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Given a hyperbolic homeomorphism on a compact metric space, consider the space of linear cocycles over this base dynamics which are Hölder continuous and whose projective actions are partially hyperbolic dynamical systems. We prove that locally near any typical cocycle, the Lyapunov exponents are Hölder continuous functions relative to the uniform topology. This result is obtained as a consequence of a uniform large deviations type estimate in the space of cocycles. As a byproduct of our approach, we also establish other statistical properties for the iterates of such cocycles, namely a central limit theorem and a large deviations principle.

In this paper we show that a “typical” random product of conservative surface diffeomorphism has positive Lyapunov exponents. We prove that for any compact oriented surface S, any r≥1 , and any d≥2 , there exists a C1 -open and C1 -dense subset of Diffvolr(S)d such that if (f1,…,fd) belongs to this subset, the random product generated by them has positive Lyapunov exponents. Our proof also allows us to deal with more general skew products, for example skew products with a volume preserving Anosov diffeomorphism on the basis, or with a subshift of finite type on the basis preserving a measure with product structure. In these cases we prove the C1 -density and Cr -openness of the existence of positive Lyapunov exponents.

We study the u -Gibbs measures of a certain class of uniformly hyperbolic skew products on T 4 . These systems have a strong unstable and a weak unstable direction. Among such skew products, we show the existence of a subset which is C r -dense and C 2 -open for which every u -Gibbs measure is SRB. In particular, there is only one such measure. As an application, we obtain the minimality of the strong unstable foliation.

We study theu -Gibbs measures of a certain class of uniformly hyperbolic skew products on𝕋⁴ . These systems have a strong unstable and a weak unstable directions. We show thatCʳ -dense andC² -open in this set everyu -Gibbs measure is SRB, in particular, there is only one such measure. As an application of this, we can obtain the minimality of the strong unstable foliation.

We prove that for Anosov maps of the 3-torus if the Lyapunov exponents of absolutely continuous measures in every direction are equal to the geometric growth of the invariant foliations then f is C1 conjugated to its linear part.

For C^1+ maps, possibly non-invertible and with singularities, we prove that the homoclinic class of each ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study the finiteness/uniqueness of such measures in several different settings: finite horizon dispersing billiards, codimension one partially hyperbolic endomorphisms with “large” entropy, robustly non-uniformly hyperbolic volume-preserving endomorphisms as in Andersson–Carrasco–Saghin (2025), and Viana maps (1997).

We prove a Livšic-type theorem for Hölder continuous and matrix-valued cocycles over non-uniformly hyperbolic systems. More precisely, we prove that whenever (f,μ) is a non-uniformly hyperbolic system and A:M→GL(d,R) is an α -Hölder continuous map satisfying A(fn-1(p))…A(p)=Id for every p∈Fix(fn) and n∈N , there exists a measurable map P:M→GL(d,R) satisfying A(x)=P(f(x))P(x)-1 for μ -almost every x∈M . Moreover, we prove that whenever the measure μ has local product structure the transfer map P is α -Hölder continuous in sets with arbitrary large measure.

We prove that generic fiber-bunched and Hölder continuous linear cocycles over a non-uniformly hyperbolic system endowed with a $u$-Gibbs measure have simple Lyapunov spectrum. This gives an affirmative answer to a conjecture proposed by Viana in the context of fiber-bunched linear cocycles.

In this survey we talk about what is known as Invariance Principle in dynamical systems. It states that the disintegration of measures with zero center Lyapunov exponents admits some extra invariance by holonomies. We focus on explaining the basic definitions and ideas behind a series of results about the Invariance Principle and give some basic applications on how this is used in dynamical systems.