In this paper we consider C1 diffeomorphisms on compact Riemannian manifolds admitting a dominated splitting Ecs⊕Ecu. First, we prove that the smallest Lyapunov exponent along Ecu, computed with respect to the Lebesgue measure, is computable using observable measures. Then we show that if the Lyapunov exponents along Ecu are positive Lebesgue almost everywhere and Ecu admits a finest 1-dominated splitting on the support of an ergodic observable measure then f is non-uniformly expanding along Ecu. As a byproduct, every C1+α diffeomorphism exhibiting a dominated splitting Es⊕Ecu where Ecu fulfills the previous assumptions admits an SRB measure.

We develop a higher-dimensional extension of multifractal analysis for typical fiber-bunched linear cocycles. Our main result is a relative variational principle, which shows that the topological entropy of Lyapunov exponent level sets can be approximated by the metric entropy of ergodic measures fully concentrated on those level sets, addressing a question posed by Breuillard and Sert. We also establish a variational principle for the generalized singular value function. As an application to dynamically defined linear cocycles, we obtain a multifractal formalism for open sets of C^1+α repellers and Anosov diffeomorphisms.

We consider the one-step matrix cocycle generated by a particular pair of non-negative parabolic matrices and study the equilibrium measures for tlog & Vert;A & Vert; as t runs over the reals. We show that there is a freezing first order phase transition at t=-2 so that for t <=-2 the equilibrium measure is non-unique and supported on the two fixed points, while for t>-2, the equilibrium measure is unique, non-atomic and fully supported. The phase transition closely resembles the classical Hofbauer example. In particular, our example shows that there may be non-unique equilibrium measures for negative t even if the cocycle is strongly irreducible and proximal.

We contribute to the thermodynamic formalism of Hölder continuous fiber-bunched matrix cocycles, Anosov diffeomorphisms, and hyperbolic repellers. Specifically, we prove that 1-typical fiber-bunched cocycles 𝒜 over topologically mixing subshifts of finite type admit a unique Gibbs equilibrium state μ_t associated with the non-additive family of potentials {t log𝒜^n}_n ∈ℕ, for a range of parameters t ∈ (-t_*, +∞), where t_* > 0. Furthermore, these equilibrium states are ψ-mixing, therefore weak Bernoulli. In addition, these results allow us to derive consequences for the thermodynamic formalism of open sets of hyperbolic repellers and Anosov diffeomorphisms. In particular, it provides a positive answer to a conjecture posed by Gatzouras and Peres for C^1-open sets of α-fiber-bunched hyperbolic repellers.

We study cocycles taking values in the mapping class group of closed surfaces and investigate their leading topological Lyapunov exponent. Under a natural closing property, we show that the top topological Lyapunov exponent can be approximated by periodic orbits. We also extend the notion of the joint spectral radius to this setting, interpreting it via the exponential growth of curves under iterated mapping classes. Our approach connects ideas from ergodic theory, Teichmüller geometry, and spectral theory, and suggests a broader framework for similar results.

In this paper we study ergodic optimization problems for typical cocycles. We consider one-step SL(2,R)-cocycles that satisfy pinching and twisting conditions (in the sense of [8]). We prove that the Lyapunov maximizing measures have zero entropy under additional assumptions that the maps e1 and e2 are one-to-one on the Mather set, thus extending a result by Bochi and Rams [6].

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine iterated function systems. Also, we study Besicovitch-Eggleston sets for finite generalized Luroth series number systems with redundancy. The redundancy refers to the fact that each number x is an element of [0, 1] has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

In this paper, we study the size of the level sets of all Lyapunov exponents. For typical cocycles, we establish a variational relation between the topological entropy of the level sets of Lyapunov exponents and the topological pressure of the generalized singular value function.

In this paper, we show that a locally constant cocycle $\mathcal{A}$ is $k$-quasi multiplicative under the irreducibility assumption. More precisely, we show that if $\mathcal{A}^t$ and $\mathcal{A}^{\wedge m}$ are irreducible for every $t \mid d$ and $1\leq m \leq d-1$, then $\mathcal{A}$ is $k$-uniformly spannable for some $k\in \mathbb{N}$, which implies that $\mathcal{A}$ is $k$-quasi multiplicative. We apply our results to show that the unique subadditive equilibrium Gibbs state is $\psi$-mixing and calculate the Hausdorff dimension of cylindrical shrinking target and recurrence sets.

In this paper, we study the multifractal formalism of Lyapunov exponents for typical cocycles. We establish a variational relation between the Legendre transform of topological pressure of the generalized singular value function and measure-theoretic entropies. As a consequence, we show that the restricted variational principle of Lyapunov exponents holds for typical cocycles.

We survey a collection of results in the multifractal analysis for the topological entropy of the level sets of Lyapunov exponents. We discuss the most recent results in the area as well as the main difficulties in developing a general theory. Due to the nonconformality, the Lyapunov exponents are averages of nonadditive sequences of potentials, and thus one cannot use Birkhoff’s ergodic theorem or the classical thermodynamic formalism. The results are formulated in terms of Legendre–Fenchel transforms of topological pressures as well as in terms of restricted variational principles of entropies of invariant measures with given Lyapunov exponents.

In this paper we study ergodic optimization and multifractal behavior of Lyapunov exponents for matrix cocycles. We show that the restricted variational principle holds for generic cocycles [in the sense of (Bonatti and Viana in Ergod Theory Dyn Syst 24(5):1295–1330, 2004)] over mixing subshifts of finite type. We also show that the Lyapunov spectrum is equal to the closure of the set where the entropy spectrum is positive for such cocycles. Moreover, we show the continuity of the entropy spectrum at boundary of Lyapunov spectrum in the sense that $$h_{top}(E(\alpha _{t}))\ \rightarrow h_{top}(E(\beta ({\mathcal {A}}))$$ , where $$E(\alpha )=\{x\in X: \lim _{n\rightarrow \infty }\frac{1}{n}\log \Vert {\mathcal {A}}^{n}(x)\Vert =\alpha \}$$ , for such cocycles. We prove the continuity of the lower joint spectral radius for linear cocycles under the assumption that linear cocycles satisfy a cone condition.

In this paper we consider C diffeomorphisms on compact Riemannian manifolds of arbitrary dimension that admit a dominated splitting E⊕E. We prove that if the Lyapunov exponents along E are positive for Lebesgue almost every point, then a map f is non-uniformly expanding along E under the additional assumption that the Lyapunov spectrum has a uniform 1-gap property. As a result, there exists a physical SRB measure for a C diffeomorphism map f that admits a dominated splitting E ⊕ E under assumptions that f has non-zero Lyapunov exponents for Lebesgue almost every point and the Lyapunov spectrum has a uniform 1-gap property. Our result provides an affirmative answer to a question posed by Alves, Bonatti and Viana (Invent. Math. 140(2): 351-398, 2000).