# Historic behavior in non-hyperbolic homoclinic classes

## TL;DR

This paper proves that for generic diffeomorphisms in dimensions three and higher, certain homoclinic classes exhibit widespread historic behavior, meaning orbits do not settle into regular patterns.

## Contribution

It demonstrates that in high-dimensional manifolds, homoclinic classes with saddles of different indices generically contain points with historic behavior.

## Key findings

- Residual subset of points with historic behavior in such classes
- Historic behavior is typical for generic diffeomorphisms in these settings
- Results apply to manifolds of dimension three or higher

## Abstract

We show that $C^1$-generically for diffeomorphisms of manifolds of dimension $d\geq3$, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior.