In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

We give new tools for homotopy Brouwer theory. In particular, we describe a canonical reducing set called the set of walls, which splits the plane into maximal translation areas and irreducible areas. We then focus on Brouwer mapping classes relative to four orbits and describe them explicitly by adding a tangle to Handel’s diagram and to the set of walls. This is essentially an isotopy class of simple closed curves in the cylinder minus two points.

We prove a generalization of the fixed point theorem of Cartwright and Littlewood. Namely, suppose that $h:\mathbb{R}^{2}\rightarrow \mathbb{R}^{2}$ is an orientation preserving planar homeomorphism, and let $C$ be a continuum such that $h^{-1}(C)\cup C$ is acyclic. If there is a $c\in C$ such that $\{h^{-i}(c):i\in \mathbb{N}\}\subseteq C$ , or $\{h^{i}(c):i\in \mathbb{N}\}\subseteq C$ , then $C$ also contains a fixed point of $h$ . Our approach is based on Brown’s short proof of the result of Cartwright and Littlewood. In addition, making use of a linked periodic orbits theorem of Bonino, we also prove a counterpart of the aforementioned result for orientation reversing homeomorphisms, that guarantees a $2$ -periodic orbit in $C$ if it contains a $k$ -periodic orbit ( $k>1$ ).

We consider for each $t$ the set $K(t)$ of points of the circle whose forward orbit for the doubling map does not intersect $(0,t)$ , and look at the dimension function $\unicode[STIX]{x1D702}(t):=\text{H.dim}\,K(t)$ . We prove that at every bifurcation parameter $t$ , the local Hölder exponent of the dimension function equals the value of the function $\unicode[STIX]{x1D702}(t)$ itself. A similar statement holds for general expanding maps of the circle: namely, we consider the topological entropy of the map restricted to the survival set, and obtain bounds on its local Hölder exponent in terms of the value of the function.

Let $\unicode[STIX]{x1D6E4}$ be a lattice in a simply connected nilpotent Lie group $G$ . Given an infinite measure-preserving action $T$ of $\unicode[STIX]{x1D6E4}$ and a ‘direction’ in $G$ (i.e. an element $\unicode[STIX]{x1D703}$ of the projective space $P(\mathfrak{g})$ of the Lie algebra $\mathfrak{g}$ of $G$ ), some notions of recurrence and rigidity for $T$ along $\unicode[STIX]{x1D703}$ are introduced. It is shown that the set of recurrent directions ${\mathcal{R}}(T)$ and the set of rigid directions for $T$ are both $G_{\unicode[STIX]{x1D6FF}}$ . In the case where $G=\mathbb{R}^{d}$ and $\unicode[STIX]{x1D6E4}=\mathbb{Z}^{d}$ , we prove that (a) for each $G_{\unicode[STIX]{x1D6FF}}$ -subset $\unicode[STIX]{x1D6E5}$ of $P(\mathfrak{g})$ and a countable subset $D\subset \unicode[STIX]{x1D6E5}$ , there is a rank-one action $T$ such that $D\subset {\mathcal{R}}(T)\subset \unicode[STIX]{x1D6E5}$ and (b) ${\mathcal{R}}(T)=P(\mathfrak{g})$ for a generic infinite measure-preserving action $T$ of $\unicode[STIX]{x1D6E4}$ . This partly answers a question from a recent paper by Johnson and Şahin. Some applications to the directional entropy of Poisson actions are discussed. In the case where $G$ is the Heisenberg group $H_{3}(\mathbb{R})$ and $\unicode[STIX]{x1D6E4}=H_{3}(\mathbb{Z})$ , a rank-one $\unicode[STIX]{x1D6E4}$ -action $T$ is constructed for which ${\mathcal{R}}(T)$ is not invariant under the natural ‘adjoint’ $G$ -action.

A necessary and sufficient condition is given for the existence of an embedding of an irreducible subshift of finite type into the Fibonacci–Dyck shift.

Consider a Riccati foliation whose monodromy representation is non-elementary and parabolic and consider a non-invariant section of the fibration whose associated developing map is onto. We prove that any holonomy germ from any non-invariant fibre to the section can be analytically continued along a generic Brownian path. To prove this theorem, we prove a dual result about complex projective structures. Let $\unicode[STIX]{x1D6F4}$ be a hyperbolic Riemann surface of finite type endowed with a branched complex projective structure: such a structure gives rise to a non-constant holomorphic map ${\mathcal{D}}:\tilde{\unicode[STIX]{x1D6F4}}\rightarrow \mathbb{C}\mathbb{P}^{1}$ , from the universal cover of $\unicode[STIX]{x1D6F4}$ to the Riemann sphere $\mathbb{C}\mathbb{P}^{1}$ , which is $\unicode[STIX]{x1D70C}$ -equivariant for a morphism $\unicode[STIX]{x1D70C}:\unicode[STIX]{x1D70B}_{1}(\unicode[STIX]{x1D6F4})\rightarrow \mathit{PSL}(2,\mathbb{C})$ . The dual result is the following. If the monodromy representation $\unicode[STIX]{x1D70C}$ is parabolic and non-elementary and if ${\mathcal{D}}$ is onto, then, for almost every Brownian path $\unicode[STIX]{x1D714}$ in $\tilde{\unicode[STIX]{x1D6F4}}$ , ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not have limit when $t$ goes to $\infty$ . If, moreover, the projective structure is of parabolic type, we also prove that, although ${\mathcal{D}}(\unicode[STIX]{x1D714}(t))$ does not converge, it converges in the Cesàro sense.

We study tangent sets of strictly self-affine sets in the plane. If a set in this class satisfies the strong separation condition and projects to a line segment for sufficiently many directions, then for each generic point there exists a rotation ${\mathcal{O}}$ such that all tangent sets at that point are either of the form ${\mathcal{O}}((\mathbb{R}\times C)\cap B(0,1))$ , where $C$ is a closed porous set, or of the form ${\mathcal{O}}((\ell \times \{0\})\cap B(0,1))$ , where $\ell$ is an interval.

We show that minimal shifts with zero topological entropy are topologically conjugate to interval exchange transformations, which are generally infinite. When these shifts have linear factor complexity (linear block growth), the conjugate interval exchanges are proved to satisfy strong finiteness properties.

The main result of the paper is classification of free multidimensional Borel flows up to Lebesgue orbit equivalence, by which we mean an orbit equivalence that preserves the Lebesgue measure on each orbit. Two non-smooth $\mathbb{R}^{d}$ -flows are shown to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.

In this paper, we study the dynamics of the family of rational maps with two parameters

$$\begin{eqnarray}f_{a,b}(z)=z^{n}+\frac{a^{2}}{z^{n}-b}+\frac{a^{2}}{b},\end{eqnarray}$$

$n\geq 2$

$a,b\in \mathbb{C}^{\ast }$

$f_{a,b}$