arXiv subjects

Yuto Nakajima

Publications and source records attributed to Yuto Nakajima.

18 recordsLinked to original sources

arXiv2026

Coloring in anyon superconductivity

The recently observed signatures of superconductivity proximate to a fractional quantum anomalous Hall (FQAH) state in a twisted MoTe$_2$ bilayer has revitalized interest in quantum phases of matter induced by anyon dynamics. Here we show how a panoply of anyon-driven phases associated with doping the lattice ${\nu=2/3}$ FQAH state can be realized as competing instabilities of a Fermi surface of charge-$e/3$ ``quarks'' coupled to a $\mathrm{SU}(3)_{-1}$ Chern-Simons gauge field, which is dual to the more conventional $\mathrm{U}(1)_3$ Chern-Simons-Ginzburg-Landau theory of quasiholes. For example, a range of electronic superconductors emerge from \emph{color superconductivity}, under which the Fermi surface experiences a pairing instability mediated by gauge fluctuations. These include SC$\star$ phases -- where superconductivity coexists with topological order -- as well as topological superconductors displaying half-integer chiral central charges when the quarks are weakly paired. One example is a $p+ip$ ``color-valley-locked'' superconductor, a topological analogue of the color superconductor familiar in quantum chromodynamics. On the other hand, both superconducting and non-Fermi liquid phases can emerge when the quarks form an itinerant ferromagnet, polarizing the Fermi surface to a particular combination of colors. Finally, our framework naturally accommodates the possibility of anyonic bound state formation, allowing access to phases induced by doping anyons of charge $2e/3$ as opposed to $e/3$ within the same model. Our work unifies many earlier proposed anyonic phases as instabilities of a single parent \emph{quark metal} phase, distilling their emergence into a competition between superconductivity and itinerant color ferromagnetism.

cond-mat.str-el↗

arXiv2026

Exact dimensionality of projected measures for expanding rational semigroups

We study the dimension theory of expanding rational semigroups. An expanding rational semigroup is naturally described by a skew product on the product of the symbolic space and the Riemann sphere. For an invariant Borel probability measure of this skew product, we consider its disintegration over a symbolic factor and study the push-forwards of the conditional measures under the second coordinate projection. We prove a Ledrappier--Young type formula for the projected conditional measures. In particular, if the original invariant measure is ergodic, then these projected conditional measures are exact dimensional for almost every fiber. Applying this result to the trivial factor yields exact dimensionality of the projected invariant measure. This result can be viewed as a backward analogue of the result of [D.-J. Feng and H. Hu. \textit{Comm. Pure Appl. Math.} \textbf{62} (2009), no. 11, 1435--1500].

math.DS↗

arXiv2026

Fractal transference principle for continued fractions of Laurent series

We establish a fractal transference principle for continued fraction expansions over the field of Laurent series. Let $S$ be an infinite subset of the set of all polynomials over a finite field of $q$ elements of positive degree with growth density exponent $\alpha \ge 1$, and let $U \subset S$ be a subset of positive relative upper density. We prove that there exists a subset $E_{S,U}$ of the set of points whose continued fraction digits are pairwise distinct and belong to $S$ such that \[\dim_{\rm H} E_{S,U}=\frac{1}{2\alpha}.\] Moreover, the set of digits appearing in the continued fraction expansions of points in $E_{S,U}$ recovers the relative upper density of $U$ in $S$. We also show that the same construction preserves the relative upper density of the corresponding degree sets in $\mathbb N$. As a consequence, combinatorial statements for subsets of $\mathbb N$ of positive upper density can be transferred to degree sets arising from continued fraction expansions of Laurent series on sets of optimal Hausdorff dimension.

math.DS↗

arXiv2026

Functional correlation bound for random Lasota--Yorke maps with holes and its applications to conditional normal approximations

This paper investigates the statistical properties of random open dynamical systems generated by families of Lasota--Yorke maps. Open systems, in which trajectories may escape through `holes', model transient phenomena and present additional difficulties for statistical analysis because the underlying ensemble loses mass over time. We show that the framework of functional correlation bounds (FCB), originally developed for closed systems, can also be adapted to this random open setting. The extension requires new ingredients based on Lasota--Yorke type inequalities in order to control the effect of escaping trajectories. We establish an FCB with exponential decay and combine it with the abstract normal-approximation results of \cite{LNN25,LS20} to obtain a conditional CLT with rates in Wasserstein distance and a conditional functional CLT with a rate in an integral distance over Barbour's class of smooth test functions. Additionally, we adapt Tikhomirov's method to obtain a bound in Kolmogorov distance for the conditional CLT.

math.DS↗

arXiv2026

Topology of slices through the Sierpi\'nski tetrahedron

We investigate slices of the Sierpi\'nski tetrahedron from a topological viewpoint. For each $c\in[0,1]$, we study the \v{C}ech (co)homology group of the slice at height $c$. We show that the topology of the slice exhibits a sharp dichotomy. If $c$ is a dyadic rational, then the slice has finitely many connected components, infinite first \v{C}ech homology, and trivial higher homology. If $c$ is not a dyadic rational, then the slice is totally disconnected and all positive-degree \v{C}ech homology groups vanish.

math.DS↗

arXiv2026

How many points contain homothetic copies in their Hurwitz continued fraction expansion?

We prove that the set of complex irrationals whose partial quotients in their Hurwitz continued fraction expansion are naturally regarded as subsets of $\mathbb Z^2$ and contain infinitely many homothetic copies of any finite subset of $\mathbb Z^2$ is of Hausdorff dimension $1$. Our result provides a clear and concrete example of multidimensional pattern emergence in number-theoretic expansions.

math.DS↗

arXiv2026

Hausdorff dimension of sets of numbers whose continued fractions contain arbitrarily long arithmetic progressions

Continued fractions with prescribed structures on sequences of their partial quotients have been intensively studied in the literature. As far as an integer sequence, especially a randomly generated one is concerned, an attractive question is whether it contains arbitrarily long arithmetic progressions. In this paper we study the fractal structure of irrational numbers whose sequences of partial quotients are strictly increasing and contain arbitrarily long, quantified arithmetic progressions.

math.NT↗

arXiv2025

Error bounds in a smooth metric for Brownian approximation of dynamical systems via Stein's method

We adapt Stein's method of diffusion approximations, developed by Barbour, to the study of chaotic dynamical systems. We establish an error bound in the functional central limit theorem with respect to an integral probability metric of smooth test functions under a functional correlation decay bound. For systems with a sufficiently fast polynomial rate of correlation decay, the error bound is of order $O(N^{-1/2})$, under an additional condition on the linear growth of variance. Applications include a family of interval maps with neutral fixed points and unbounded derivatives, and two-dimensional dispersing Sinai billiards.

math.DS↗

arXiv2025

On the topology of the limit sets of non-autonomous iterated function systems

Since Mandelbrot's seminal work, there has been growing interest in the geometric nature of fractals. While the topological properties of the limit sets of IFSs have been studied -- notably in the pioneering work of Hata -- many aspects remain poorly understood, specially in the non-autonomous setting. In this paper, we investigate the topology of limit sets arising from randomly generated non-autonomous IFSs. To this end, we develop a simplicial-homological framework that makes their topological structure accessible to rigorous analysis. We apply our abstract theory to the concrete analysis of the so-called fractal squares, and provide an answer to a variant of Mandelbrot's percolation problem. Moreover, for the non-autonomous fractal squares considered here, we prove that the Betti numbers of the finite-stage approximations grow exponentially at a rate equal to the natural symbolic entropy of the system. This reveals a quantitative link between topology across scales and dynamical complexity.

math.DS↗

arXiv2025

Density combinatorics theorems in fractal dimension theory of continued fractions

We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of subsets of irrationals in $(0,1)$ for which the set $\{a_n(x)\colon n\in\mathbb N\}$ of partial quotients induces an injection $n\in\mathbb N\mapsto a_n(x)\in\mathbb N$. Let $(*)$ be a certain property that holds for any subset of $\mathbb N$ with positive upper density. The principle asserts that for any subset $S$ of $\mathbb N$ with positive upper density, there exists a set $E_S$ of Hausdorff dimension $1/2$ such that the set $\bigcup_{n\in\mathbb N}\bigcap_{x\in E_S}\{a_n(x)\}\cap S$ has the same upper density as that of $S$, and thus inherits property $(*)$. Examples of $(*)$ include the existence of arithmetic progressions of arbitrary lengths and the existence of arbitrary polynomial progressions, known as Szemerédi's and Bergelson-Leibman's theorems respectively. In the same spirit, we establish a relativized version of the principle applicable to the primes, to the primes of the form $y^2+z^2+1$, to the sets given by the Piatetski-Shapiro sequences.

math.NT↗

arXiv2025

Thermodynamics of dilute anyon gases from fusion constraints

Recent measurements on 2d materials tuning between fractional quantum anomalous Hall phases and a plethora of correlated electronic states call for a detailed understanding of the dynamics of anyons. Here we develop a general theory of the statistical mechanics of anyon gases at finite temperature, valid in regimes where the anyons are sufficiently dilute and can be treated as weakly interacting particles. We find that with a minimal set of universal braiding and fusion data, along with information about the hierarchy of anyon gaps, it is possible to construct a distribution function for any dilute anyon gas, as well as derive thermodynamic observables. Our results are built on an anyon exclusion principle manifesting as a constraint on fusion outcomes of physical states. Our approach unifies and streamlines a range of results for itinerant anyon models, from solvable lattice Hamiltonians to large-N field theories.

cond-mat.str-el↗

arXiv2025

A problem of Hirst for the Hurwitz continued fraction and the Hausdorff dimension of sets with restricted slowly growing digits

We address the problem of determining the Hausdorff dimension of sets consisting of complex irrationals whose complex continued fraction digits satisfy prescribed restrictions and growth conditions. For the Hurwitz continued fraction, we confirm Hirst's conjecture, as a complex analogue of the result of Wang and Wu [Bull. Lond. Math. Soc. {\bf 40} (2008), no. 1, 18--22] for the regular continued fraction. We also prove a complex analogue of the second-named author's result on the Hausdorff dimension of sets with restricted slowly growing digits [Proc. Amer. Math. Soc. {\bf 151} (2023), no. 9, 3645--3653]. To these ends, we exploit an infinite conformal iterated function system associated with the Hurwitz continued fraction.

math.DS↗

arXiv2024

Hausdorff dimension of sets with restricted, slowly growing partial quotients in semi-regular continued fractions

We determine the Hausdorff dimension of sets of irrationals in $(0,1)$ whose partial quotients in semi-regular continued fractions obey certain restrictions and growth conditions. This result substantially generalizes that of the second author [Proc. Amer. Math. Soc. {\bf 151} (2023), 3645--3653] and the solution of Hirst's conjecture [B.-W. Wang and J. Wu, Bull. London Math. Soc. {\bf 40} (2008), 18--22], both previously obtained for the regular continued fraction. To prove the result, we construct non-autonomous iterated function systems well-adapted to the given restrictions and growth conditions on partial quotients, estimate the associated pressure functions, and then apply Bowen's formula.

math.DS↗

arXiv2024

Local Polyakov-loop fluctuation and center domains in quark-gluon plasma with many colors

The deconfinement transition in non-Abelian gauge theory is understood as spontaneous breaking of $\mathbb{Z}_N$ symmetry at high temperatures. Accordingly, quark-gluon plasma generally includes some partial cells called center domains, each with a homogeneous Polyakov-loop expectation value. In this work, constructing an effective action describing the deconfinement vacuum of Yang-Mills theory with $N$ colors, we discuss the properties of center domains. First, we evaluate the spatial correlation of local Polyakov-loop fluctuation and demonstrate that some fluctuation becomes a Nambu-Goldstone-like mode in the large-$N$ limit. We also discuss surface tension between two $\mathbb{Z}_N$ center domains. Second, we estimate the global vacuum-to-vacuum transition in a single center domain. We find that some threshold volume exists, where a domain larger than this volume is stable, and vice versa. Identifying the threshold as the lower bound of a stable center domain volume, we quantitatively argue the typical volume scale of center domains.

hep-lat↗

arXiv2023

Transversal family of non-autonomous conformal iterated function systems

We study Non-autonomous Iterated Function Systems (NIFSs) with overlaps. A NIFS on a compact subset $X\subset\mathbb{R}^m$ is a sequence $Φ=(\{ϕ^{(j)}_{i}\}_{i\in I^{(j)}})_{j=1}^{\infty}$ of collections of uniformly contracting maps $ϕ^{(j)}_{i}: X\rightarrow X$, where $I^{(j)}$ is a finite set. In comparison to usual iterated function systems, we allow the contractions $ϕ^{(j)}_{i}$ applied at each step $j$ to depend on $j$. In this paper, we focus on a family of parameterized NIFSs on $\mathbb{R}^m$. Here, we do not assume the open set condition. We show that if a $d-$parameter family of such systems satisfies the transversality condition, then for almost every parameter value the Hausdorff dimension of the limit set is the minimum of $m$ and the Bowen dimension. Moreover, we give an example of a family $\{Φ_t\}_{t\in U}$ of parameterized NIFSs such that $\{Φ_t\}_{t\in U}$ satisfies the transversality condition but $Φ_t$ does not satisfy the open set condition for any $t\in U$.

math.DS↗

arXiv2023

Mandelbrot set for fractal $n$-gons and zeros of power series

We give a framework to study the connectedness of the set of zeros of power series with coefficients in a finite subset $G\subset \mathbb{C}$. We prove that the set of zeros in the unit disk is connected and locally connected if some graph on the set $G$ of coefficients is connected. Furthermore, we apply this result to the study of the Mandelbrot set $\mathcal{M}_n$ for fractal $n$-gons. We prove that $\mathcal{M}_n$ is connected and locally connected for any $n$.

math.DS↗

arXiv2023

$\mathbb{Z}_N$ structure of deconfinement vacuum in SU($N$) Yang-Mills theory: emergence of Nambu-Goldstone mode in large-$N$ limit

Using the Polyakov-loop effective action, we investigate the structure of spontaneously broken $\mathbb{Z}_N$ symmetry in the deconfinement vacuum in the SU($N$) Yang-Mills theory with finite $N$. First, we examine the Polyakov-loop fluctuation around a $\mathbb{Z}_N$-broken vacuum and calculate the spatial correlation of its phase variable. We show that the phase variable of the Polyakov loop becomes a Nambu-Goldstone mode in the large-$N$ limit. Second, we estimate the global vacuum-to-vacuum transition rate in a finite-volume domain of the quark-gluon plasma. Based on our estimation, we state that some threshold volume exists, a domain larger than which is stable, and vice versa. Identifying the threshold as the lower bound of a stable center domain volume, we find the typical volume scale of center domains.

hep-th↗

arXiv2020

Self-similar fractals related to regular tetrahedron and imaginary cubes

We consider self-similar sets in three-dimensional Euclidean space related to a regular tetrahedron. Sierpi${\rm \acute{n}}$ski tetrahedron is one such self-similar set. In this paper, we study the whole family of those sets. Our motivation is to obtain three-dimensional analogues of the fractal $n$-gons. In particular, we focus on the geometric properties of those sets from a viewpoint of ``imaginary cube''. An imaginary cube is a set $A$ for which there is some cube $C$ such that the projections of $A$ in the directions of the faces of $C$ equal these projections of $C$. It is already known that the Sierpi${\rm \acute{n}}$ski tetrahedron is an imaginary cube. We obtain a criterion for self-similar sets to be imaginary cubes. Furthermore, we show some properties of those sets which are imaginary cubes from a viewpoint of rotational symmetry or connectedness.

math.DS↗