Characterising blenders via covering relations and cone conditions

TL;DR

This paper introduces a new method to characterize blenders using topological and cone conditions, which can be verified through a single iterate and extended to higher dimensions and heterodimensional cycles, with computer-assisted validation.

Contribution

It provides a novel, verifiable characterization of blenders applicable in any dimension, extending to heterodimensional cycles with a flexible, computer-assisted approach.

Findings

Conditions can be checked via a single iterate of the diffeomorphism.

Method applies to arbitrary dimensions.

Supports computer-assisted validation using interval arithmetic.

Abstract

We present a characterisation of a blender based on the topological alignment of certain sets in phase space in combination with cone conditions. Importantly, the required conditions can be verified by checking properties of a single iterate of the diffeomorphism, which is achieved by finding finite series of sets that form suitable sequences of alignments. This characterisation is applicable in arbitrary dimension. Moreover, the approach naturally extends to establishing C1-persistent heterodimensional cycles. Our setup is flexible and allows for a rigorous, computer-assisted validation based on interval arithmetic.