boundary regular fixed points both in terms of the boundary behavior of infinitesimal generators and in terms of pluripotential theory.
We prove a Julia-Wolff-Caratheodory type theorem for infinitesimal generators on the unit ball in C^n. Moreover, we study jets expansions at the boundary and give necessary and sufficient conditions on such jets for an infinitesimal generator to generate a group of automorphisms of the ball.
We study infinitesimal generators of one-parameter semigroups in the unit disk $\mathbb D$ having prescribed boundary regular fixed points. Using an explicit representation of such infinitesimal generators in combination with Krein-Milman Theory we obtain new sharp inequalities relating spectral values at the fixed points with other important quantities having dynamical meaning.vWe also give a new proof of the classical Cowen-Pommerenke inequalities for univalent self-maps of $\mathbb D$.
We prove a Julia-Wolff-Carathédory theorem on angular derivatives of infinitesimal generators of one-parameter semigroups of holomorphic self-maps of the unit ball $B^n\subset\mathbb{C}^n$, starting from results recently obtained by Bracci and Shoikhet.
We prove the multisummability of the infinitesimal generator of unfoldings of finite codimension tangent to the identity 1-dimensional local complex analytic diffeomorphisms. We also prove the multisummability of Fatou coordinates and extensions of the Ecalle-Voronin invariants associated to these unfoldings. The quasi-analytic nature is related to the parameter variable. As an application we prove an isolated zeros theorem for the analytic conjugacy problem. The proof is based on good asymptotics of Fatou coordinates and the introduction of a new auxiliary tool, the so called multi-transversa…
We characterize the infinitesimal generator of a semigroup of linear fractional self-maps of the unit ball in $\mathbb C^n$, $n\geq 1$. For the case $n=1$ we also completely describe the associated Koenigs function and we solve the embedding problem from a dynamical point of view, proving, among other things, that a generic semigroup of holomorphic self-maps of the unit disc is a semigroup of linear fractional maps if and only if it contains a linear fractional map for some positive time.
In this technical report the $C_0$ semigroup of Frobenius-Perron operators on $L^1(X)$ is considered, where the underlying dynamical system is such that trajectories may leave the state space $X$ and terminate. We introduce a discrete infinitesimal generator and show, that the operator semigroup generated by this discrete generator converges in $L^1(X)$ pointwise to the Frobenius-Perron operator of the system.
In this paper I give a condition, in terms of boundary behaviour, that forces the infinitesimal generator of a one-parameter semigroup of holomorphic maps to vanish.
Koopman-Based Learning of Infinitesimal Generators without Operator Logarithm
针对基于离散观测估计连续非线性系统转移特性的难题，提出无需算子对数、不破坏Koopman学习框架的无穷小生成元学习方法，可兼容同类验证工具，经数值实验验证在系统辨识和稳定性预测上有效。
We prove that any compact manifold without boundary admits a pair of diffeomorphisms that generates $C^1$ robustly minimal dynamics. We apply the results to the construction of blenders and robustly transitive skew product diffeomorphisms.
We provide a complete system of analytic invariants for unfoldings of non-linearizable resonant complex analytic diffeomorphisms as well as its geometrical interpretation. In order to fulfill this goal we develop an extension of the Fatou coordinates with controlled asymptotic behavior in the neighborhood of the fixed points. The classical constructions are based on finding regions where the dynamics of the unfolding is topologically stable. We introduce a concept of infinitesimal stability leading to Fatou coordinates reflecting more faithfully the analytic nature of the unfolding. These impr…
We establish the notion of a ``projective analytic vector'', whose defining requirements are weaker than the usual ones of an analytic vector, and use it to prove generation theorems for one-parameter groups on locally convex spaces. More specifically, we give a characterization of the generators of strongly continuous one-parameter groups which arise as the result of a projective limit procedure, in which the existence of a dense set of projective analytic vectors plays a central role. An application to strongly continuous Lie group representations on Banach spaces is given, with a focused an…
Flow primitives and infinitesimal generators of Perron-Frobenius and Koopman operators
本文提出一种基于无穷小生成元指数近似的计算方法，通过李括号刻画向量场微扰对Perron-Frobenius和Koopman算子的影响，实现从流基元构造算子及处理参数扰动。
On the structure of the infinitesimal generators of semigroups with discrete Lyapunov functionals
本文刻画了在$C^0[0,1]$或$C^1[0,1]$上具有离散Lyapunov函数的半流的无穷小生成元，证明前者对应抛物型偏微分方程，后者对应单调反馈时滞微分方程的推广。
In this paper we extend the Lumer-Phillips theorem to the context of two--parameter C_0-semigroup of contractions. That is, we characterize the infinitesimal generators of two--parameter C_0-semigroups of contractions. Conditions on the behavior of the resolvent of operators, which are necessary and sufficient for the pair of operators to be the infinitesimal generator of a C_0-semigroup of contractions are given.
If $G$ is a finitely generated group with generators $\{g_1,..., g_s\}$, we say an infinite-order element $f \in G$ is a distortion element of $G$ provided that $\displaystyle \liminf_{n \to \infty} \frac{|f^n|}{n} = 0$, where $|f^n|$ is the word length of $f^n$ with respect to the given generators. Let $S$ be a compact orientable surface, possibly with boundary, and let $\Diff(S)_0$ denote the identity component of the group of $C^1$ diffeomorphisms of $S$. Our main result is that if $S$ has genus at least two, and $f$ is a distortion element in some finitely generated subgroup of $\Diff(S)_0…
We study boundary singularities which can appear for infinitesimal generators of one-parameter semigroups of holomorphic self-maps in the unit disc. We introduce "regular" fractional singularities and characterize them in terms of the behavior of the associated semigroups and Koenigs functions. We also provide necessary and sufficient geometric criteria on the shape of the image of the Koenigs function for having such singularities. In order to do this, we study contact points of semigroups and prove that any contact (not fixed) point of a one-parameter semigroup corresponds to a maximal arc o…
Optimal transport over nonlinear systems via infinitesimal generators on graphs
本文提出了一种基于图的集定向计算框架，用于连续时间非线性动力系统中的最优传输，通过最小化二次控制成本，恢复可证明最优的控制律，以在有限时间内将初始分布引导到指定的最终分布，并探讨了不变子流形、叶动力学和近似不变集在有限时间内高效传输分布中的作用。
A pseudospectral method for investigating the stability of linear population models with two physiological structures
针对含两个生理结构的一阶双曲线性种群模型，本文通过 Carathéodory 绝对连续函数空间重构和二元配置伪谱离散，将无穷小生成元谱逼近为矩阵特征值问题，并验证了收敛性及其对系数正则性的依赖。
Introducing the notions of (inner) $σ$-derivation, (inner) $σ$-endomorphism and one-parameter group of $σ$-endomorphisms ($σ$-dynamics) on a Banach algebra, we correspond to each $σ$-dynamics a $σ$-derivation named as its $σ$-infinitesimal generator. We show that the $σ$-infinitesimal generator of a $σ$-dynamics of inner $σ$-endomorphisms is an inner $σ$-derivation and deal with the converse. We also establish a nice generalized Leibniz formula and extend the Kleinenckr--Sirokov theorem for $σ$-derivations under certain conditions.
A strong inspiration for studying perturbation theory for fractional evolution equations comes from the fact that they have proven to be useful tools in modeling many physical processes. In this paper, we study fractional evolution equations of order $α\in (1,2]$ associated with the infinitesimal generator of an operator fractional cosine function generated by bounded time-dependent perturbations in a Banach space. We show that the abstract fractional Cauchy problem associated with the infinitesimal generator $A$ of a strongly continuous fractional cosine function remains uniformly well-posed …
We give an analytic description for the infinitesimal generator constructed by Applebaum-Estrade for Lévy flights on a broad class of closed Riemannian manifolds including all negatively-curved manifolds, the flat torus and the sphere. Various properties of the associated semigroup and the asymptotics of the expected stopping time for Lévy flight based random searches for small targets, also known as the narrow capture problem, are then obtained using our newfound understanding of the infinitesimal generator. Our study also relates to the Lévy flight foraging hypothesis in the field of biology…
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