Ergodicity Of Partially Hyperbolic Endomorphisms

TL;DR

The paper proves that for $C^2$, volume-preserving, partially hyperbolic, center-bunched endomorphisms with constant Jacobian, essential accessibility implies ergodicity. The authors adapt the Burns–Wilkinson Hopf argument to the noninvertible setting by lifting dynamics to the space of orbits $M_f$, constructing fake invariant foliations and new $rsc$-juliennes to control density points along stable, center, unstable, and fiber directions. A key innovation is proving that the Lebesgue-density points of the projected set are bisaturated via stable and unstable saturation arguments, which, combined with essential accessibility, yields ergodicity; a parallel K-property result is provided by adapting invertible arguments to $M_f$. These methods extend Pugh–Shub-type ergodicity results to noninvertible systems and introduce tools (fake foliations, juliennes in the fiber direction) applicable to broader noninvertible partially hyperbolic dynamics.

Abstract

We prove that for volume preserving, partially hyperbolic, center bunched endomorphisms with constant Jacobian, essential accessibility implies ergodicity.

Figures (8)

• Figure 1: Unlike the unstable foliation in $M_f$, the stable foliation projects down by $\pi$ to the unique unstable foliation on $M$.
• Figure 2: Multiple unstable manifolds passing through a point $x_0 \in M$, each corresponding to a different past orbit of $x_0$ in the space of orbits.
• Figure 3: In the strong partially hyperbolic diffeomorphism case, an su-path connects $x$ to $y$ by alternating stable and unstable manifolds. In the weak partially hyperbolic endomorphism case, an u-path connects $x_0$ to $y_0$, inside the base manifold $M$, using only unstable curves. In the strong partially hyperbolic endomorphism case, an su-path connects $t_0$ to $w_0$, inside the base manifold $M$.
• Figure 4: The projection $\pi$ relates the notions of accessibility on the space of orbits $M_f$ and on the manifold $M$. Note that this is the weakly partially hyperbolic endomorphism case.
• Figure 5: An $su$-path in the base manifold $M$ lifted to the space of orbits where it becomes a path with stable, unstable and "$r$" segments.

Theorems & Definitions (64)

• Theorem 1.1: Main
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Proposition 2.3
• Lemma 2.4
• proof
• Proposition 3.1
• proof : Proof of Proposition \ref{['prop:fakefoln']}