Pith Number

pith:2HHYKOM3

pith:2026:2HHYKOM3SGOX6SPMVE3WBC5BIP

not attested not anchored not stored refs resolved

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

Junjie Miao, Minghui Xu

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

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

Open paper page JSON Open Graph Bundle Merged state Verified badge What is a Pith Number?

Add to your LaTeX paper

\usepackage{pith}
\pithnumber{2HHYKOM3SGOX6SPMVE3WBC5BIP}

Prints a linked badge after your title and injects PDF metadata. Compiles on arXiv. Learn more · Embed verified badge

Record completeness

1 Bitcoin timestamp

2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

✓ Portable graph bundle live · download bundle · merged state

The bundle contains the canonical record plus signed events. A mirror can host it anywhere and recompute the same current state with the deterministic merge algorithm.

Claims

C1strongest claim

We provide a complete solution to the determination of the general dimension formulas of invariant sets for reverse iterated function systems, determining the upper and lower mass dimensions, the Beurling dimension, and the discrete Hausdorff dimension of its forward orbits and invariant sets.

C2weakest assumption

The orbit is non-overlapping and uniformly discrete when applying renewal theory to obtain the precise asymptotic central density (explicit constant in non-arithmetic case, multiplicatively periodic in arithmetic case).

C3one line summary

Reverse IFS invariant sets are unions of forward orbits whose dimensions equal those of the dual contractive attractor, with explicit asymptotic densities from renewal theory in non-arithmetic and arithmetic cases, and matching p-adic box dimensions.

References

48 extracted · 48 resolved · 0 Pith anchors

[1] S. Akiyama and V. Komornik. Discrete spectra and Pisot numbers.J. Number Theory133(2013), 375–390 2013

[2] M. R. Allen, G. S. H. Cruttwell, K. E. Hare, and J.-O. R¨ onning. Dimensions of fractals in the large.Chaos Solitons Fractals31(2007), 5–13 2007

[3] Asmussen.Applied Probability and Queues 2019

[4] A. Baker. Linear forms in the logarithms of algebraic numbers.Mathematika13(1966), 204–216 1966

[5] B´ ar´ any, M 2019

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-18T03:08:58.562983Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

d1cf85399b919d7f49eca937608ba143edc643b619456e9fdfb21e8c46e9f310

Aliases

arxiv: 2605.13085 · arxiv_version: 2605.13085v1 · doi: 10.48550/arxiv.2605.13085 · pith_short_12: 2HHYKOM3SGOX · pith_short_16: 2HHYKOM3SGOX6SPM · pith_short_8: 2HHYKOM3

Agent API

Resolver JSON Graph JSON Events JSON Schema Signing key

Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/2HHYKOM3SGOX6SPMVE3WBC5BIP \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: d1cf85399b919d7f49eca937608ba143edc643b619456e9fdfb21e8c46e9f310

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "7f572168991b139782d76491ce8da00a4442afc174dcae08061320a82177c9df",
    "cross_cats_sorted": [
      "math.FA"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-05-13T06:54:59Z",
    "title_canon_sha256": "3fd8415e0a376ba89f29784afd5995879a4b3a32a03cf29e57dbe50b76de0861"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.13085",
    "kind": "arxiv",
    "version": 1
  }
}