# Historic behaviour for nonautonomous contraction mappings

## TL;DR

This paper demonstrates that non-i.i.d. noise can induce historic behaviour in contraction mappings, which is not observable under i.i.d. noise, highlighting the impact of noise structure on dynamical properties.

## Contribution

It shows that non-i.i.d. noise can create observable historic behaviour in contraction mappings, contrasting with the effects of i.i.d. noise.

## Key findings

- Non-i.i.d. noise can induce historic behaviour in contraction mappings.
- Historic behaviour set can have positive Lebesgue measure under non-i.i.d. noise.
- Under i.i.d. noise, historic behaviour is not observable in these systems.

## Abstract

We consider a parametrised perturbation of a $\mathscr C^r$ diffeomorphism on a closed smooth Riemannian manifold with $r\geq 1$, modeled by nonautonomous dynamical systems. A point without time averages for a (nonautonomous) dynamical system is said to have historic behaviour. It is known that for any $\mathscr C^r$ diffeomorphism, the observability of historic behaviour, in the sense of the existence of a positive Lebesgue measure set consisting of points with historic behaviour, disappears under absolutely continuous, independent and identically distributed (i.i.d.) noise. On contrast, we show that the observability of historic behaviour can appear by a non-i.i.d. noise: we consider a contraction mapping for which the set of points with historic behaviour is of zero Lebesgue measure and provide an absolutely continuous, non-i.i.d. noise under which the set of points with historic behaviour is of positive Lebesgue measure.