# Upper bound on the regularity of the Lyapunov exponent for random products of matrices

@inproceedings{Bezerra2022UpperBO, title={Upper bound on the regularity of the Lyapunov exponent for random products of matrices}, author={Jamerson Bezerra and Pedro Duarte}, year={2022} }

We prove that if µ is a ﬁnitely supported measure on SL 2 ( R ) with positive Lyapunov exponent but not uniformly hyperbolic, then the Lyapunov exponent function is not α -H¨older around µ for any α exceeding the Shannon entropy of µ over the Lyapunov exponent of µ .

## 2 Citations

### A dynamical Thouless formula

- Mathematics
- 2022

. In this paper we establish an abstract, dynamical Thouless-type formula for aﬃne families of GL(2 , R ) cocycles. This result extends the classical formula relating, via the Hilbert transform, the…

### Continuity of the Lyapunov exponents of non-invertible random cocycles with constant rank

- Mathematics
- 2022

In this paper we establish uniform large deviations estimates of exponential type and Hölder continuity of the Lyapunov exponents for random non-invertible cocycles with constant rank.

## References

SHOWING 1-10 OF 31 REFERENCES

### Approximating Lyapunov Exponents and Stationary Measures

- MathematicsJournal of Dynamics and Differential Equations
- 2019

We give a new proof of E. Le Page’s theorem on the Hölder continuity of the first Lyapunov exponent in the class of irreducible Bernoulli cocycles. This suggests an algorithm to approximate the first…

### Moduli of continuity for the Lyapunov exponents of random 𝐺𝐿(2)-cocycles

- MathematicsTransactions of the American Mathematical Society
- 2019

The Lyapunov exponents of i.i.d. random
G
L
(
2
)
\mathrm {GL}(2)
-cocycles are Hölder continuous functions of the underlying probability distribution at each point with a simple…

### Continuity of Lyapunov exponents for random two-dimensional matrices

- MathematicsErgodic Theory and Dynamical Systems
- 2016

The Lyapunov exponents of locally constant $\text{GL}(2,\mathbb{C})$ -cocycles over Bernoulli shifts vary continuously with the cocycle and the invariant probability measure.

### Density of positive Lyapunov exponents for SL(2,R) cocycles

- Mathematics
- 2010

We show that SL(2,R) cocycles with a positive Lyapunov exponent are dense in all regularity classes and for all non-periodic dynamical systems. For Schr\"odinger cocycles, we show prevalence of…

### Large Deviations for Products of Random Two Dimensional Matrices

- MathematicsCommunications in Mathematical Physics
- 2019

We establish large deviation type estimates for i.i.d. products of two dimensional random matrices with finitely supported probability distribution. The estimates are stable under perturbations and…

### Lyapunov Exponents of Linear Cocycles: Continuity via Large Deviations

- Mathematics
- 2016

Introduction.- Estimates on Grassmann Manifolds.- Abstract Continuity of Lyapunov Exponents.- The Oseledets Filtration and Decomposition.- Large Deviations for Random Cocycles.- Large Deviations for…

### Lectures on Lyapunov Exponents

- Mathematics
- 2014

Preface 1. Introduction 2. Linear cocycles 3. Extremal Lyapunov exponents 4. Multiplicative ergodic theorem 5. Stationary measures 6. Exponents and invariant measures 7. Invariance principle 8.…

### Random Walks on Reductive Groups

- Mathematics
- 2016

We apply the previous results to random walks in products of algebraic reductive groups over local fields. We prove the Law of Large Numbers for both the Iwasawa cocycle and the Cartan projection. We…

### Harmonic analysis on SL(2,R) and smoothness of the density of states in the one-dimensional Anderson model

- Mathematics
- 1985

We consider infinite Jacobi matrices with ones off-diagonal, and independent identically distributed random variables with distributionF(v)dv on-diagonal. IfF has compact support and lies in some…