# Quasi-multiplicativity of typical cocycles

## TL;DR

This paper proves that typical fiber-bunched cocycles over subshifts are uniformly quasi-multiplicative, ensuring continuity of singular value pressure and enabling multifractal analysis of Lyapunov exponents.

## Contribution

It establishes quasi-multiplicativity for typical cocycles, proving continuity of pressure and analyzing the Lyapunov spectrum, advancing understanding of their multifractal properties.

## Key findings

- Proves uniform quasi-multiplicativity for typical cocycles.
- Shows the singular value pressure is continuous and has a unique equilibrium state.
- Demonstrates the Lyapunov spectrum is closed and convex, enabling multifractal analysis.

## Abstract

We show that typical (in the sense of Bonatti-Viana) H\"{o}lder and fiber-bunched $GL_d(\mathbb{R})$-valued cocycles over a subshift of finite type are uniformly quasi-multiplicative with respect to all singular value potentials. We prove the continuity of the singular value pressure and its corresponding (necessarily unique) equilibrium state for such cocycles, and apply this result to repellers. Moreover, we show that the pointwise Lyapunov spectrum is closed and convex, and establish partial multifractal analysis on the level sets of pointwise Lyapunov exponents for such cocycles.