We investigate the entropy for a class of upper semi-continuous set-valued functions, called Markov set-valued functions, that are a generalization of single-valued Markov interval functions. It is known that the entropy of a Markov interval function can be found by calculating the entropy of an associated shift of finite type. In this paper, we construct a similar shift of finite type for Markov set-valued functions and use this shift space to find upper and lower bounds on the entropy of the set-valued function.

We prove that all transcendental entire functions have infinite topological entropy.

There is much research on the dynamical complexity on irregular sets and level sets of ergodic average from the perspective of density in base space, the Hausdorff dimension, Lebesgue positive measure, positive or full topological entropy (and topological pressure), etc. However, this is not the case from the viewpoint of chaos. There are many results on the relationship of positive topological entropy and various chaos. However, positive topological entropy does not imply a strong version of chaos, called DC1. Therefore, it is non-trivial to study DC1 on irregular sets and level sets. In this paper, we will show that, for dynamical systems with specification properties, there exist uncountable DC1-scrambled subsets in irregular sets and level sets. Meanwhile, we prove that several recurrent level sets of points with different recurrent frequency have uncountable DC1-scrambled subsets. The major argument in proving the above results is that there exists uncountable DC1-scrambled subsets in saturated sets.

Let $(X,{\mathcal{B}},\unicode[STIX]{x1D707},T)$ be an ergodic measure-preserving system, let $A\in {\mathcal{B}}$ and let $\unicode[STIX]{x1D716}>0$. We study the largeness of sets of the form

$$\begin{eqnarray}S=\{n\in \mathbb{N}:\unicode[STIX]{x1D707}(A\cap T^{-f_{1}(n)}A\cap T^{-f_{2}(n)}A\cap \cdots \cap T^{-f_{k}(n)}A)>\unicode[STIX]{x1D707}(A)^{k+1}-\unicode[STIX]{x1D716}\}\end{eqnarray}$$

$\{f_{1},\ldots ,f_{k}\}$

$f_{i}:\mathbb{N}\rightarrow \mathbb{N}$

$k\leq 3$

$f_{i}(n)=if(n)$

$S$

$f(n)=q(p_{n})$

$q\in \mathbb{Z}[x]$

$q(1)$

$q(-1)=0$

$p_{n}$

$n$

$f$

$T^{q}$

$q\in \mathbb{N}$

$r\in \mathbb{Z}$

$S$

$f(n)=qn+r$

$f_{i}(n)=a_{i}n$

$a_{i}$

$S$

$k\geq 4$

$k=3$

$S$

$f_{i}$

We study compact group actions with finite Rokhlin dimension, particularly in relation to crossed products. For example, we characterize the duals of such actions, generalizing previous partial results for the Rokhlin property. As an application, we determine the ideal structure of their crossed products. Under the assumption of so-called commuting towers, we show that taking crossed products by such actions preserves a number of relevant classes of $C^{\ast }$-algebras, including: $D$-absorbing $C^{\ast }$-algebras, where $D$ is a strongly self-absorbing $C^{\ast }$-algebra; stable $C^{\ast }$-algebras; $C^{\ast }$-algebras with finite nuclear dimension (or decomposition rank); $C^{\ast }$-algebras with finite stable rank (or real rank); and $C^{\ast }$-algebras whose $K$-theory is either trivial, rational, or $n$-divisible for $n\in \mathbb{N}$. The combination of nuclearity and the universal coefficient theorem (UCT) is also shown to be preserved by these actions. Some of these results are new even in the well-studied case of the Rokhlin property. Additionally, and under some technical assumptions, we show that finite Rokhlin dimension with commuting towers implies the (weak) tracial Rokhlin property. At the core of our arguments is a certain local approximation of the crossed product by a continuous $C(X)$-algebra with fibers that are stably isomorphic to the underlying algebra. The space $X$ is computed in some cases of interest, and we use its description to construct a $\mathbb{Z}_{2}$-action on a unital AF-algebra and on a unital Kirchberg algebra satisfying the UCT, whose Rokhlin dimensions with and without commuting towers are finite but do not agree.

In this work we deal with extreme value theory in the context of continued fractions using techniques from probability theory, ergodic theory and real analysis. We give an upper bound for the rate of convergence in the Doeblin–Iosifescu asymptotics for the exceedances of digits obtained from the regular continued fraction expansion of a number chosen randomly from $(0,1)$ according to the Gauss measure. As a consequence, we significantly improve the best known upper bound on the rate of convergence of the maxima in this case. We observe that the asymptotics of order statistics and the extremal point process can also be investigated using our methods.

We establish the limiting distribution of certain subsets of Farey sequences, i.e., sequences of primitive rational points, on expanding horospheres in covers $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ of $\text{SL}(n+1,\mathbb{Z})\backslash \text{SL}(n+1,\mathbb{R})$, where $\unicode[STIX]{x1D6E5}$ is a finite-index subgroup of $\text{SL}(n+1,\mathbb{Z})$. These subsets can be obtained by projecting to the hyperplane $\{(x_{1},\ldots ,x_{n+1})\in \mathbb{R}^{n+1}:x_{n+1}=1\}$ sets of the form $\mathbf{A}=\bigcup _{j=1}^{J}\mathbf{a}_{j}\unicode[STIX]{x1D6E5}$, where for all $j$, $\mathbf{a}_{j}$ is a primitive lattice point in $\mathbb{Z}^{n+1}$. Our method involves applying the equidistribution of expanding horospheres in quotients of $\text{SL}(n+1,\mathbb{R})$ developed by Marklof and Strömbergsson, and more precisely understanding how the full Farey sequence distributes in $\unicode[STIX]{x1D6E5}\backslash \text{SL}(n+1,\mathbb{R})$ when embedded on expanding horospheres as done in previous work by Marklof. For each of the Farey sequence subsets, we extend the statistical results by Marklof regarding the full multidimensional Farey sequences, and solutions by Athreya and Ghosh to Diophantine approximation problems of Erdős–Szüsz–Turán and Kesten. We also prove that Marklof’s result on the asymptotic distribution of Frobenius numbers holds for sets of primitive lattice points of the form $\mathbf{A}$.

We study dynamical systems that have bounded complexity with respect to three kinds metrics: the Bowen metric $d_{n}$, the max-mean metric $\hat{d}_{n}$ and the mean metric $\bar{d}_{n}$, both in topological dynamics and ergodic theory. It is shown that a topological dynamical system $(X,T)$ has bounded complexity with respect to $d_{n}$ (respectively $\hat{d}_{n}$) if and only if it is equicontinuous (respectively equicontinuous in the mean). However, we construct minimal systems that have bounded complexity with respect to $\bar{d}_{n}$ but that are not equicontinuous in the mean. It turns out that an invariant measure $\unicode[STIX]{x1D707}$ on $(X,T)$ has bounded complexity with respect to $d_{n}$ if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-equicontinuous. Meanwhile, it is shown that $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\hat{d}_{n}$ if and only if $\unicode[STIX]{x1D707}$ has bounded complexity with respect to $\bar{d}_{n}$, if and only if $(X,T)$ is $\unicode[STIX]{x1D707}$-mean equicontinuous and if and only if it has discrete spectrum.

We show that the finitely generated simple left orderable groups $G_{\!\unicode[STIX]{x1D70C}}$ constructed by the first two authors in Hyde and Lodha [Finitely generated infinite simple groups of homeomorphisms of the real line. Invent. Math. (2019), doi:10.1007/s00222-019-00880-7] are uniformly perfect—each element in the group can be expressed as a product of three commutators of elements in the group. This implies that the group does not admit any homogeneous quasimorphism. Moreover, any non-trivial action of the group on the circle, which lifts to an action on the real line, admits a global fixed point. It follows that any faithful action on the real line without a global fixed point is globally contracting. This answers Question 4 of the third author [A. Navas. Group actions on 1-manifolds: a list of very concrete open questions. Proceedings of the International Congress of Mathematicians, Vol. 2. Eds. B. Sirakov, P. Ney de Souza and M. Viana. World Scientific, Singapore, 2018, pp, 2029–2056], which asks whether such a group exists. This question has also been answered simultaneously and independently, using completely different methods, by Matte Bon and Triestino [Groups of piecewise linear homeomorphisms of flows. Preprint, 2018, arXiv:1811.12256]. To prove our results, we provide a characterization of elements of the group $G_{\!\unicode[STIX]{x1D70C}}$ which is a useful new tool in the study of these examples.

We show that various classes of products of manifolds do not support transitive Anosov diffeomorphisms. Exploiting the Ruelle–Sullivan cohomology class, we prove that the product of a negatively curved manifold with a rational homology sphere does not support transitive Anosov diffeomorphisms. We extend this result to products of finitely many negatively curved manifolds of dimension at least three with a rational homology sphere that has vanishing simplicial volume. As an application of this study, we obtain new examples of manifolds that do not support transitive Anosov diffeomorphisms, including certain manifolds with non-trivial higher homotopy groups and certain products of aspherical manifolds.

Let $X$ be a finite-dimensional connected compact abelian group equipped with the normalized Haar measure $\unicode[STIX]{x1D707}$. We obtain the following mean ergodic theorem over ‘thin’ phase sets. Fix $k\geq 1$ and, for every $n\geq 1$, let $A_{n}$ be a subset of $\mathbb{Z}^{k}\cap [-n,n]^{k}$. Assume that $(A_{n})_{n\geq 1}$ has $\unicode[STIX]{x1D714}(1/n)$ density in the sense that $\lim _{n\rightarrow \infty }(|A_{n}|/n^{k-1})=\infty$. Let $T_{1},\ldots ,T_{k}$ be ergodic automorphisms of $X$. We have

$$\begin{eqnarray}\frac{1}{|A_{n}|}\mathop{\sum }_{(n_{1},\ldots ,n_{k})\in A_{n}}f_{1}(T_{1}^{n_{1}}(x))\cdots f_{k}(T_{k}^{n_{k}}(x))\stackrel{L_{\unicode[STIX]{x1D707}}^{2}}{\longrightarrow }\int f_{1}\,d\unicode[STIX]{x1D707}\cdots \int f_{k}\,d\unicode[STIX]{x1D707},\end{eqnarray}$$

$f_{1},\ldots ,f_{k}\in L_{\unicode[STIX]{x1D707}}^{\infty }$

$T_{i}$

$A_{n}$

$[0,n]^{k}$

$n$

$A_{i}$

Let $f$ be an $n$-dimensional holomorphic map defined in a neighborhood of the origin such that the origin is an isolated fixed point of all of its iterates, and let ${\mathcal{N}}_{M}(f)$ denote the number of periodic orbits of $f$ of period $M$ hidden at the origin. Gorbovickis gives an efficient way of computing ${\mathcal{N}}_{M}(f)$ for a large class of holomorphic maps. Inspired by Gorbovickis’ work, we establish a similar method for computing ${\mathcal{N}}_{M}(f)$ for a much larger class of holomorphic germs, in particular, having arbitrary Jordan matrices as their linear parts. Moreover, we also give another proof of the result of Gorbovickis [On multi-dimensional Fatou bifurcation. Bull. Sci. Math.138(3)(2014) 356–375] using our method.

The notion of quantized characters was introduced in our previous paper as a natural quantization of characters in the context of asymptotic representation theory for quantum groups. As in the case of ordinary groups, the representation associated with any extreme quantized character generates a von Neumann factor. From the viewpoint of operator algebras (and measurable dynamical systems), it is natural to ask what is the Murray–von Neumann–Connes type of the resulting factor. In this paper, we give a complete solution to this question when the inductive system is of quantum unitary groups $U_{q}(N)$.

Let $m\in \mathbb{N}$ and $\mathbf{X}=(X,{\mathcal{X}},\unicode[STIX]{x1D707},(T_{\unicode[STIX]{x1D6FC}})_{\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}})$ be a measure-preserving system with an $\mathbb{R}^{m}$-action. We say that a Borel measure $\unicode[STIX]{x1D708}$ on $\mathbb{R}^{m}$ is weakly equidistributed for $\mathbf{X}$ if there exists $A\subseteq \mathbb{R}$ of density 1 such that, for all $f\in L^{\infty }(\unicode[STIX]{x1D707})$, we have

$$\begin{eqnarray}\lim _{t\in A,t\rightarrow \infty }\int _{\mathbb{R}^{m}}f(T_{t\unicode[STIX]{x1D6FC}}x)\,d\unicode[STIX]{x1D708}(\unicode[STIX]{x1D6FC})=\int _{X}f\,d\unicode[STIX]{x1D707}\end{eqnarray}$$

$\unicode[STIX]{x1D707}$

$x\in X$

$W(\mathbf{X})$

$\unicode[STIX]{x1D6FC}\in \mathbb{R}^{m}$

$\mathbb{R}$

$(T_{t\unicode[STIX]{x1D6FC}})_{t\in \mathbb{R}}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\mathbf{X}$

$\unicode[STIX]{x1D708}(W(\mathbf{X})+\unicode[STIX]{x1D6FD})=0$

$\unicode[STIX]{x1D6FD}\in \mathbb{R}^{m}$

$\unicode[STIX]{x1D708}$

$\mathbb{R}^{m}$

$\unicode[STIX]{x1D708}(\ell )=0$

$\ell$

$\mathbb{R}^{m}$

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.