We present a new method of proving the Diophantine extremality of various dynamically defined measures, vastly expanding the class of measures known to be extremal. This generalizes and improves the celebrated theorem of Kleinbock and Margulis [Logarithm laws for flows on homogeneous spaces. Invent. Math.138(3) (1999), 451–494] resolving Sprindžuk’s conjecture, as well as its extension by Kleinbock, Lindenstrauss, and Weiss [On fractal measures and Diophantine approximation. Selecta Math.10 (2004), 479–523], hereafter abbreviated KLW. As applications we prove the extremality of all hyperbolic measures of smooth dynamical systems with sufficiently large Hausdorff dimension, and of the Patterson–Sullivan measures of all nonplanar geometrically finite groups. The key technical idea, which has led to a plethora of new applications, is a significant weakening of KLW’s sufficient conditions for extremality. In the first of this series of papers [Extremality and dynamically defined measures, part I: Diophantine properties of quasi-decaying measures. Selecta Math.24(3) (2018), 2165–2206], we introduce and develop a systematic account of two classes of measures, which we call quasi-decaying and weakly quasi-decaying. We prove that weak quasi-decay implies strong extremality in the matrix approximation framework, as well as proving the ‘inherited exponent of irrationality’ version of this theorem. In this paper, the second of the series, we establish sufficient conditions on various classes of conformal dynamical systems for their measures to be quasi-decaying. In particular, we prove the above-mentioned result about Patterson–Sullivan measures, and we show that equilibrium states (including conformal measures) of nonplanar infinite iterated function systems (including those which do not satisfy the open set condition) and rational functions are quasi-decaying.

Let $M$ be a closed, oriented, and connected Riemannian $n$ -manifold, for $n\geq 2$ , which is not a rational homology sphere. We show that, for a non-constant and non-injective uniformly quasiregular self-map $f:M\rightarrow M$ , the topological entropy $h(f)$ is $\log \deg f$ . This proves Shub’s entropy conjecture in this case.

Let $\unicode[STIX]{x1D6F4}$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $\unicode[STIX]{x1D6E4}$ of $\unicode[STIX]{x1D6F4}$ acts on the $\mathsf{SU}(3)$ -character variety of $\unicode[STIX]{x1D6F4}$ . We show that the action is ergodic with respect to the natural symplectic measure on the character variety.

Furstenberg [Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math. Syst. Theory1 (1967), 1–49] calculated the Hausdorff and Minkowski dimensions of one-sided subshifts in terms of topological entropy. We generalize this to $\mathbb{Z}^{2}$ -subshifts. Our generalization involves mean dimension theory. We calculate the metric mean dimension and the mean Hausdorff dimension of $\mathbb{Z}^{2}$ -subshifts with respect to a subaction of $\mathbb{Z}$ . The resulting formula is quite analogous to Furstenberg’s theorem. We also calculate the rate distortion dimension of $\mathbb{Z}^{2}$ -subshifts in terms of Kolmogorov–Sinai entropy.

Let $(G,\unicode[STIX]{x1D6EC})$ be a self-similar $k$ -graph with a possibly infinite vertex set $\unicode[STIX]{x1D6EC}^{0}$ . We associate a universal C*-algebra ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ to $(G,\unicode[STIX]{x1D6EC})$ . The main purpose of this paper is to investigate the ideal structures of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . We prove that there exists a one-to-one correspondence between the set of all $G$ -hereditary and $G$ -saturated subsets of $\unicode[STIX]{x1D6EC}^{0}$ and the set of all gauge-invariant and diagonal-invariant ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Under some conditions, we characterize all primitive ideals of ${\mathcal{O}}_{G,\unicode[STIX]{x1D6EC}}$ . Moreover, we describe the Jacobson topology of some concrete examples, which includes the C*-algebra of the product of odometers. On the way to our main results, we study self-similar $P$ -graph C*-algebras in depth.

Let $\operatorname{Homeo}_{+}(D_{n}^{2})$ be the group of orientation-preserving homeomorphisms of $D^{2}$ fixing the boundary pointwise and $n$ marked points as a set. The Nielsen realization problem for the braid group asks whether the natural projection $p_{n}:\operatorname{Homeo}_{+}(D_{n}^{2})\rightarrow B_{n}:=\unicode[STIX]{x1D70B}_{0}(\operatorname{Homeo}_{+}(D_{n}^{2}))$ has a section over subgroups of $B_{n}$ . All of the previous methods use either torsion or Thurston stability, which do not apply to the pure braid group $PB_{n}$ , the subgroup of $B_{n}$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside the area-preserving homeomorphisms using rotation numbers.

Given a ${\mathcal{C}}^{\infty }$ expanding map $T$ of the circle, we construct a Hilbert space ${\mathcal{H}}$ of smooth functions on which the transfer operator ${\mathcal{L}}$ associated to $T$ acts as a compact operator. This result is made quantitative (in terms of singular values of the operator ${\mathcal{L}}$ acting on ${\mathcal{H}}$ ) using the language of Denjoy–Carleman classes. Moreover, the nuclear power decomposition of Baladi and Tsujii can be performed on the space ${\mathcal{H}}$ , providing a bound on the growth of the dynamical determinant associated to ${\mathcal{L}}$ .

We study fast approximation of integrals with respect to stationary probability measures associated to iterated function systems on the unit interval. We provide an algorithm for approximating the integrals under certain conditions on the iterated function system and on the function that is being integrated. We apply this technique to estimate Hausdorff moments, Wasserstein distances and Lyapunov exponents of stationary probability measures.

We study Smale skew product endomorphisms (introduced in Mihailescu and Urbański [Skew product Smale endomorphisms over countable shifts of finite type. Ergod. Th. & Dynam. Sys. doi: 10.1017/etds.2019.31. Published online June 2019]) now over countable graph-directed Markov systems, and we prove the exact dimensionality of conditional measures in fibers, and then the global exact dimensionality of the equilibrium measure itself. Our results apply to large classes of systems and have many applications. They apply, for instance, to natural extensions of graph-directed Markov systems. Another application is to skew products over parabolic systems. We also give applications in ergodic number theory, for example to the continued fraction expansion, and the backward fraction expansion. In the end we obtain a general formula for the Hausdorff (and pointwise) dimension of equilibrium measures with respect to the induced maps of natural extensions ${\mathcal{T}}_{\unicode[STIX]{x1D6FD}}$ of $\unicode[STIX]{x1D6FD}$ -maps $T_{\unicode[STIX]{x1D6FD}}$ , for arbitrary $\unicode[STIX]{x1D6FD}>1$ .

The space of convex projective structures has been well studied with respect to the topological entropy. But, to better understand the geometry of the structure, we study the entropy of the Sinai–Ruelle–Bowen measure and show that it is a continuous function on the space of strictly convex real projective structures.

We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the $2$ -torus and the modular surface.

Let $\unicode[STIX]{x1D6E4}$ denote the mapping class group of the plane minus a Cantor set. We show that every action of $\unicode[STIX]{x1D6E4}$ on the circle is either trivial or semiconjugate to a unique minimal action on the so-called simple circle.

In this article, we will prove a full topological version of Popa’s measurable cocycle superrigidity theorem for full shifts [Popa, Cocycle and orbit equivalence superrigidity for malleable actions of $w$ -rigid groups. Invent. Math. 170(2) (2007), 243–295]. Let $G$ be a finitely generated group that has one end, undistorted elements and sub-exponential divergence function. Let $H$ be a target group that is complete and admits a compatible bi-invariant metric. Then, every Hölder continuous cocycle for the full shifts of $G$ with value in $H$ is cohomologous to a group homomorphism via a Hölder continuous transfer map. Using the ideas of Behrstock, Druţu, Mosher, Mozes and Sapir [Divergence, thick groups, and short conjugators. Illinois J. Math. 58(4) (2014), 939–980; Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity. Math. Ann. 344(3) (2009), 543–595; Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Amer. Math. Soc. 362(5) (2010), 2451–2505; Tree-graded spaces and asymptotic cones of groups. Topology 44(5) (2005), 959–1058], we show that the class of our acting groups is large including wide groups having undistorted elements and one-ended groups with strong thick of finite orders. As a consequence, irreducible uniform lattices of most of higher rank connected semisimple Lie groups, mapping class groups of $g$ -genus surfaces with $p$ -punches, $g\geq 2,p\geq 0$ ; Richard Thompson groups $F,T,V$ ; $\text{Aut}(F_{n})$ , $\text{Out}(F_{n})$ , $n\geq 3$ ; certain (two-dimensional) Coxeter groups; and one-ended right-angled Artin groups are in our class. This partially extends the main result in Chung and Jiang [Continuous cocycle superrigidity for shifts and groups with one end. Math. Ann. 368(3–4) (2017), 1109–1132].

We focus on various dynamical invariants associated to monomial correspondences on toric varieties, using algebraic and arithmetic geometry. We find a formula for their dynamical degrees, relate the exponential growth of the degree sequences to a strict log-concavity condition on the dynamical degrees and compute the asymptotic rate of the growth of heights of points of such correspondences.

The aim of the article is to provide a characterization of the Haagerup property for locally compact, second countable groups in terms of actions on $\unicode[STIX]{x1D70E}$ -finite measure spaces. It is inspired by the very first definition of amenability, namely the existence of an invariant mean on the algebra of essentially bounded, measurable functions on the group.

We define a new class of shift spaces which contains a number of classes of interest, like Sturmian shifts used in discrete geometry. We show that this class is closed under two natural transformations. The first one is called conjugacy and is obtained by sliding block coding. The second one is called the complete bifix decoding, and typically includes codings by non-overlapping blocks of fixed length.

Kifer, Peres, and Weiss proved in [A dimension gap for continued fractions with independent digits. Israel J. Math.124 (2001), 61–76] that there exists $c_{0}>0$ , such that $\dim \unicode[STIX]{x1D707}\leq 1-c_{0}$ for any probability measure $\unicode[STIX]{x1D707}$ , which makes the digits of the continued fraction expansion independent and identically distributed random variables. In this paper we prove that amongst this class of measures, there exists one whose dimension is maximal. Our results also apply in the more general setting of countable branched systems.

We consider the dynamics of complex rational maps on $\widehat{\mathbb{C}}$ . We prove that, after reducing their orbits to a fixed number of positive values representing the Fubini–Study distances between finitely many initial elements of the orbit and the origin, ergodic properties of the rational map are preserved.

Fix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$ . The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit

$$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$

$q\in (1,M+1)$

$f(q)>0$

$q\in \overline{\mathscr{U}}\backslash \mathscr{C}$

$\mathscr{C}$

$f$

$\mathscr{U}$

$\mathscr{U}$

For $G$ a Polish group, we consider $G$ -flows which either contain a comeager orbit or have all orbits meager. We single out a class of flows, the maximally highly proximal (MHP) flows, for which this analysis is particularly nice. In the former case, we provide a complete structure theorem for flows containing comeager orbits, generalizing theorems of Melleray, Nguyen Van Thé, and Tsankov and of Ben Yaacov, Melleray, and Tsankov. In the latter, we show that any minimal MHP flow with all orbits meager has a metrizable factor with all orbits meager, thus ‘reflecting’ complicated dynamical behavior to metrizable flows. We then apply this to obtain a structure theorem for Polish groups whose universal minimal flow is distal.