Monotone quotients of surface diffeomorphisms

One-parameter families of circle diffeomorphisms with strictly monotone rotation number

We show that if $f \colon S^1 \times S^1 \to S^1 \times S^1$ is $C^2$, with $f(x, t) = (f_t(x), t)$, and the rotation number of $f_t$ is equal to $t$ for all $t \in S^1$, then $f$ is topologically conjugate to the linear Dehn twist of the torus $(1&1 0&1)$. We prove a differentiability result where the assumption that the rotation number of $f_t$ is $t$ is weakened to say that the rotation number is strictly monotone in $t$.

Partially hyperbolic diffeomorphisms with uniformly center foliation: the quotient dynamics

We show that a partially hyperbolic $C^1$ -diffeomorphism $f : M \to M$ with a uniformly compact $f$ -invariant center foliation $F^c$ is dynamically coherent. Further, the induced homeomorphism $F : M/F^c \to M/F^c$ on the quotient space of the center foliation has the shadowing property, i.e. for every $\varepsilon> 0$ there exists $δ> 0$ such that every $δ$-pseudo orbit of center leaves is $\varepsilon$-shadowed by an orbit of center leaves. Although the shadowing orbit is not necessarily unique, we prove the density of periodic center leaves inside the chain recurrent set of the quotient d…

Partially hyperbolic diffeomorphisms with a trapping property

We study partially hyperbolic diffeomorphisms satisfying a trapping property which makes them look as if they were Anosov at large scale. We show that, as expected, they share several properties with Anosov diffeomorphisms. We construct an expansive quotient of the dynamics and study some dynamical consequences related to this quotient.

关于一般\(C^1\)族圆周微分同胚的旋转数不是赫尔德连续的

The Rotation Number for a Generic $C^1$ Family of Circle Diffeomorphisms is Not Hölder

研究圆周微分同胚单调单参数族中旋转数随参数的正则性，证明\(C^2\)单调族中最优赫尔德指数为\(1/2\)，\(C^{1+\alpha}\)族中旋转数正则性不超过\(\alpha/(1+\alpha)\)-赫尔德，得出一般\(C^1\)族旋转数非赫尔德连续。

Codimension one partially hyperbolic diffeomorphisms with a uniformly compact center foliation

We consider a partially hyperbolic C1-diffeomorphism f on a smooth compact manifold M with a uniformly compact f-invariant center foliation. We show that if the unstable bundle is one-dimensional and oriented, then the holonomy of the center foliation vanishes everywhere, the quotient space of the center foliation is a torus and f induces a hyperbolic automorphism on it, in particular, f is centrally transitive. We actually obtain further interesting results without restrictions on the unstable, stable and center dimension: we prove a kind of spectral decomposition for the chain recurrent set …

Normal subgroups of diffeomorphism and homeomorphism groups of R^n and other open manifolds

We determine all the normal subgroups of the group of C^r diffeomorphisms of R^n, r = 1,2,...,infinity, except when r=n+1 or n=4, and also of the group of homeomorphisms of R^n (r=0). We also study the group A_0 of diffeomorphisms of an open manifold M that are isotopic to the identity. If M is the interior of a compact manifold with nonempty boundary, then the quotient of A_0 by the normal subgroup of diffeomorphisms that coincide with the identity near to a given end e of M is simple.