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 show that for a minimal system $(X,T)$, the set of saturated points along cubes with respect to its maximal $\infty $-step pro-nilfactor $X_\infty $ has a full measure. As an application, it is shown that if a minimal system $(X,T)$ has no non-trivial $(k+1)$-tuples with arbitrarily long finite IP-independence sets, then it has only at most k ergodic measures and is an almost $k'$ to one extension of $X_\infty $ for some $k'\leqslant k$. In particular, for $k=1$, we prove that $(X,T)$ is uniquely ergodic (even regular with respect to $X_\infty $), which answers a conjecture stated by Dong et al [Infinite-step nilsystems, independence and complexity. Ergod. Th. & Dynam. Sys. 33(1) (2013), 118–143].

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu $ defined on the completion of the Borel $\sigma $-algebra, and $\mathcal F$ a f-invariant one-dimensional continuous foliation of M by $C^1$-leaves. Then, if f preserves a continuous $\mathcal {F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu $ along $\mathcal F$, namely: (i) they are atomic for almost every leaf; or (ii) for almost every leaf, they are equivalent to the measure $\unicode{x3bb} _x$ induced by the invariant arc-length system over $\mathcal F$; or (iii) for almost every leaf, their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

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.

We construct various novel and elementary examples of dynamics with metric attractors that have intermingled basins. A main ingredient is the introduction of random walks along orbits of a given dynamical system. We develop the theory and use it in particular to provide examples of thick metric attractors with intermingled basins.

Let f be a rational map of degree $d\geq 2$. The moduli space $\mathcal {M}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting of all quasiconformal conjugacy classes of f. For f that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:\mathcal {M}_f\to X_f$ such that $i(\mathcal {M}_f)$ is precompact in $X_f$. In particular, $\mathcal {M}_f$ is Carathéodory hyperbolic (that is, bounded holomorphic functions separate points in $\mathcal {M}_f$), provided that f is not flexible Lattès. This solves a conjecture of McMullen. When $d\geq 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.

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.

Let $\Omega \in \mathbb C$ be a domain such that $K:= \mathbb C \setminus \Omega $ is compact and non-polar. Let $(q_k)_{k>0}$ be a sequence of polynomials with $n_k$, the degree of $q_k$ satisfying $n_k \to \infty $, and let $(q_k^{(m)})_k$ denote the sequence of mth derivatives. We provide conditions which ensure that the preimages $(q_k^{(m)})^{-1}(\{a\})$ uniformly equidistribute on $\partial \Omega $, as $k \to \infty $, for every $a \in \mathbb C$ and every $m = 0, 1, \ldots .$

We consider $L^q$-spectra of planar graph-directed self-affine measures generated by diagonal or anti-diagonal matrices. Assuming the directed graph is strongly connected and the system satisfies the rectangular open set condition, we obtain a general closed form expression for the $L^q$-spectra. Consequently, we obtain a closed form expression for box dimensions of associated planar graph-directed box-like self-affine sets. We also provide a precise answer to a question posed by Fraser [On the $L^q$-spectrum of planar self-affine measures. Trans. Amer. Math. Soc. 368(8) (2016), 5579–5620] concerning the $L^q$-spectra of planar self-affine measures generated by diagonal matrices. An interesting observation of the closed form expression is that it is possible to calculate the $L^q$-spectrum of a measure without involving the exact $L^q$-spectra of its projections to the axes.