Research

Self-similar sets.
Connected questions.

I study how self-similar sets connect, how their topology changes across parameter space, and how this geometry relates to roots of integer polynomials.

The doctoral thesis ↗Download bibliography

Publications & preprints

2013–2026

2026

Explicit interpolations among the Sierpiński, Rauzy, and Apollonian gaskets

Preprint

Bernat EspigulearXiv:2606.23278

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigule2026gaskets,
  author = {Espigule, Bernat},
  title = {Explicit interpolations among the {Sierpi\'nski}, {Rauzy}, and {Apollonian} gaskets},
  year = {2026},
  eprint = {2606.23278},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.23278},
  url = {https://arxiv.org/abs/2606.23278}
}

2026

Infinitely many holes in connectedness loci for collinear affine iterated function systems

Preprint

Bernat EspigulearXiv:2606.01467

arXiv ↗PDF ↗

Cite

@misc{espigule2026holes,
  author = {Espigule, Bernat},
  title = {Infinitely many holes in connectedness loci for collinear affine iterated function systems},
  year = {2026},
  eprint = {2606.01467},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2606.01467},
  url = {https://arxiv.org/abs/2606.01467}
}

2026

Finite capture and the closure of roots of restricted polynomials

Preprint

Bernat Espigule, David JuherarXiv:2603.07397

arXiv ↗PDF ↗Explorer ↗

Cite

@misc{espigulejuher2026capture,
  author = {Espigule, Bernat and Juher, David},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  eprint = {2603.07397},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.2603.07397},
  url = {https://arxiv.org/abs/2603.07397}
}

2024

Collinear Fractals and Bandt's Conjecture

Journal article

Bernat Espigule, David Juher, Joan SaldañaFractal and Fractional 8(12), 725

DOI ↗arXiv ↗PDF ↗

Cite

@article{espigulejuhersaldana2024bandt,
  author = {Espigule, Bernat and Juher, David and Salda{\~n}a, Joan},
  title = {Collinear Fractals and {Bandt}'s Conjecture},
  year = {2024},
  journal = {Fractal and Fractional},
  volume = {8},
  number = {12},
  pages = {725},
  doi = {10.3390/fractalfract8120725},
  url = {https://doi.org/10.3390/fractalfract8120725}
}

2019

Families of connected self-similar sets generated by complex trees

Preprint

Bernat EspigulearXiv:1902.11282

arXiv ↗PDF ↗

Cite

@misc{espigule2019families,
  author = {Espigule, Bernat},
  title = {Families of connected self-similar sets generated by complex trees},
  year = {2019},
  eprint = {1902.11282},
  archivePrefix = {arXiv},
  primaryClass = {math.DS},
  note = {Preprint},
  doi = {10.48550/arXiv.1902.11282},
  url = {https://arxiv.org/abs/1902.11282}
}

2019

A Family of Fern-like Ternary Complex Trees

Proceedings article

Bernat EspiguléBridges 2019, pp. 147–154

Publication ↗PDF ↗

Cite

@inproceedings{espigule2019ferns,
  author = {Espigul{\'e}, Bernat},
  title = {A Family of Fern-like Ternary Complex Trees},
  year = {2019},
  booktitle = {Proceedings of Bridges 2019: Mathematics, Art, Music, Architecture, Education, Culture},
  pages = {147--154},
  publisher = {Tessellations Publishing},
  address = {Phoenix, Arizona},
  editor = {Goldstine, Susan and McKenna, Douglas and Fenyvesi, Krist{\'o}f},
  url = {https://archive.bridgesmathart.org/2019/bridges2019-147.html}
}

2013

Generalized self-contacting symmetric fractal trees

Journal article

Bernat Espigulé PonsSymmetry: Culture and Science 24(1–4), 320–338

Publication ↗PDF ↗

Cite

@article{espigule2013symmetry,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Generalized self-contacting symmetric fractal trees},
  year = {2013},
  journal = {Symmetry: Culture and Science},
  volume = {24},
  number = {1--4},
  pages = {320--338},
  url = {https://journal-scs.symmetry.hu/content-pages/volume-24-numbers-1-4-pages-001-512-2013/}
}

2013

Unfolding Symmetric Fractal Trees

Proceedings article

Bernat Espigulé PonsBridges 2013, pp. 295–302

Publication ↗PDF ↗

Cite

@inproceedings{espigule2013unfolding,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Unfolding Symmetric Fractal Trees},
  year = {2013},
  booktitle = {Proceedings of Bridges 2013: Mathematics, Music, Art, Architecture, Culture},
  pages = {295--302},
  publisher = {Tessellations Publishing},
  address = {Phoenix, Arizona},
  editor = {Hart, George W. and Sarhangi, Reza},
  url = {https://archive.bridgesmathart.org/2013/bridges2013-295.html}
}

Citations use the titles and author credits of the linked public records. LaTeX citations containing \url require the url or hyperref package.

Theses

2026

Collinear Fractals and Connectedness Loci: Topology, Finite Capture, and Restricted Polynomial Roots

Doctoral thesis · Universitat de Girona
Corrected circulation edition, 15 September 2026 · defence pending

Thesis page ↗

Cite

@unpublished{espigule2026thesis,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Collinear Fractals and Connectedness Loci: Topology, Finite Capture, and Restricted Polynomial Roots},
  year = {2026},
  url = {https://complextrees.com/research/thesis/},
  note = {Doctoral thesis, Universitat de Girona. Corrected circulation edition, 15 September 2026. Thesis deposited 9 September 2026; defence pending}
}

2018

Complex trees and their families of connected self-similar sets

Master’s thesis · Universitat de Barcelona
Supervisors: Núria Fagella and Xavier Jarque

Institutional repository ↗

Cite

@mastersthesis{espigule2018master,
  author = {Espigul{\'e} Pons, Bernat},
  title = {Complex trees and their families of connected self-similar sets},
  school = {Universitat de Barcelona},
  year = {2018},
  url = {https://hdl.handle.net/2445/129385}
}

Talks & materials

2026

Finite capture and the closure of roots of restricted polynomials

Recorded talk

Bernat EspiguleInstitut Henri Poincaré · recorded 27 March 2026

DOI ↗

Cite

@misc{espigule2026ihptalk,
  author = {Espigule, Bernat},
  title = {Finite capture and the closure of roots of restricted polynomials},
  year = {2026},
  howpublished = {Institut Henri Poincar{\'e}, audiovisual resource},
  note = {Recorded 27 March 2026},
  doi = {10.57987/IHP.2026.T1.WS3.016},
  url = {https://doi.org/10.57987/IHP.2026.T1.WS3.016}
}

Jun 2026

One World Numeration SeminarResearch seminar presentation

Request slides

2025

ICERM · Illustrating Mathematics Reunion/ExpansionOnline research presentation

Math Cartography ↗

Jun 2024

Collinear fractals · GSD-UABResearch talk, 10 June 2024

Slides PDF ↗

2015

Experiences in 3D Printing FractalsEuropean Wolfram Technology Conference

Watch recording ↗