This paper is concerned with the study of random (Bernoulli and Markovian) products of matrices on a compact space of symbols. We establish the analyticity of the maximal Lyapunov exponent as a function of the transition probabilities, thus extending the results and methods of Y. Peres from a finite to an infinite (but compact) space of symbols. Our approach combines the spectral properties of the associated Markov operator with the theory of holomorphic functions in Banach spaces.

Given some integer $m \geq 3$, we find the first explicit collection of countably many intervals in $(1,2)$ such that for any q in one of these intervals, the set of points with exactly m base q expansions is non-empty and, moreover, has positive Hausdorff dimension. Our method relies on an application of a theorem proved by Falconer and Yavicoli [Intersections of thick compact sets in ${\mathbb{R}}^d$. Math. Z. 301(3) (2022), 2291–2315, Theorem 6], which guarantees that the intersection of a family of compact subsets of $\mathbb {R}^d$ has positive Hausdorff dimension under certain conditions.

It is well known that the dynamical behavior of a rational map $f:\widehat {\mathbb C}\to \widehat {\mathbb C}$ is governed by the forward orbits of the critical points of f. The map f is said to be postcritically finite if every critical point has finite forward orbit, or equivalently, if every critical point eventually maps into a periodic cycle of f. We encode the orbits of the critical points of f with a finite directed graph called a ramification portrait. In this article, we study which graphs arise as ramification portraits. We prove that every abstract polynomial ramification portrait is realized as the ramification portrait of a postcritically finite polynomial and classify which abstract polynomial ramification portraits can only be realized by unobstructed maps.

Let $ ([0,1]^d,T,\mu ) $ be a measure-preserving dynamical system so that the correlations decay exponentially for Hölder continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under suitable conditions on the underlying dynamical system, we obtain a strong dynamical Borel–Cantelli lemma for recurrence: for any sequence $ \{R_n\} $ of hyperrectangles centered at the origin, with sides parallel to the axes and diameter going to $0$ as $n\to \infty $, where $ \mathbf {x}\in [0,1]^d $ and $ R_n+\mathbf {x} $ is the translation of $ R_n $. The result applies to the Gauss map, $\beta $-transformations, and expanding toral endomorphisms.

We introduce a correlation number for two strictly positive locally Hölder continuous independent potentials with strong entropy gaps at infinity on a topologically mixing countable state Markov shift with big images and pre-images (BIP) property. We define in this way a correlation number for pairs of cusped Hitchin representations. Furthermore, we explore the connection between the correlation number and the Manhattan curve, along with several rigidity properties of this correlation number.

We examine the convergence of ergodic averages along polynomials in Toeplitz systems and prove that it is possible for averages along one polynomial to converge, and along another to diverge. We also study the density of the polynomial orbits in Toeplitz systems—we show that it implies equidistribution of the polynomial orbits in the class of regular Toeplitz systems, but not in the class of strictly ergodic ones.

We investigate the thermodynamic formalism for Viana maps—skew products obtained by coupling an expanding circle map with a slightly perturbed quadratic family on the fibers. For every Hölder potential $\varphi $ whose oscillation is below an explicit threshold, we show that an equilibrium state not only exists, but is unique and satisfies an upper level-2 large-deviation principle. All of these conclusions persist under sufficiently small perturbations of the reference map.

Let $X=G/\Gamma $ be the quotient of a semisimple Lie group G by its non-cocompact arithmetic lattice. Let H be a reductive algebraic subgroup of G acting on X. We give several equivalent algebraic conditions on H for the existence of a fixed compact set in X intersecting every H-orbit. This generalizes previous results concerning certain special reductive group action on X in this setting. When G is of real rank one, $\Gamma $ is a non-cocompact lattice of G, and $H<G$ is an algebraic group, we also obtain an algebraic condition on H which is equivalent to the return of every H-orbit to a single compact set in X. This complements our results in the case of an arithmetic lattice.