A remark on the central model method for the weak Palis conjecture of higher dimensional singular flows

TL;DR

This paper analyzes the limitations of the central model method in proving the weak Palis conjecture for higher-dimensional singular flows, showing it cannot create horseshoes and introducing a simplified blowup approach.

Contribution

It demonstrates the insufficiency of the central model method for higher-dimensional singular flows and provides a simplified blowup construction and second-order derivative computations.

Findings

Central model cannot produce horseshoes in this setting.

A simplified blowup construction is proposed.

Extended rescaled Poincaré map computed up to second order derivatives.

Abstract

For a generic vector field robustly without horseshoes, and an aperiodic chain recurrent class with singularities whose saddle values have different signs, the extended rescaled Poincar\'e map is associated with a central model. We estimate such central model and show it must have chain recurrent central segments over the singularities. This obstructs the application of central model to create horseshoes, and indicates that, differing from $C^1$ diffeomorphisms, solo using central model method is insufficient as a strategy to prove weak Palis conjecture for higher dimensional ($\geq 4$) singular flows. Our computation is actually based on simplified way of addressing blowup construction. As a byproduct, we are applicable to directly compute the extended rescaled Poincar\'e map upto second order derivatives, which we believe has its independent interests.