Partially hyperbolic diffeomorphisms that are center fixing

Abstract

We show that every transitive dynamically coherent partially hyperbolic diffeomorphism with a one-dimensional center foliation $\W^c$ satisfying that $f(W)=W$ for every leaf $W\in \W^c$ is a discretized Anosov flow.

Figures (4)

• Figure 1: A large iterate of $f$ sends the $\mathcal{W}^c$-foliation box $U$ transverse to itself, so the iterate of some plaque $I_{z_0}$ of $U$ has to intersect itself. As a consequence, $\mathcal{W}^c(x_0)$ needs to be compact by the center fixing property.
• Figure 2: As close as wanted to a point $x$ where the function $z \mapsto d_c(z,f(z))$ is not continuous one can find a point $z_0$ such that $f(z_0)$ is not given by a center holonomy map along the center curve from $x$ to $f(x)$
• Figure 3: At $r=\frac{1}{2}$ the function $x\mapsto d_\mathcal{W}(x,f(x))$ tends to 1 from the left and to 2 from the right.
• Figure 4: The function $x\mapsto d_\mathcal{W}(x,f(x))$ is locally unbounded at $r=0$ on each annulus $\theta=const.$

Theorems & Definitions (60)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Proposition 2.2: Mar
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Lemma 2.5
• proof