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arxiv: 2605.13085 · v1 · submitted 2026-05-13 · 🧮 math.DS · math.FA

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· Lean Theorem

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

Junjie Miao, Minghui Xu

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Pith reviewed 2026-05-14 18:46 UTC · model grok-4.3

classification 🧮 math.DS math.FA
keywords reverse iterated function systemsforward orbitsinvariant setsmass dimensionBeurling dimensiondiscrete Hausdorff dimensionrenewal theoryp-adic systems
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The pith

Reverse iterated function systems have explicit dimension formulas for their forward orbits and invariant sets.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper resolves an open problem from 1996 on the dimensions of invariant sets generated by reverse iterated function systems of expanding maps. It shows that non-empty invariant sets are unions of forward orbits and gives formulas for the upper and lower mass dimensions, the Beurling dimension, and the discrete Hausdorff dimension of these orbits and sets. These quantities equal the box-counting and similarity dimensions of the attractor of the dual contractive system obtained by inverting the maps. Renewal theory supplies the precise asymptotic central density of the orbits when they are non-overlapping and uniformly discrete, producing an explicit constant in the non-arithmetic case and a multiplicatively periodic function in the arithmetic case. The same dimension relations hold for the analogous systems over the p-adic numbers.

Core claim

In reverse iterated function systems of expanding maps, the non-empty invariant sets are unions of forward orbits whose upper and lower mass dimensions, Beurling dimension, and discrete Hausdorff dimension equal the box-counting and similarity dimensions of the attractor of the dual contractive iterated function system; renewal theory yields the exact asymptotic central density under the non-overlapping and uniformly discrete assumptions, with the results carrying over to p-adic systems where the mass dimension matches the p-adic box dimension.

What carries the argument

Forward orbits under the expanding maps of the RIFS, connected by dimension equalities to the attractor of the inverse contractive IFS and analyzed with renewal theory for asymptotic densities.

If this is right

  • Upper and lower mass dimensions of forward orbits and invariant sets equal the box dimension of the dual attractor.
  • Beurling dimension and discrete Hausdorff dimension of the orbits are likewise determined by the dual attractor dimensions.
  • In non-arithmetic cases the central density converges to an explicit computable constant.
  • In arithmetic cases the density approaches a multiplicatively periodic function.
  • In p-adic systems the mass dimension of a forward orbit equals the p-adic box dimension of the corresponding attractor.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The reduction to dual contractive systems may allow reuse of existing dimension results for contractive IFS when studying expanding ones.
  • The density formulas could be tested numerically on concrete expanding maps with known orbits to verify convergence rates.
  • Similar orbit-union and dimension-link arguments might extend to other locally compact metric spaces beyond the reals and p-adics.

Load-bearing premise

The forward orbits must be non-overlapping and uniformly discrete to obtain the precise asymptotic central density via renewal theory.

What would settle it

A specific non-overlapping uniformly discrete RIFS on the reals in which the mass dimension of a forward orbit differs from the box-counting dimension of the attractor of the inverse system.

read the original abstract

In 1996, Strichartz introduced reverse iterated function systems (RIFS) $\mathcal{F}=\{f_i(x)=r_i x+b_i\}_{i=1}^m$ of expanding mappings on $\mathbb{Z}$ and left the determination of the general dimension formulas of invariant sets as an open problem. In this paper we study the topological and geometric properties as well as the dimensions of the forward orbits generated by such systems, thereby providing a complete solution. We first work in a general locally compact complete metric space to show that the non-empty invariant sets of $\mathcal{F}$ are unions of forward orbits, along with giving necessary and sufficient conditions for their existence. Specialising to the RIFS $\mathcal{F}$ on $\mathbb{R}$, we determine the upper and lower mass dimensions, the Beurling dimension, and the discrete Hausdorff dimension of its forward orbits and invariant sets. Moreover, we establish a fundamental connection with the box-counting and similarity dimensions of the attractor generated by the dual contractive IFS $\mathcal{F}^{-1}=\{f_i^{-1}(x)=r_i^{-1}(x-b_i)\}_{i=1}^m$. Under the assumptions that the orbit is non-overlapping and uniformly discrete, renewal theory yields the precise asymptotic central density: in the non-arithmetic case it converges to an explicitly computable constant, while in the arithmetic case it approaches a multiplicatively periodic function. Finally, an analogous treatment is given for $p$-adic systems, where the mass dimension of a forward orbit equals the $p$-adic box dimension of the corresponding $p$-adic attractor.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper addresses Strichartz's 1996 open problem on dimension formulas for invariant sets of reverse iterated function systems (RIFS) F = {f_i(x) = r_i x + b_i} on Z. In general locally compact complete metric spaces it shows non-empty invariant sets are unions of forward orbits and gives existence conditions. Specializing to R, it computes upper/lower mass dimensions, Beurling dimension, and discrete Hausdorff dimension of forward orbits and invariant sets, and links these to the box-counting and similarity dimensions of the attractor of the dual contractive IFS F^{-1}. Under the assumptions that orbits are non-overlapping and uniformly discrete, renewal theory supplies explicit asymptotic central density (constant in non-arithmetic case, multiplicatively periodic in arithmetic case). An analogous treatment for p-adic RIFS equates the mass dimension of a forward orbit to the p-adic box dimension of the corresponding attractor.

Significance. If the non-overlapping and uniformly discrete assumptions can be verified or the results suitably restricted, the work would resolve an open problem by supplying explicit dimension formulas for RIFS invariant sets and forward orbits together with a direct link to the dual contractive IFS. The use of renewal theory for precise asymptotics and the p-adic extension constitute genuine contributions to the geometric theory of expanding maps.

major comments (3)
  1. [Abstract] Abstract: the claim of a 'complete solution to the determination of the general dimension formulas of invariant sets' is not supported, because the precise asymptotic central density (hence the exact mass and Beurling dimensions) is obtained from renewal theory only 'under the assumptions that the orbit is non-overlapping and uniformly discrete.' No argument is supplied showing these properties hold for arbitrary r_i > 1 and b_i in R or Z.
  2. [Renewal theory application (RIFS on R)] The section applying renewal theory to obtain the asymptotic central density: the explicit constant (non-arithmetic case) and multiplicatively periodic function (arithmetic case) are derived only after invoking the non-overlapping and uniformly discrete hypotheses; without a proof that these hold in general, the dimension formulas remain conditional rather than general.
  3. [Connection to dual IFS] The paragraph establishing the connection between RIFS dimensions and box/similarity dimensions of the dual IFS attractor: the stated equivalence inherits the same non-overlapping/uniformly-discrete restriction, so the claimed fundamental connection is limited in scope; the manuscript does not clarify for which parameter choices the link is unconditional.
minor comments (2)
  1. [Introduction / definitions] Notation for the various dimensions (mass, Beurling, discrete Hausdorff) should be introduced with explicit definitions or references to standard sources before their first use.
  2. [p-adic extension] The p-adic section would benefit from a brief comparison table or statement highlighting which results carry over verbatim from the real case and which require new arguments.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments, which help clarify the scope of our results. We address each major comment below and will revise the manuscript accordingly to ensure the conditional nature of the dimension formulas is unambiguous.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claim of a 'complete solution to the determination of the general dimension formulas of invariant sets' is not supported, because the precise asymptotic central density (hence the exact mass and Beurling dimensions) is obtained from renewal theory only 'under the assumptions that the orbit is non-overlapping and uniformly discrete.' No argument is supplied showing these properties hold for arbitrary r_i > 1 and b_i in R or Z.

    Authors: We acknowledge that the abstract's reference to a 'complete solution' risks overstating the unconditional generality of the results. Although the assumptions are stated explicitly when renewal theory is applied in the body of the paper, we agree the abstract should be revised to qualify this claim. We will modify the abstract to state that the dimension formulas and asymptotic densities are obtained under the non-overlapping and uniformly discrete assumptions. The manuscript does not contain a general proof that these properties hold for arbitrary parameters, and the revision will reflect this scope. revision: yes

  2. Referee: [Renewal theory application (RIFS on R)] The section applying renewal theory to obtain the asymptotic central density: the explicit constant (non-arithmetic case) and multiplicatively periodic function (arithmetic case) are derived only after invoking the non-overlapping and uniformly discrete hypotheses; without a proof that these hold in general, the dimension formulas remain conditional rather than general.

    Authors: The relevant section already states the hypotheses before deriving the asymptotic densities via renewal theory. To address the concern directly, we will insert a short clarifying paragraph at the beginning of the section emphasizing that the explicit formulas are conditional on these properties. This revision will prevent any impression that the formulas are unconditional. revision: yes

  3. Referee: [Connection to dual IFS] The paragraph establishing the connection between RIFS dimensions and box/similarity dimensions of the dual IFS attractor: the stated equivalence inherits the same non-overlapping/uniformly-discrete restriction, so the claimed fundamental connection is limited in scope; the manuscript does not clarify for which parameter choices the link is unconditional.

    Authors: We will revise the paragraph on the connection to the dual IFS to state explicitly that the equivalence of dimensions holds under the non-overlapping and uniformly discrete assumptions. The revision will also note that the link applies whenever these hypotheses are satisfied, without claiming it is unconditional for all parameter choices. revision: yes

Circularity Check

0 steps flagged

No significant circularity; dimensions derived from external renewal theory and dual IFS connection

full rationale

The paper derives mass/Beurling/discrete Hausdorff dimensions of RIFS forward orbits and invariant sets by connecting them to box-counting and similarity dimensions of the dual contractive IFS attractor, using standard IFS duality. Asymptotics for central density are obtained by applying renewal theory under the explicitly stated assumptions that the orbit is non-overlapping and uniformly discrete; the resulting explicit constant (non-arithmetic) or periodic function (arithmetic) is not fitted to the dimension formulas but follows from the expansion ratios. The p-adic case is treated analogously. No self-definitional reductions, fitted-input predictions, or load-bearing self-citations appear in the derivation chain; the central claims remain independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on standard background from metric space theory and renewal theory without introducing new free parameters or postulated entities; all constants arise from the given expansion ratios and the dual contractive system.

axioms (2)
  • standard math Locally compact complete metric spaces admit non-empty invariant sets under expanding maps that are unions of forward orbits
    Invoked in the general setting to establish existence and structure of invariant sets.
  • standard math Renewal theory applies to the counting function of non-overlapping uniformly discrete orbits
    Used to derive asymptotic central density in arithmetic and non-arithmetic cases.

pith-pipeline@v0.9.0 · 5596 in / 1440 out tokens · 47993 ms · 2026-05-14T18:46:45.405907+00:00 · methodology

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