For $ \beta>1 $, let $ T_\beta $ be the $\beta $-transformation on $ [0,1) $. Let $ \beta _1,\ldots ,\beta _d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define

$$ \begin{align*}W(\mathcal P)=\{\mathbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_d})^n(\mathbf{x})\in P_n\text{ infinitely often}\}.\end{align*} $$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}}$

$ P_n $

$\dim _{\mathrm {H}} W(\mathcal P)$

$\dim _{\mathrm {H}} W(\mathcal P)$

We define balanced self-similar quasi-round carpets and compare the carpet moduli of some path families relating to such a carpet. Then, using some known results on quasiconformal geometry of carpets, we prove that the group of quasisymmetric self-homeomorphisms of every balanced self-similar quasi-round carpet is finite. Furthermore, we prove that some balanced self-similar carpets in the unit square with strong geometric symmetry are quasisymmetrically rigid by using the quasisymmetry of weak tangents of carpets.

We study the multifractal properties of the uniform approximation exponent and asymptotic approximation exponent in continued fractions. As a corollary, we calculate the Hausdorff dimension of the uniform Diophantine set

$$ \begin{align*} {\mathcal{U}(\hat{\nu})}= &\ \{x\in[0,1)\colon \text{for all }N\gg1,\text{ there exists }n\in[1,N],\\&\ \ \text{ such that }|T^{n}(x)-y| < |I_{N}(y)|^{\hat{\nu}}\} \end{align*} $$

$y\in [0,1)$

For $\lambda \in (0,\,1/2]$ let $K_\lambda \subset \mathbb {R}$ be a self-similar set generated by the iterated function system $\{\lambda x,\, \lambda x+1-\lambda \}$. Given $x\in (0,\,1/2)$, let $\Lambda (x)$ be the set of $\lambda \in (0,\,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda (x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\,\ldots,\, y_p\in (0,\,1/2)$ there exists a full Hausdorff dimensional set of $\lambda \in (0,\,1/2]$ such that $y_1,\,\ldots,\, y_p \in K_\lambda$.

We compare the dimension of a non-invertible self-affine set to the dimension of the respective invertible self-affine set. In particular, for generic planar self-affine sets, we show that the dimensions coincide when they are large and differ when they are small. Our study relies on thermodynamic formalism where, for dominated and irreducible matrices, we completely characterise the behaviour of the pressures.

Feng and Huang [Variational principle for weighted topological pressure. J. Math. Pures Appl. (9) 106 (2016), 411–452] introduced weighted topological entropy and pressure for factor maps between dynamical systems and established its variational principle. Tsukamoto [New approach to weighted topological entropy and pressure. Ergod. Th. & Dynam. Sys. 43 (2023), 1004–1034] redefined those invariants quite differently for the simplest case and showed via the variational principle that the two definitions coincide. We generalize Tsukamoto’s approach, redefine the weighted topological entropy and pressure for higher dimensions, and prove the variational principle. Our result allows for an elementary calculation of the Hausdorff dimension of affine-invariant sets such as self-affine sponges and certain sofic sets that reside in Euclidean space of arbitrary dimension.

Let $[a_1(x),a_2(x),a_3(x),\ldots ]$ be the continued fraction expansion of an irrational number $x\in [0,1)$. We are concerned with the asymptotic behaviour of the product of consecutive partial quotients of x. We prove that, for Lebesgue almost all $x\in [0,1)$,

$$ \begin{align*} \liminf_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n} = 0\quad \text{and}\quad \limsup_{n \to \infty} \frac{\log (a_n(x)a_{n+1}(x))}{\log n}=1. \end{align*} $$

We also study the Baire category and the Hausdorff dimension of the set of points for which the above liminf and limsup have other different values and similarly analyse the weighted product of consecutive partial quotients.

Let $K\subset {\mathbb {R}}^d$ be a self-similar set generated by an iterated function system $\{\varphi _i\}_{i=1}^m$ satisfying the strong separation condition and let f be a contracting similitude with $f(K)\subseteq K$. We show that $f(K)$ is relatively open in K if all $\varphi _i$ share a common contraction ratio and orthogonal part. We also provide a counterexample when the orthogonal parts are allowed to vary. This partially answers a question of Elekes, Keleti and Máthé [Ergod. Th. & Dynam. Sys. 30 (2010), 399–440]. As a byproduct of our argument, when $d=1$ and K admits two homogeneous generating iterated function systems satisfying the strong separation condition but with contraction ratios of opposite signs, we show that K is symmetric. This partially answers a question of Feng and Wang [Adv. Math. 222 (2009), 1964–1981].

Let $\{b_n\}_{n=1}^{\infty }$ be a sequence of integers larger than 1. We will study the harmonic analysis of the equal-weighted Moran measures $\mu _{\{b_n\},\{{\mathcal D}_n\}}$ with ${\mathcal D}_n=\{0,1,2,\ldots ,q_n-1\}$, where $q_n$ divides $b_n$ for all $n\geq 1.$ In this paper, we first characterize all the maximal orthogonal sets of $L^2(\mu _{\{b_n\},\{{\mathcal D}_n\}})$ via a tree mapping. By this characterization, we give some sufficient conditions for the maximal orthogonal set to be an orthonormal basis.

We show that the Hausdorff dimension of any slice of the graph of the Takagi function is bounded above by the Assouad dimension of the graph minus one, and that the bound is sharp. The result is deduced from a statement on more general self-affine sets, which is of independent interest. We also prove that Marstrand’s slicing theorem on the graph of the Takagi function extends to all slices if and only if the upper pointwise dimension of every projection of the length measure on the x-axis lifted to the graph is at least one.

We determine almost sure limits of rescaled intrinsic volumes of the construction steps of fractal percolation in ${\mathbb R}^d$ for any dimension $d\geq 1$. We observe a factorization of these limit variables which allows one, in particular, to determine their expectations and covariance structure. We also show the convergence of the rescaled expectations and variances of the intrinsic volumes of the construction steps to the expectations and variances of the limit variables, and we give rates for this convergence in some cases. These results significantly extend our previous work, which addressed only limits of expectations of intrinsic volumes.

We present an elementary proof that the Rauzy gasket has Hausdorff dimension strictly smaller than two.

In this paper, we study the dimension of planar self-affine sets, of which generating iterated function system (IFS) contains non-invertible affine mappings. We show that under a certain separation condition the dimension equals to the affinity dimension for a typical choice of the linear-parts of the non-invertible mappings, furthermore, we show that the dimension is strictly smaller than the affinity dimension for certain choices of parameters.

Given a Borel probability measure µ on $\mathbb{R}^n$ and a real matrix $R\in M_n(\mathbb{R})$. We call R a spectral eigenmatrix of the measure µ if there exists a countable set $\Lambda\subset \mathbb{R}^n$ such that the sets $E_\Lambda=\big\{{\rm e}^{2\pi i \langle\lambda,x\rangle}:\lambda\in \Lambda\big\}$ and $E_{R\Lambda}=\big\{{\rm e}^{2\pi i \langle R\lambda,x\rangle}:\lambda\in \Lambda\big\}$ are both orthonormal bases for the Hilbert space $L^2(\mu)$. In this paper, we study the structure of spectral eigenmatrix of the planar self-affine measure $\mu_{M,D}$ generated by an expanding integer matrix $M\in M_2(2\mathbb{Z})$ and the four-elements digit set $D = \{(0,0)^t,(1,0)^t,(0,1)^t,(-1,-1)^t\}$. Some sufficient and/or necessary conditions for R to be a spectral eigenmatrix of $\mu_{M,D}$ are given.

This paper seeks to build on the extensive connections that have arisen between automata theory, combinatorics on words, fractal geometry, and model theory. Results in this paper establish a characterization for the behavior of the fractal geometry of “k-automatic” sets, subsets of $[0,1]^d$ that are recognized by Büchi automata. The primary tools for building this characterization include the entropy of a regular language and the digraph structure of an automaton. Via an analysis of the strongly connected components of such a structure, we give an algorithmic description of the box-counting dimension, Hausdorff dimension, and Hausdorff measure of the corresponding subset of the unit box. Applications to definability in model-theoretic expansions of the real additive group are laid out as well.

The purpose of this study is two-fold. First, the Hausdorff dimension formula of the multidimensional multiplicative subshift (MMS) in $\mathbb {N}^d$ is presented. This extends the earlier work of Kenyon et al [Hausdorff dimension for fractals invariant under multiplicative integers. Ergod. Th. & Dynam. Sys. 32(5) (2012), 1567–1584] from $\mathbb {N}$ to $\mathbb {N}^d$. In addition, the preceding work of the Minkowski dimension of the MMS in $\mathbb {N}^d$ is applied to show that their Hausdorff dimension is strictly less than the Minkowski dimension. Second, the same technique allows us to investigate the multifractal analysis of multiple ergodic average in $\mathbb {N}^d$. Precisely, we extend the result of Fan et al, [Multifractal analysis of some multiple ergodic averages. Adv. Math. 295 (2016), 271–333] of the multifractal analysis of multiple ergodic average from $\mathbb {N}$ to $\mathbb {N}^d$.

Let G be a real Lie group, $\Lambda <G$ a lattice and $H\leqslant G$ a connected semisimple subgroup without compact factors and with finite center. We define the notion of H-expanding measures $\mu $ on H and, applying recent work of Eskin–Lindenstrauss, prove that $\mu $-stationary probability measures on $G/\Lambda $ are homogeneous. Transferring a construction by Benoist–Quint and drawing on ideas of Eskin–Mirzakhani–Mohammadi, we construct Lyapunov/Margulis functions to show that H-expanding random walks on $G/\Lambda $ satisfy a recurrence condition and that homogeneous subspaces are repelling. Combined with a countability result, this allows us to prove equidistribution of trajectories in $G/\Lambda $ for H-expanding random walks and to obtain orbit closure descriptions. Finally, elaborating on an idea of Simmons–Weiss, we deduce Birkhoff genericity of a class of measures with respect to some diagonal flows and extend their applications to Diophantine approximation on similarity fractals to a nonconformal and weighted setting.

Several authors have shown that Kusuoka’s measure κ on fractals is a scalar Gibbs measure; in particular, it maximizes a pressure. There is also a different approach, in which one defines a matrix-valued Gibbs measure µ, which induces both Kusuoka’s measure κ and Kusuoka’s bilinear form. In the first part of the paper, we show that one can define a ‘pressure’ for matrix-valued measures; this pressure is maximized by µ. In the second part, we use the matrix-valued Gibbs measure µ to count periodic orbits on fractals, weighted by their Lyapounov exponents.

In this paper, we investigate the regularity properties and determine the almost sure multifractal spectrum of a class of random functions constructed as sums of pulses with random dilations and translations. In addition, the continuity moduli of the sample paths of these stochastic processes are investigated.