arXiv subjects

Junjie Miao

Publications and source records attributed to Junjie Miao.

8 recordsLinked to original sources

arXiv2026

Dimensions of Bedford-McMullen type sets in $\mathbb Z^2$

We study forward orbits in $\mathbb Z^2$ generated by the expanding affine maps $(x,y)\mapsto(mx+i,ny+j)$, where $m\ge n\ge2$ are integers and $(i,j)$ ranges over a nonempty digit set $Λ\subseteq\{0,\ldots,m-1\}\times\{0,\ldots,n-1\}$. We obtain explicit formulae for the mass, Beurling, discrete packing, and Assouad dimensions, the discrete Hausdorff dimension and its lower variant, and the lower entropy index. When $m=n$, all these dimensions equal $\log_m\#Λ$. When $m>n$, they can differ, and several depend on the starting point through the sizes of the endpoint columns of $Λ$. The two Hausdorff dimensions coincide and admit a pressure variational formula. Our proofs use finite symbolic models and approximate squares to relate digit counts to coverings in centered windows, translated windows, and annuli. We also characterize invariant lattice sets and determine their dimensions. Finally, we compare the orbit dimensions with semigroup growth and the dimensions of the compact dual attractor. In particular, the mass and Beurling dimensions need not equal the semigroup growth exponent, in contrast to the corresponding one-dimensional theory.

math.DS↗

arXiv2026

Reverse Iterated Function Systems: Density, Dimensions, and $p$-adic Extension

Strichartz initiated the study of reverse iterated function systems (RIFSs) on integer lattices and the dimensions of their invariant sets. We develop a dimension theory for RIFSs through forward orbits, first on locally compact complete metric spaces and then for one-dimensional affine systems. For an affine RIFS $\mathcal F=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ on $\mathbb R$, we introduce the semigroup-growth exponent\[ d_{\mathcal F}=\lim_{R\to\infty}\frac{\log\#\{g\in G^*(\mathcal F):|g'|\le R\}}{\log R}, \] where $G^*(\mathcal F)$ is the semigroup of distinct maps generated by $\mathcal F$. We prove that this limit exists and that $\dim_{\mathrm B}K(\mathcal F^{-1})\le d_{\mathcal F}\le s$. Every forward orbit has mass dimension $d_{\mathcal F}$ if it is locally finite and infinity otherwise; its Beurling dimension is $d_{\mathcal F}$ if it is uniformly locally finite and infinity otherwise. These formulas extend to suitably discrete invariant sets. Moreover, $d_{\mathcal F}=s$ if and only if the dual IFS has no exact overlaps, while $d_{\mathcal F}=0$ if and only if $\mathcal F$ is degenerate. Under finite overlaps, we obtain two-sided polynomial orbit-counting estimates. For non-overlapping uniformly locally finite orbits, renewal theory gives a limiting central density in the non-arithmetic case and a multiplicatively periodic asymptotic profile in the arithmetic case. We also determine the lower and discrete Hausdorff dimensions of rational lattice orbits with finite overlaps. Finally, for expansion ratios that are signed powers of a fixed prime $p$, we identify the Euclidean mass and Beurling dimensions of every rational orbit with the $p$-adic box-counting dimensions of the orbit and its attractor, without any separation assumption.

math.DS↗

arXiv2025

Dimensions and dimension spectra of Non-autonomous iterated function systems

Non-autonomous iterated function systems are a generalization of iterated function systems. If the contractions in the system are conformal mappings, it is called a non-autonomous conformal iterated function system, and its attractor is called a non-autonomous conformal set. In this paper, we study intermediate dimension spectra of non-autonomous conformal sets which provide a unifying framework for Hausdorff and box-counting dimensions. First, we obtain the intermediate dimension spectra formula of non-autonomous conformal sets by using upper and lower topological pressures. As a consequence, we obtain simplified forms of their Hausdorff, packing and box dimensions. Finally, we explore the Hausdorff dimensions of the non-autonomous infinite conformal iterated function systems which consists of countably many conformal mappings at each level, and we provide the Hausdorff dimension formula under certain conditions.

math.DS↗

arXiv2025

Assouad and quasi-Assouad dimensions of Moran sets

Moran sets are a non-autonomous generalization of self-similar sets. In this paper, we study the quasi-Assouad and Assouad dimensions of Moran sets in $\mathbb{R}^{d}$. First we provide quasi-Assouad dimension formulae for Moran sets satisfying $c_*>0$. Then, we provide the upper and lower bounds for quasi-Assouad dimension formulae for Moran sets without assuming $c_*>0$. To obtain the exact dimension formulae in this case, we define quasi-normal and normal Moran sets, and provide quasi-Assouad dimension formulae for these sets.

math.DS↗

arXiv2025

Existence, equivalence and spectrality of infinite convolutions in $\R^d$

In this paper, we study existence, equivalence and spectrality of infinite convolutions which may not be compactly supported in $d$-dimensional Euclidean space by manipulating various techniques in probability theory. First, we define the equivalent sequences, and we prove that the infinite convolutions converges simultaneously if they are generated by equivalent sequences. Moreover, the equi-positivity keeps unchanged for infinite convolutions generated by equivalent sequences. Next, we study the spectrality of infinite convolutions generated by admissible pairs, and we show such infinite convolutions have the same spectrum if they are generated by the equivalent sequences. Finally, we provide some sufficient conditions for the existence and spectral properties of infinite convolutions in higher dimensions.

math.FA↗

arXiv2025

Existence and spectrality of infinite convolutions generated by infinitely many admissible pairs

In this paper, we study the spectrality of infinite convolutions generated by infinitely many admissible pairs which may not be compactly supported, where the spectrality means the corresponding square integrable function space admits a family of exponential functions as an orthonormal basis. First, we prove that the infinite convolution exists and is a spectral measure if the sequence of admissible pairs satisfies the remainder bounded condition, and it has a subsequence consisting of general consecutive sets. Then we show that the subsequence of general consecutive sets may be replaced by a general assumption, named $θ$-bounded condition. Finally, we investigate the infinite convolutions generated by special subsequences, and give a sufficient condition for the spectrality of such infinite convolutions.

math.FA↗

arXiv2025

Existence and Spectrality of random measures generated by infinite convolutions

In this paper, we construct a class of random measures $μ^{\mathbf{n}}$ by infinite convolutions. Given infinitely many admissible pairs $\{(N_{k}, B_{k})\}_{k=1}^{\infty}$ and a positive integral sequence $\boldsymbol{n}=\{n_{k}\}_{k=1}^{\infty}$, for every $\boldsymbolω\in \mathbb{N}^{\mathbb{N}}$, we write $μ^{\mathbf{n}}(\boldsymbolω) = δ_{N_{ω_{1}}^{-n_{1}}B_{ω_{1}}} * δ_{N_{ω_{1}}^{-n_{1}}N_{ω_{2}}^{-n_{2}}B_{ω_{2}}} * \cdots$. If $n_{k}=1$ for $k\geq 1$, write $μ(\boldsymbolω)=μ^{\mathbf{n}}(\boldsymbolω)$. First, we show that the mapping $μ^{\mathbf{n}}: (\boldsymbolω, B) \mapsto μ^{\mathbf{n}}(\boldsymbolω)(B)$ is a random measure if the family of Borel probability measures $\{μ(\boldsymbolω) : \boldsymbolω \in \mathbb{N}^{\mathbb{N}}\}$ is tight. Then, for every Bernoulli measure $\mathbb{P}$ on $\mathbb{N}^{\mathbb{N}}$, the random measure $μ^{\mathbf{n}}$ is also a spectral measure $\mathbb{P}$-a.e.. If the positive integral sequence $\boldsymbol{n}$ is unbounded, the random measure $μ^{\mathbf{n}}$ is a spectral measure regardless of the measures on the sequence space $\mathbb{N}^{\mathbb{N}}$. Moreover, we provide some sufficient conditions for the existence of the random measure $μ^{\boldsymbol{n}}$. Finally, we verify that random measures have the intermediate-value property.

math.FA↗

arXiv2024

Intermediate dimensions of Moran sets and their visualization

Intermediate dimensions are a class of new fractal dimensions which provide a spectrum of dimensions interpolating between the Hausdorff and box-counting dimensions. In this paper, we study the intermediate dimensions of Moran sets. Moran sets may be regarded as a generalization of self-similar sets generated by using different class of similar mappings at each level with unfixed translations, and this causes the lack of ergodic properties on Moran set. Therefore, the intermediate dimensions do not necessarily exist, and we calculate the upper and lower intermediate dimensions of Moran sets. In particular, we obtain a simplified intermediate dimension formula for homogeneous Moran sets. Moreover, we study the visualization of the upper intermediate dimensions for some homogeneous Moran sets, and we show that their upper intermediate dimensions are given by Mobius transformations.

math.DS↗