We study the dynamics of post-critically finite endomorphisms of $\mathbb{P}^{k}(\mathbb{C})$. We prove that post-critically finite endomorphisms are always post-critically finite all the way down under a regularity condition on the post-critical set. We study the eigenvalues of periodic points of post-critically finite endomorphisms. Then, under a transversality condition and assuming Kobayashi hyperbolicity of the complement of the post-critical set, we prove that the only possible Fatou components are super-attracting basins; thus, partially extending to any dimension is a result of Fornaess–Sibony and Rong holding in the case $k=2$.

In this work we consider translation-bounded measures over a locally compact Abelian group $\mathbb{G}$, with a particular interest in their so-called diffraction. Given such a measure $\unicode[STIX]{x1D714}$, its diffraction $\widehat{\unicode[STIX]{x1D6FE}}$ is another measure on the Pontryagin dual $\widehat{\mathbb{G}}$, whose decomposition into the sum $\widehat{\unicode[STIX]{x1D6FE}}=\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}+\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$ of its atomic and continuous parts is central in diffraction theory. The problem we address here is whether the above decomposition of $\widehat{\unicode[STIX]{x1D6FE}}$ lifts to $\unicode[STIX]{x1D714}$ itself, that is to say, whether there exists a decomposition $\unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{\text{p}}+\unicode[STIX]{x1D714}_{\text{c}}$, where $\unicode[STIX]{x1D714}_{\text{p}}$ and $\unicode[STIX]{x1D714}_{\text{c}}$ are translation-bounded measures having diffraction $\widehat{\unicode[STIX]{x1D6FE}}_{\text{p}}$ and $\widehat{\unicode[STIX]{x1D6FE}}_{\text{c}}$, respectively. Our main result here is the almost sure existence, in a sense to be made precise, of such a decomposition. It will also be proved that a certain uniqueness property holds for the above decomposition. Next, we will be interested in the situation where translation-bounded measures are weighted Meyer sets. In this context, it will be shown that the decomposition, whether it exists, also consists of weighted Meyer sets. We complete this work by discussing a natural generalization of the considered problem.

We study invariant random subgroups (IRSs) of semidirect products $G=A\rtimes \unicode[STIX]{x1D6E4}$. In particular, we characterize all IRSs of parabolic subgroups of $\text{SL}_{d}(\mathbb{R})$, and show that all ergodic IRSs of $\mathbb{R}^{d}\rtimes \text{SL}_{d}(\mathbb{R})$ are either of the form $\mathbb{R}^{d}\rtimes K$ for some IRS of $\text{SL}_{d}(\mathbb{R})$, or are induced from IRSs of $\unicode[STIX]{x1D6EC}\rtimes \text{SL}(\unicode[STIX]{x1D6EC})$, where $\unicode[STIX]{x1D6EC}<\mathbb{R}^{d}$ is a lattice.

Let $f:X\rightarrow X$ be a continuous dynamical system on a compact metric space $X$ and let $\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$ be an $m$-dimensional continuous potential. The (generalized) rotation set $\text{Rot}(\unicode[STIX]{x1D6F7})$ is defined as the set of all $\unicode[STIX]{x1D707}$-integrals of $\unicode[STIX]{x1D6F7}$, where $\unicode[STIX]{x1D707}$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $\unicode[STIX]{x210B}(w)$ to each $w\in \text{Rot}(\unicode[STIX]{x1D6F7})$. In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $f$ is a subshift of finite type. We prove that $\text{Rot}(\unicode[STIX]{x1D6F7})$ is computable and that $\unicode[STIX]{x210B}(w)$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $\unicode[STIX]{x210B}$ is not continuous on the boundary of the rotation set when considered as a function of $\unicode[STIX]{x1D6F7}$ and $w$. In particular, $\unicode[STIX]{x210B}$ is, in general, not computable at the boundary of $\text{Rot}(\unicode[STIX]{x1D6F7})$.

We show that for almost every $(P,\unicode[STIX]{x1D706})$, where $P$ is a convex polygon and $\unicode[STIX]{x1D706}\in (0,1)$, the corresponding outer billiard about $P$ with contraction $\unicode[STIX]{x1D706}$ is asymptotically periodic, i.e., has a finite number of periodic orbits and every orbit is attracted to one of them.

In this paper, we establish a non-commutative analogue of Calderón’s transference principle, which allows us to deduce the non-commutative maximal ergodic inequalities from the special case—operator-valued maximal inequalities. As applications, we deduce the non-commutative Stein–Calderón maximal ergodic inequality and the dimension-free estimates of the non-commutative Wiener maximal ergodic inequality over Euclidean spaces. We also show the corresponding individual ergodic theorems. To show Wiener’s pointwise ergodic theorem, following a somewhat standard way we construct a dense subset on which pointwise convergence holds. To show Jones’ pointwise ergodic theorem, we use again the transference principle together with the Littlewood–Paley method, which is different from Jones’ original variational method that is still unavailable in the non-commutative setting.

We consider a class of dynamical systems generated by finite sets of Möbius transformations acting on the unit disc. Compositions of such Möbius transformations give rise to sequences of transformations that are used in the theory of continued fractions. In that theory, the distinction between sequences of limit-point type and sequences of limit-disc type is of central importance. We prove that sequences of limit-disc type only arise in exceptional circumstances, and we give necessary and sufficient conditions for a sequence to be of limit-disc type. We also calculate the Hausdorff dimension of the set of sequences of limit-disc type in some significant cases. Finally, we obtain strong and complete results on the convergence of these dynamical systems.

Let $(X,\mathfrak{B},\unicode[STIX]{x1D707})$ be a Borel probability space. Let $T_{n}:X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $\unicode[STIX]{x1D708}$ be a probability measure on $X$ such that $(1/N)\sum _{n=1}^{N}(T_{n})_{\ast }\unicode[STIX]{x1D708}\rightarrow \unicode[STIX]{x1D707}$ in the weak-$\ast$ topology. Under general conditions, we show that for $\unicode[STIX]{x1D708}$ almost every $x\in X$, the measures $(1/N)\sum _{n=1}^{N}\unicode[STIX]{x1D6FF}_{T_{n}x}$ become equidistributed towards $\unicode[STIX]{x1D707}$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prove equidistribution of exponentially sparse orbits of the horocycle flow on quotients of $\text{SL}(2,\mathbb{R})$, starting from every point in almost every direction.

We show that every uniformly recurrent subgroup of a locally compact group is the family of stabilizers of a minimal action on a compact space. More generally, every closed invariant subset of the Chabauty space is the family of stabilizers of an action on a compact space on which the stabilizer map is continuous everywhere. This answers a question of Glasner and Weiss. We also introduce the notion of a universal minimal flow relative to a uniformly recurrent subgroup and prove its existence and uniqueness.

Analyticity results of expected pressure and invariant densities in the context of random dynamics of transcendental functions are established. These are obtained by a refinement of work by Rugh [On the dimension of conformal repellors, randomness and parameter dependency. Ann. of Math. (2) 168(3) (2008), 695–748] leading to a simple approach to analyticity. We work under very mild dynamical assumptions. Just the iterates of the Perron–Frobenius operator are assumed to converge. We also provide Bowen’s formula expressing the almost sure Hausdorff dimension of the radial fiberwise Julia sets in terms of the zero of an expected pressure function. Our main application establishes real analyticity for the variation of this dimension for suitable hyperbolic random systems of entire or meromorphic functions.

We study the finite versus infinite nature of C$^{\ast }$-algebras arising from étale groupoids. For an ample groupoid $G$, we relate infiniteness of the reduced C$^{\ast }$-algebra $\text{C}_{r}^{\ast }(G)$ to notions of paradoxicality of a K-theoretic flavor. We construct a pre-ordered abelian monoid $S(G)$ which generalizes the type semigroup introduced by Rørdam and Sierakowski for totally disconnected discrete transformation groups. This monoid characterizes the finite/infinite nature of the reduced groupoid C$^{\ast }$-algebra of $G$ in the sense that if $G$ is ample, minimal, topologically principal, and $S(G)$ is almost unperforated, we obtain a dichotomy between the stably finite and the purely infinite for $\text{C}_{r}^{\ast }(G)$. A type semigroup for totally disconnected topological graphs is also introduced, and we prove a similar dichotomy for these graph $\text{C}^{\ast }$-algebras as well.

In this paper, we consider the spectrum of discrete quasi-periodic Schrödinger operators on $\ell ^{2}(\mathbb{Z})$ with the potentials $v\in C^{k}(\mathbb{T})$. For sufficiently large $k$, we show that the Lebesgue measure of the spectrum at irrational frequencies is the limit of the Lebesgue measure of the spectrum of its periodic approximants. This gives a partial answer to the problem proposed in Jitomirskaya and Mavi [Continuity of the measure of the spectrum for quasiperiodic schrödinger operator with rough potentials. Comm. Math. Phys.325 (2014), 585–601]. Our results are based on a generalization of the rigidity theorem in Avila and Krikorian [Reducibility or non-uniform hyperbolicity for quasiperiodic Schrödinger cocycles. Ann. of Math. (2)164 (2006), 911–940]; more precisely, we prove that in the $C^{k}$ case, for almost every frequency $\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$ and for almost every $E$, the corresponding quasi-periodic Schrödinger cocycles are either reducible or non-uniformly hyperbolic.