# Examples of diffeomorphism group cocycles with no periodic approximation

## TL;DR

This paper constructs examples of diffeomorphism group cocycles with no periodic approximation, revealing complex dynamical behaviors and providing new examples of Banach cocycles that defy periodic approximation.

## Contribution

It introduces a finitely generated subgroup of diffeomorphisms with exponential growth of derivatives and constructs new cocycles lacking periodic approximation.

## Key findings

- Constructed a subgroup of Diff^∞(S^3×S^1) with exponential growth of derivatives.
- Provided examples of Banach cocycles without periodic approximation.
- Demonstrated complex dynamical behaviors in diffeomorphism group actions.

## Abstract

We construct a finitely generated subgroup of $\text{Diff}^{\infty}(\mathbb{S}^3 \times \mathbb{S}^1)$ where every element is conjugate to an isometry but such that the group action itself is far from isometric (the group has "exponential growth of derivatives"). As a corollary, one obtains a locally constant $\text{Diff}^{\infty}(\mathbb{S}^3 \times \mathbb{S}^1)$ valued cocycle over a hyperbolic dynamical system which has elliptic behavior over its periodic orbits but which preserves a measure with non-zero top Fiber Lyapunov exponent. Additionally, we provide new examples of Banach cocycles not satisfying the periodic approximation property as first shown by Kalinin-Sadovskaya.