# The generalized recurrent set, explosions and Lyapunov functions

## TL;DR

This paper investigates explosion phenomena in the generalized recurrent set of homeomorphisms on compact metric spaces, providing conditions to prevent explosions, exploring their relation to cycles, and applying findings to Lyapunov stability.

## Contribution

It introduces new conditions to avoid explosions in the generalized recurrent set and links explosion phenomena to cycles, enhancing understanding of stability under perturbations.

## Key findings

- Explosions can occur in the generalized recurrent set, unlike in the chain recurrent set.
- Sufficient conditions are provided to prevent explosions.
- Explosion phenomena are characterized in terms of cycles on manifolds.

## Abstract

We consider explosions in the generalized recurrent set for homeomorphisms on a compact metric space. We provide multiple examples to show that such explosions can occur, in contrast to the case for the chain recurrent set. We give sufficient conditions to avoid explosions and discuss their necessity. Moreover, we explain the relations between explosions and cycles for the generalized recurrent set. In particular, for a compact topological manifold with dimension greater or equal $2$, we characterize explosion phenomena in terms of existence of cycles. We apply our results to give sufficient conditions for stability, under $\mathscr{C}^0$ perturbations, of the property of admitting a continuous Lyapunov function which is not a first integral.