[Paper Review] Anomalous partially hyperbolic diffeomorphisms I: dynamically coherent examples

Christian Bonatti, Kamlesh Parwani+1 more|arXiv (Cornell University)|Nov 5, 2014

Mathematical Dynamics and Fractals10 references22 citations

TL;DR

This paper constructs the first known example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism on a closed 3-manifold with exponential growth in its fundamental group, where no iterate is isotopic to the identity. The construction combines a non-transitive Anosov flow's time-$t$ map with a carefully chosen Dehn twist, contradicting a long-standing conjecture that such systems must be leaf conjugate to standard models like Anosov flows or skew products.

ABSTRACT

We build an example of a non-transitive, dynamically coherent partially hyperbolic diffeomorphism $f$ on a closed $3$-manifold with exponential growth in its fundamental group such that $f^n$ is not isotopic to the identity for all $n\ eq 0$. This example contradicts a conjecture in \\cite{HHU}. The main idea is to consider a well-understood time-$t$ map of a non-transitive Anosov flow and then carefully compose with a Dehn twist.

Motivation & Objective

• To construct a dynamically coherent, non-transitive partially hyperbolic diffeomorphism on a closed 3-manifold with exponential fundamental group growth.
• To demonstrate that such a diffeomorphism exists where no iterate is isotopic to the identity, contradicting a widely held conjecture.
• To provide a counterexample to the conjecture that all dynamically coherent partially hyperbolic diffeomorphisms are, up to finite cover and iterate, leaf conjugate to one of three standard models: Anosov automorphisms, skew products, or time-one maps of Anosov flows.
• To show that center leaves are not fixed under any iterate, despite the center-stable and center-unstable foliations being invariant.

Proposed method

• Start with a non-transitive Anosov flow on a closed 3-manifold with exponential fundamental group growth.
• Take the time-$t$ map $X_N$ of this flow, which is partially hyperbolic and preserves the strong stable and unstable foliations.
• Perform a Dehn twist $G$ along a torus $T_1$ transverse to the flow, modifying the dynamics in a neighborhood of $T_1$.
• Compose the time-$t$ map $X_N$ with the Dehn twist $G$ to define the new diffeomorphism $f = G \circ X_N$.
• Ensure the Dehn twist is supported in a neighborhood of $T_1$ and acts nontrivially on homology, preserving the dynamical coherence of the system.
• Use homological arguments to show that $f^k$ acts nontrivially on $H_1(M, \mathbb{Z})$ for all $k \neq 0$, proving it is not isotopic to the identity.

Experimental results

Research questions

• RQ1 Can a dynamically coherent, non-transitive partially hyperbolic diffeomorphism exist on a 3-manifold with exponential fundamental group growth?
• RQ2 Is it possible for such a diffeomorphism to have no iterate isotopic to the identity?
• RQ3 Can a partially hyperbolic diffeomorphism be constructed that is not leaf conjugate to any of the three standard models (Anosov automorphism, skew product, time-one Anosov flow)?
• RQ4 Do center leaves remain invariant under iterates even when the center-stable and center-unstable foliations are invariant?
• RQ5 Can a Dehn twist be used to modify an Anosov flow's time-map to break isotopy triviality while preserving dynamical coherence?

Key findings

• The constructed diffeomorphism $f$ is dynamically coherent, non-transitive, and partially hyperbolic on a closed 3-manifold with exponential fundamental group growth.
• No iterate $f^n$ for $n \neq 0$ is isotopic to the identity, as shown by nontrivial action on first homology.
• The center-stable and center-unstable foliations are invariant under $f$, but the center leaves (their intersections) are not fixed by any iterate of $f$.
• The example contradicts the conjecture that all dynamically coherent partially hyperbolic diffeomorphisms are, up to finite cover and iterate, leaf conjugate to one of the three standard models.
• The construction shows that the center foliation can be non-invariant under iterates even when the strong foliations are preserved, highlighting a subtle failure of global invariance.
• The fundamental group's exponential growth prevents the manifold from supporting Anosov diffeomorphisms or skew products, ruling out two of the three standard models.