Pith Number

pith:IPAQLX7T

pith:2026:IPAQLX7TC4DO3TA5RX2DXWDANC

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Complexity of Billiards in Polygons Associated to Hyperbolic $(p,q)$-Tilings

Jane Wang, Sunrose T. Shrestha

Billiard languages in hyperbolic (p,q)-polygons have explicit exponential growth rates for even q and complete grammar rules for realizable words.

arxiv:2605.14030 v1 · 2026-05-13 · math.DS · math.GT

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\pithnumber{IPAQLX7TC4DO3TA5RX2DXWDANC}

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Record completeness

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2 Internet Archive

3 Author claim open · sign in to claim

4 Citations open

5 Replications open

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Claims

C1strongest claim

we compute these exponential growth rates explicitly when q is even and give bounds when q is odd. Additionally, for the q even case, we give complete grammar rules that establish when a word (finite, infinite or bi-infinite) in p letters is realized by a billiard path.

C2weakest assumption

That the new methods relating to minimal tiling paths suffice to give a complete, rigorous characterization of realizable words, and that the previously known equality between language growth rate and topological entropy continues to hold for these specific polygons.

C3one line summary

Explicit growth rates for billiard word complexity in hyperbolic (p,q)-polygons for even q, plus complete grammar rules for realizable paths using minimal tiling paths.

References

92 extracted · 92 resolved · 2 Pith anchors

[1] Katok, A. , TITLE =. Comm. Math. Phys. , FJOURNAL =. 1987 , NUMBER = 1987

[2] Nagar, Anima and Singh, Pradeep , TITLE =. J. Aust. Math. Soc. , FJOURNAL =. 2024 , NUMBER =. doi:10.1017/S1446788723000174 , URL = 2024 · doi:10.1017/s1446788723000174

[3] Troubetzkoy, S. , TITLE =. Chaos , FJOURNAL =. 1998 , NUMBER =. doi:10.1063/1.166301 , URL = 1998 · doi:10.1063/1.166301

[4] Davis, Diana , TITLE =. New York J. Math. , FJOURNAL =. 2014 , PAGES = 2014

[5] Smillie, John and Ulcigrai, Corinna , TITLE =. Proc. Lond. Math. Soc. (3) , FJOURNAL =. 2011 , NUMBER =. doi:10.1112/plms/pdq018 , URL = 2011 · doi:10.1112/plms/pdq018

Formal links

2 machine-checked theorem links

Receipt and verification

First computed	2026-05-17T23:39:12.855019Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

43c105dff31706edcc1d8df43bd86068b28b7b40740cb9f4ea5326a56d8f6898

Aliases

arxiv: 2605.14030 · arxiv_version: 2605.14030v1 · doi: 10.48550/arxiv.2605.14030 · pith_short_12: IPAQLX7TC4DO · pith_short_16: IPAQLX7TC4DO3TA5 · pith_short_8: IPAQLX7T

Agent API

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/IPAQLX7TC4DO3TA5RX2DXWDANC \
  | 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: 43c105dff31706edcc1d8df43bd86068b28b7b40740cb9f4ea5326a56d8f6898

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "63e144a2cf2b6e266b9d39f7578b9a26a9e21e52bc6fc6666052efcc672c0b4d",
    "cross_cats_sorted": [
      "math.GT"
    ],
    "license": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
    "primary_cat": "math.DS",
    "submitted_at": "2026-05-13T18:42:31Z",
    "title_canon_sha256": "cc86a128c343ee4d001e143b12a237aaf857c47900d16d3c97acadeaf81fffa0"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.14030",
    "kind": "arxiv",
    "version": 1
  }
}