# A note on two types of Lyapunov exponents and entropies for $\mathbb{Z}^k$-actions

## TL;DR

This paper explores two types of Lyapunov exponents and entropies for smooth ^k-actions, analyzing their relationships and providing formulas based on the generators' Lyapunov exponents.

## Contribution

It introduces and compares random and directional Lyapunov exponents and entropies for ^k-actions, offering formulas linking these quantities to generator Lyapunov exponents.

## Key findings

- Established relations among the two types of Lyapunov exponents and entropies.
- Derived formulas expressing these quantities via generator Lyapunov exponents.
- Analyzed the close connections between these dynamical invariants.

## Abstract

In this paper, two types of Lyapunov exponents: random Lyapunov exponents and directional Lyapunov exponents, and the corresponding entropies: random entropy and directional entropy, are considered for smooth $\mathbb{Z}^k$-actions. The close relation among these quantities are investigated and the formulas of them are given via the Lyapunov exponents of the generators.