arXiv subjects

Nathan Dalaklis

Publications and source records attributed to Nathan Dalaklis.

6 recordsLinked to original sources

arXiv2026

Complex Diophantine Approximations and Cusp Excursions

We study the Hausdorff dimension spectrum of asymptotic approximation rates of complex Diophantine approximation and that of the asymptotic average excursion time of cusp excursions on the Bianchi orbifold $\mathbb{H}^3/\operatorname{PSL}(2,\mathbb {Z}[i])$ via a unified approach using the Hurwitz map. In particular, we construct a conformal graph directed system (CGDS) for the Hurwitz map and show that the Lyapunov exponent of the Hurwitz CGDS simultaneously captures the asymptotic approximation rate and the the asymptotic average excursion time. Applying the multifractal analysis of Lyapunov exponents for this system, we obtain a formula and real-analyticity for the Hausdorff dimension spectrum functions.

math.DS↗

arXiv2025

Multifractal Analysis of Equilibrium States of Endomorphisms of $\mathbb{P}^k$

Let $f$ be a holomorphic endomorphism of $\mathbb{C}\mathbb{P}^k$ of algebraic degree at least $2$ and let $X \subseteq \mathbb{C}\mathbb{P}^k$ be an uniformly expanding set. In this paper, we study multifractal analysis of equilibrium states of H\"older continuous functions for the non-conformal dynamical system $f : X \to X$. In lieu of Hausdorff dimensions, we use a new dimension theory (i.e., the volume dimension theory) to define various local dimension multifractal spectra and show that each of these spectra form a Legendre transform pair with the temperature function as in the conformal case. As an application of our main theorems, we also prove a conditional variational principle for such dimension multifractal spectra.

math.DS↗

arXiv2025

The higher order partial derivatives of Okamoto's function with respect to the parameter

Let $\{F_a: a\in(0,1)\}$ be Okamoto's family of continuous self-affine functions, introduced in [{\em Proc. Japan Acad. Ser. A Math. Sci.} {\bf 81} (2005), no. 3, 47--50]. This family includes well-known ``pathological" examples such as Cantor's devil's staircase and Perkins' continuous but nowhere differentiable function. It is well known that $F_a(x)$ is real analytic in $a$ for every $x\in[0,1]$. We introduce the functions \[ M_{k,a}(x):=\frac{\partial^k}{\partial a^k}F_a(x), \qquad k\in\mathbb{N}, \quad x\in[0,1]. \] We compute the box-counting dimension of the graph of $M_{k,a}$, characterize its differentiability, and investigate in detail the set of points where $M_{k,a}$ has an infinite derivative. While some of our results are similar to the known facts about Okamoto's function, there are also some notable differences and surprising new phenomena that arise when considering the higher order partial derivatives of $F_a$.

math.CA↗

arXiv2024

The $D$-Variant of Transfinite Hausdorff Dimension

We assign every metric space $X$ the value $t_{D}HD(X)$, an ordinal number or one of the symbols $-1$ or $\Omega$, and we call it the $D$-variant of transfinite Hausdorff dimension of $X$. This ordinal assignment is primarily constructed by way of the $D$-dimension, a transfinite dimension function consistent with the large inductive dimension on finite dimensional metric spaces while also addressing shortcomings of the large transfinite inductive dimension. Similar to Hausdorff dimension, $t_{D}HD(\cdot)$ is monotone with respect to subspaces, and is a bi-Lipschitz invariant. It is also non-increasing with respect to Lipschitz maps and satisfies a coarse intermediate dimension property. We also show that this new transfinite Hausdorff dimension function addresses the primary goal of transfinite Hausdorff dimension functions; to classify metric spaces with infinite Hausdorff dimension. In particular, we show that if $t_{D}HD\geq \omega_0$, then $HD(X) = \infty$. $t_{D}HD(X)<\omega_1$ for any separable metric space, and that one can find a metrizable space with $t_{D}HD(X)$ bounded between a given ordinal and it's successive cardinal with topological dimension $0$.

math.GN↗

arXiv2024

Multifractal Analysis of F-exponents for Finitely Irreducible Conformal Graph Directed Markov Systems

Let $\Phi = \{\phi_e\}_{e\in E}$ be a finitely irreducible conformal graph directed Markov system (CGDMS) with symbolic representation $E_A^{\infty}$ and limit set $J$. Under a mild condition on the system, we give a multifractal analysis of level sets of Birkhoff averages with respect to Hausdorff dimension for a large family of functions. We then apply these results to a few examples in the case of both $E$ finite and $E$ countably infinite.

math.DS↗

arXiv2021

The partial derivative of Okamoto's functions with respect to the parameter

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.

math.CA↗