Given $\beta \in (1,2]$, let $T_{\beta }$ be the $\beta $-transformation on the unit circle $[0,1)$ such that $T_{\beta }(x)=\beta x\pmod 1$. For each $t\in [0,1)$, let $K_{\beta }(t)$ be the survivor set consisting of all $x\in [0,1)$ whose orbit $\{T^{n}_{\beta }(x): n\ge 0\}$ never hits the open interval $(0,t)$. Kalle et al [Ergod. Th. & Dynam. Sys. 40(9) (2020) 2482–2514] proved that the Hausdorff dimension function $t\mapsto \dim _{H} K_{\beta }(t)$ is a non-increasing Devil’s staircase. So there exists a critical value $\tau (\beta )$ such that $\dim _{H} K_{\beta }(t)>0$ if and only if $t<\tau (\beta )$. In this paper, we determine the critical value $\tau (\beta )$ for all $\beta \in (1,2]$, answering a question of Kalle et al (2020). For example, we find that for the Komornik–Loreti constant $\beta \approx 1.78723$, we have $\tau (\beta )=(2-\beta )/(\beta -1)$. Furthermore, we show that (i) the function $\tau : \beta \mapsto \tau (\beta )$ is left continuous on $(1,2]$ with right-hand limits everywhere, but has countably infinitely many discontinuities; (ii) $\tau $ has no downward jumps, with $\tau (1+)=0$ and $\tau (2)=1/2$; and (iii) there exists an open set $O\subset (1,2]$, whose complement $(1,2]\setminus O$ has zero Hausdorff dimension, such that $\tau $ is real-analytic, convex, and strictly decreasing on each connected component of O. Consequently, the dimension $\dim _{H} K_{\beta }(t)$ is not jointly continuous in $\beta $ and t. Our strategy to find the critical value $\tau (\beta )$ depends on certain substitutions of Farey words and a renormalization scheme from dynamical systems.

For any self-similar measure $\mu $ in $\mathbb {R}$, we show that the distribution of $\mu $ is controlled by products of non-negative matrices governed by a finite or countable graph depending only on the iterated function system of similarities (IFS). This generalizes the net interval construction of Feng from the equicontractive finite-type case. When the measure satisfies the weak separation condition, we prove that this directed graph has a unique attractor. This allows us to verify the multifractal formalism for restrictions of $\mu $ to certain compact subsets of $\mathbb {R}$, determined by the directed graph. When the measure satisfies the generalized finite-type condition with respect to an open interval, the directed graph is finite and we prove that if the multifractal formalism fails at some $q\in \mathbb {R}$, there must be a cycle with no vertices in the attractor. As a direct application, we verify the complete multifractal formalism for an uncountable family of IFSs with exact overlaps and without logarithmically commensurable contraction ratios.

Let $\{R_{k}\}_{k=1}^{\infty }$ be a sequence of expanding integer matrices in $M_{n}(\mathbb {Z})$, and let $\{D_{k}\}_{k=1}^{\infty }$ be a sequence of finite digit sets with integer vectors in ${\mathbb Z}^{n}$. In this paper, we prove that under certain conditions in terms of $(R_{k},D_{k})$ for $k\ge 1$, the Moran measure

$$ \begin{align*} \mu_{\{R_{k}\},\{D_{k}\}}:=\delta_{R_{1}^{-1}D_{1}}\ast\delta_{R_{1}^{-1}R_{2}^{-1}D_{2}}\ast\cdots \end{align*} $$

$(R,D)$

We show that in the context of homogeneous Cantor sets, there are generically five possible (open and dense) structures for their arithmetic sum: a Cantor set, an L, R, M-Cantorval and a finite union of closed intervals. The dense case has been dealt with previously. In this paper, we explicitly present pairs of this space which have stable intersection, while not satisfying the generalized thickness test. Also, all the pairs of middle homogeneous Cantor sets whose arithmetic sum is a closed interval are identified.

We consider the two-dimensional shrinking target problem in beta dynamical systems (for general $\beta>1$) with general errors of approximation. Let $f, g$ be two positive continuous functions. For any $x_0,y_0\in [0,1]$, define the shrinking target set

$$ \begin{align*}E(T_\beta, f,g):=\left\{(x,y)\in [0,1]^2: \begin{array}{@{}ll@{}} \lvert T_{\beta}^{n}x-x_{0}\rvert <e^{-S_nf(x)}\\[1ex] \lvert T_{\beta}^{n}y-y_{0}\rvert < e^{-S_ng(y)} \end{array} \ {\text{for infinitely many}} \ n\in \mathbb N \right\}, \end{align*} $$

where $S_nf(x)=\sum _{j=0}^{n-1}f(T_\beta ^jx)$ is the Birkhoff sum. We calculate the Hausdorff dimension of this set and prove that it is the solution to some pressure function. This represents the first result of this kind for the higher-dimensional beta dynamical systems.

In this paper, we discuss a connection between geometric measure theory and number theory. This method brings a new point of view for some number-theoretic problems concerning digit expansions. Among other results, we show that for each integer k, there is a number $M>0$ such that if $b_{1},\ldots ,b_{k}$ are multiplicatively independent integers greater than M, there are infinitely many integers whose base $b_{1},b_{2},\ldots ,b_{k}$ expansions all do not have any zero digits.

The notion of a complete type can be generalized in a natural manner to allow assigning a value in an arbitrary Boolean algebra $\mathcal {B}$ to each formula. We show some basic results regarding the effect of the properties of $\mathcal {B}$ on the behavior of such types, and show they are particularity well behaved in the case of NIP theories. In particular, we generalize the third author’s result about counting types, as well as the notion of a smooth type and extending a type to a smooth one. We then show that Keisler measures are tied to certain Boolean types and show that some of the results can thus be transferred to measures—in particular, giving an alternative proof of the fact that every measure in a dependent theory can be extended to a smooth one. We also study the stable case. We consider this paper as an invitation for more research into the topic of Boolean types.

We give a new approach to characterising and computing the set of global maximisers and minimisers of the functions in the Takagi class and, in particular, of the Takagi–Landsberg functions. The latter form a family of fractal functions $f_\alpha:[0,1]\to{\mathbb R}$ parameterised by $\alpha\in(-2,2)$. We show that $f_\alpha$ has a unique maximiser in $[0,1/2]$ if and only if there does not exist a Littlewood polynomial that has $\alpha$ as a certain type of root, called step root. Our general results lead to explicit and closed-form expressions for the maxima of the Takagi–Landsberg functions with $\alpha\in(-2,1/2]\cup(1,2)$. For $(1/2,1]$, we show that the step roots are dense in that interval. If $\alpha\in (1/2,1]$ is a step root, then the set of maximisers of $f_\alpha$ is an explicitly given perfect set with Hausdorff dimension $1/(n+1)$, where n is the degree of the minimal Littlewood polynomial that has $\alpha$ as its step root. In the same way, we determine explicitly the minima of all Takagi–Landsberg functions. As a corollary, we show that the closure of the set of all real roots of all Littlewood polynomials is equal to $[-2,-1/2]\cup[1/2,2]$.

For any x in $[0,1)$, let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be its continued fraction. Let $\psi :\mathbb {N}\to \mathbb {R}^+$ be such that $\psi (n) \to \infty $ as $n\to \infty $. For any positive integers s and t, we study the set

$$ \begin{align*}E(\psi)=\{(x,y)\in [0,1)^2: \max\{a_{sn}(x), a_{tn}(y)\}\ge \psi(n) \ {\text{for all sufficiently large}}\ n\in \mathbb{N}\} \end{align*} $$

and determine its Hausdorff dimension.

Let $m_1 \geq m_2 \geq 2$ be integers. We consider subsets of the product symbolic sequence space $(\{0,\ldots ,m_1-1\} \times \{0,\ldots ,m_2-1\})^{\mathbb {N}^*}$ that are invariant under the action of the semigroup of multiplicative integers. These sets are defined following Kenyon, Peres, and Solomyak and using a fixed integer $q \geq 2$. We compute the Hausdorff and Minkowski dimensions of the projection of these sets onto an affine grid of the unit square. The proof of our Hausdorff dimension formula proceeds via a variational principle over some class of Borel probability measures on the studied sets. This extends well-known results on self-affine Sierpiński carpets. However, the combinatoric arguments we use in our proofs are more elaborate than in the self-similar case and involve a new parameter, namely $j = \lfloor \log _q ( {\log (m_1)}/{\log (m_2)} ) \rfloor $. We then generalize our results to the same subsets defined in dimension $d \geq 2$. There, the situation is even more delicate and our formulas involve a collection of $2d-3$ parameters.

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega $ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja et al. in [this Journal, vol. 80 (2015), pp. 1268–1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal.

We investigate stable intersections of conformal Cantor sets and their consequences to dynamical systems. First we define this type of Cantor set and relate it to horseshoes appearing in automorphisms of $\mathbb {C}^2$. Then we study limit geometries, that is, objects related to the asymptotic shape of the Cantor sets, to obtain a criterion that guarantees stable intersection between some configurations. Finally, we show that the Buzzard construction of a Newhouse region on $\mathrm{Aut}(\mathbb {C}^2)$ can be seen as a case of stable intersection of Cantor sets in our sense and give some (not optimal) estimate on how ‘thick’ those sets have to be.

We consider groupoids constructed from a finite number of commuting local homeomorphisms acting on a compact metric space and study generalized Ruelle operators and $ C^{\ast } $-algebras associated to these groupoids. We provide a new characterization of $ 1 $-cocycles on these groupoids taking values in a locally compact abelian group, given in terms of $ k $-tuples of continuous functions on the unit space satisfying certain canonical identities. Using this, we develop an extended Ruelle–Perron–Frobenius theory for dynamical systems of several commuting operators ($ k $-Ruelle triples and commuting Ruelle operators). Results on KMS states on $ C^{\ast } $-algebras constructed from these groupoids are derived. When the groupoids being studied come from higher-rank graphs, our results recover existence and uniqueness results for KMS states associated to the graphs.

This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set:

$$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$

where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$) shown by Hussain, Kleinbock, Wadleigh and Wang.

For given Boolean algebras $\mathbb {A}$ and $\mathbb {B}$ we endow the space $\mathcal {H}(\mathbb {A},\mathbb {B})$ of all Boolean homomorphisms from $\mathbb {A}$ to $\mathbb {B}$ with various topologies and study convergence properties of sequences in $\mathcal {H}(\mathbb {A},\mathbb {B})$. We are in particular interested in the situation when $\mathbb {B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb {A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin’s result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal {H}(\mathbb {A},\mathbb {B})$ for a Boolean algebra $\mathbb {B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb {A}$.

We extend the results of Hasselblatt and Schmeling [Dimension product structure of hyperbolic sets. Modern Dynamical Systems and Applications. Eds. B. Hasselblatt, M. Brin and Y. Pesin. Cambridge University Press, New York, 2004, pp. 331–345] and of Rams and Simon [Hausdorff and packing measure for solenoids. Ergod. Th. & Dynam. Sys. 23 (2003), 273–292] for $C^{1+\varepsilon }$ hyperbolic, (partially) linear solenoids $\Lambda $ over the circle embedded in $\mathbb {R}^3$ non-conformally attracting in the stable discs $W^s$ direction, to nonlinear solenoids. Under the assumptions of transversality and on the Lyapunov exponents for an appropriate Gibbs measure imposing thinness, as well as the assumption that there is an invariant $C^{1+\varepsilon }$ strong stable foliation, we prove that Hausdorff dimension $\operatorname {\mathrm {HD}}(\Lambda \cap W^s)$ is the same quantity $t_0$ for all $W^s$ and else $\mathrm {HD}(\Lambda )=t_0+1$. We prove also that for the packing measure, $0<\Pi _{t_0}(\Lambda \cap W^s)<\infty $, but for Hausdorff measure, $\mathrm {HM}_{t_0}(\Lambda \cap W^s)=0$ for all $W^s$. Also $0<\Pi _{1+t_0}(\Lambda ) <\infty $ and $\mathrm {HM}_{1+t_0}(\Lambda )=0$. A technical part says that the holonomy along unstable foliation is locally Lipschitz, except for a set of unstable leaves whose intersection with every $W^s$ has measure $\mathrm {HM}_{t_0}$ equal to 0 and even Hausdorff dimension less than $t_0$. The latter holds due to a large deviations phenomenon.

This is the second part of our study on the dimension theory of $C^1$ iterated function systems (IFSs) and repellers on $\mathbb {R}^d$. In the first part [D.-J. Feng and K. Simon. Dimension estimates for $C^1$ iterated function systems and repellers. Part I. Preprint, 2020, arXiv:2007.15320], we proved that the upper box-counting dimension of the attractor of every $C^1$ IFS on ${\Bbb R}^d$ is bounded above by its singularity dimension, and the upper packing dimension of every ergodic invariant measure associated with this IFS is bounded above by its Lyapunov dimension. Here we introduce a generalized transversality condition (GTC) for parameterized families of $C^1$ IFSs, and show that if the GTC is satisfied, then the dimensions of the IFS attractor and of the ergodic invariant measures are given by these upper bounds for almost every (in an appropriate sense) parameter. Moreover, we verify the GTC for some parameterized families of $C^1$ IFSs on ${\Bbb R}^d$.

Based on the Gale–Ryser theorem [2, 6], for the existence of suitable $(0,1)$-matrices for different partitions of a natural number, we revisit the classical result of Lorentz [4] regarding the characterization of a plane measurable set, in terms of its cross-sections, and extend it to general measure spaces.

We show that self-similar measures on $\mathbb R^d$ satisfying the weak separation condition are uniformly scaling. Our approach combines elementary ergodic theory with geometric analysis of the structure given by the weak separation condition.

We show that fractal percolation sets in $\mathbb{R}^{d}$ almost surely intersect every hyperplane absolutely winning (HAW) set with full Hausdorff dimension. In particular, if $E\subset\mathbb{R}^{d}$ is a realisation of a fractal percolation process, then almost surely (conditioned on $E\neq\emptyset$), for every countable collection $\left(f_{i}\right)_{i\in\mathbb{N}}$ of $C^{1}$ diffeomorphisms of $\mathbb{R}^{d}$, $\dim_{H}\left(E\cap\left(\bigcap_{i\in\mathbb{N}}f_{i}\left(\text{BA}_{d}\right)\right)\right)=\dim_{H}\left(E\right)$, where $\text{BA}_{d}$ is the set of badly approximable vectors in $\mathbb{R}^{d}$. We show this by proving that E almost surely contains hyperplane diffuse subsets which are Ahlfors-regular with dimensions arbitrarily close to $\dim_{H}\left(E\right)$.

We achieve this by analysing Galton–Watson trees and showing that they almost surely contain appropriate subtrees whose projections to $\mathbb{R}^{d}$ yield the aforementioned subsets of E. This method allows us to obtain a more general result by projecting the Galton–Watson trees against any similarity IFS whose attractor is not contained in a single affine hyperplane. Thus our general result relates to a broader class of random fractals than fractal percolation.