Dynamics of certain smooth one-dimensional mappings II: geometrically finite one-dimensional mappings

Dynamics of certain smooth one-dimensional mappings IV: Asymptotic geometry of Cantor sets

We study hyperbolic mappings depending on a parameter $\varepsilon $. Each of them has an invariant Cantor set. As $\varepsilon $ tends to zero, the mapping approaches the boundary of hyperbolicity. We analyze the asymptotics of the gap geometry and the scaling function geometry of the invariant Cantor set as $\varepsilon $ goes to zero. For example, in the quadratic case, we show that all the gaps close uniformly with speed $\sqrt {\varepsilon}$. There is a limiting scaling function of the limiting mapping and this scaling function has dense jump discontinuities because the limiting mapping i…

Asymptotic measures for hyperbolic piecewise smooth mappings of a rectangle

We prove the existence of Sinai-Ruelle-Bowen measures for a class of $C^2$ self-mappings of a rectangle with unbounded derivatives. The results can be regarded as a generalization of a well-known one dimensional Folklore Theorem on the existence of absolutely continuous invariant measures. In an earlier paper analogous results were stated and the proofs were sketched for the case of invertible systems. Here we give complete proofs in the more general case of noninvertible systems, and, in particular, develop the theory of stable and unstable manifolds for maps with unbounded derivatives.

Differentiability of semigroups of Lipschitz or smooth mappings

In this note we study the differentiability with respect to the time-parameter of semigroups consisting of Lipschitzian or smooth self-mappings of a domain in a Banach space.

Model order reduction methods for geometrically nonlinear structures: a review of nonlinear techniques

This paper aims at reviewing nonlinear methods for model order reduction of structures with geometric nonlinearity, with a special emphasis on the techniques based on invariant manifold theory. Nonlinear methods differ from linear based techniques by their use of a nonlinear mapping instead of adding new vectors to enlarge the projection basis. Invariant manifolds have been first introduced in vibration theory within the context of nonlinear normal modes (NNMs) and have been initially computed from the modal basis, using either a graph representation or a normal form approach to compute mappin…

Distortion and Tits alternative in smooth mapping class groups

In this article, we study the smooth mapping class group of a surface S relative to a given Cantor set, that is the group of isotopy classes of orientation-preserving smooth diffeomorphisms of S which preserve this Cantor set. When the Cantor set is the standard ternary Cantor set, we prove that the subgroup consisting of diffeomorphisms which are isotopic to the identity on S does not contain any distorted elements. Moreover, we prove a weak Tits alternative for these groups.

分数阶光滑映射下的维数失真

On the dimension distortion under fractionally smooth mappings

该研究确定了分数阶光滑连续映射在度量空间间对豪斯多夫、上闵可夫斯基及上中间维数的失真程度，扩展至加权欧几里得空间和加倍度量测度空间，并量化了拟对称映射的维数失真性质。