Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.34076

Hausdorff and Packing Measures for $β$-L{ü}roth Infinite Iterated Function Systems

Junjue Miao·Tianrui Wang
View original source ↗Read source document ↗Export metadata ↓

Abstract

We study the Hausdorff and packing measures of attractors generated by infinite $β$-Lüroth iterated function systems. Let $I\subset\mathbb N$ be infinite and let $\dim_H E_I=h$. For a regular system, we first give conditions for the positivity of $\mathcal H^h(E_I)$ and the finiteness of $\mathcal P^h(E_I)$ in terms of first-level Lüroth intervals. We then introduce tail-regular partitions and the scale $R_n=\frac{B_n^\infty}{B_n}.$ Under tail-regularity, the Hausdorff and packing conditions reduce to counting conditions for $I$ on intervals of length comparable to $R_n$, together with a tail condition in the packing case. Finally, we apply these results to regularly varying partitions and stretched-exponential partitions.

Explore related subjects

math.DS

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Junjue Miao, Tianrui Wang. 2026-10-25. Hausdorff and Packing Measures for $β$-L{ü}roth Infinite Iterated Function Systems. https://arxiv.org/abs/2610.34076

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Topological Pressure of Discontinuous Potentials and Variational Principle for Flows

Let $X$ be a compact metric space and $Φ=\{φ_t\}_{t\in\mathbb{R}}$ be a continuous flow on $X$. We introduce two types of topological pressure for family of discontinuous potentials $a=\{a_t\}_{t>0}$. First, define the topological pressure of family of measurable potentials $a=\{a_t\}_{t>0}$ on a subset $Z$ for flow and proof its invariant principle. The second topological pressure is defined on a invariant subset having a nested family of subsets, we also proof its invariant principle.

math.DS↗
arXiv2026

The Radon--Nikodym topography of acyclic measured graphs

The classical Adams dichotomy [Ada90] characterizes amenability of probability-measure-preserving treed equivalence relations in terms of the ends of a given treeing. We extend this dichotomy to the measure-class-preserving (mcp) setting by analyzing the Radon--Nikodym topography of an acyclic mcp Borel graph, i.e., the interaction between the graph geometry and the associated Radon--Nikodym cocycle. In the general mcp setting one must count not all ends, but only the nonvanishing ends: those ends which have positive upper cocycle-weight. More precisely, we prove that a locally countable acyclic mcp graph is amenable if and only if a.e. component has at most two nonvanishing ends, while it is nowhere amenable exactly when a.e. component has a nonempty perfect closed set of nonvanishing ends. At the heart of our analysis lies the study of acyclic countable-to-one Borel functions. Our critical result is that, outside of the essentially two-ended setting, all back ends in a.e. orbit are vanishing and have cocycle-finite geodesics. We also give several characterizations, including topographical and cocycle-free, of essential one-endedness of such functions, and show that the number of barytropic ends controls the essential number of ends. Our proofs utilize mass transport, end selection, and the notion of the Radon--Nikodym core for acyclic mcp graphs, a new concept that serves as a guiding framework for our topographical analysis.

math.DS↗
arXiv2026

The extended future cover and desynchronization of sofic shifts

The paper describes a cover of the future cover of a sofic shift which is canonical in the same way as the future cover itself. In some cases the cover is isomorphic to the future cover and in other it is a genuine extension. The extended future cover is then used to introduce and describe the desynchronization of sofic shifts.

math.DS↗