My current research focus is two-fold: (1) to understand naturally occurring intricate geometric structures of mathematical quasicrystals as well as fractal sets arising in number theory, and (2) to investigate applications of aperiodic order, fractal geometry, and ergodic theory in health and life sciences, battery technologies, as well as in the creation of advanced metamaterials.

Below is a selection of my publications; for pre-prints and links to published articles please visit arXiv and Google Scholar. Now and then, I also write reviews for MathSciNet and zbMATH.

Articles in review

• Daniel Herring, John Thornby, Wuttichai Somyanonthanakun, Sabrina Kombrink, Tony Samuel, Alexander Roberts and Nishal Ramadas. Electrolyte fill-level monitoring using air-coupled ultrasound.
• Richard Howat, Tony Samuel and Ayşe Yıltekin-Karataş. Cut and project schemes in the Poincaré disc: From cocompact Fuchsian groups to chaotic Delone sets. Pre-print arxiv.org/abs/abs/2603.11311.

Books (edited)

• Marc Kesseböhmer, Sabrina Kombrink, Yakov Pesin, Tony Samuel and Jörg Schmeling (2021). Thermodynamic formalism. Stoch. Dyn. 21(3). ISSN 0219-4937.
• Robert Niemeyer, Erin Pearse, John Rock and Tony Samuel (2019). Horizons of fractal geometry and complex dimensions. Contemp. Math. 731. ISBN 978-1-4704-3581-3.
• Kurt Falk, Tobias Oertel-Jäger, Marc Kesseböhmer, Jens Rademacher and Tony Samuel (2017). Diffusion on fractals and non-linear dynamics. Discrete Contin. Dyn. Syst. Ser. S 10(2). ISSN 1937-1632.

Articles in print

• Richard Howat, Andrew Mitchell and Tony Samuel. The matrix potential game and structures of self-affine sets. To appear in Int. Math. Res. Not. (2026).
• Sven van Golden, Sabrina Kombrink and Tony Samuel. On the geometry of fractal snowflakes. To appear in Recreat. Math. Mag. (2026).
• Adarsh Bura, Chris Good and Tony Samuel. The shadowing property for piecewise monotone interval maps. To appear in Amer. Math. Monthly (2026).
• Marc Kesseböhmer, Aljoscha Niemann, Tony Samuel and Hendrik Weyer. Generalised Krein-Feller operators and gap diffusions via transformations of measure spaces. To appear in Appl. Numer. Harmon. Anal. (2026).
• Philipp Gohlke, Andrew Mitchell, Dan Rust and Tony Samuel. Rauzy fractals of random substitutions. Adv. Math. 485 (2026) 110713.
• Daniel Herring, John Thornby, Wuttichai Somyanonthanakun, Sabrina Kombrink, Tony Samuel, Alexander Roberts and Nishal Ramadas. Air-coupled ultrasound spectroscopy for electrolyte fill-level detection in lithium-ion pouch cell batteries. IEEE International Ultrasonics Symposium (IUS), Utrecht, Netherlands, 1–4 (2025).
• Sven van Golden, Charlene Kalle, Sabrina Kombrink and Tony Samuel. Dimensions of infinitely generated self-affine sets and restricted digit sets for signed Lüroth expansions. Nonlinearity 38(4), 045020 (2025).
• Lauren Thomas-Seale, Barnaby Hawthorn, Sabrina Kombrink, Tony Samuel, John Bryson, Harriet Thomson and Thomas Montenegro-Johnson. Topological analysis to enhance the understanding of transdisciplinary engineering. Advances in Transdisciplinary Engineering 60, 309–317 (2024).
• Niels Langeveld and Tony Samuel. Intermediate β-shifts as greedy β-shifts with a hole. Acta Math. Hungar. 170, 269–301 (2023).
• Philipp Gohlke, Andrew Mitchell, Dan Rust and Tony Samuel. Measure theoretic entropy of random substitution subshifts. Ann. Henri Poincaré 24, 277–323 (2023).
• Sophie Abrahams, Simone Appella, Yutong Bai, Nicolas Boulle, Chris Budd, Sam Kamperis, Sabrina Kombrink, Adhiraj Mandal, Ioan Alexandru Puiu, Surajit Ray, Tony Samuel, Babooshka Shavazipour, Ann Smith, Yu Tian, Wenzhong Wang, Huining Yang, Wee Meng Yeo and Tina Zhou. V-KEMS Report – Recovery from the Pandemic: Transport Logistics. Isaac Newton Institute – International Centre for Mathematical Sciences – Knowledge Transfer Network (2022).
• Marc Kesseböhmer, Sabrina Kombrink, Yakov Pesin, Tony Samuel and Jörg Schmeling. Preface: Thermodynamic formalism. Stoch. Dyn. 21(3), i–iv (2021).
• Zoe Cooperband, Erin Pearse, Blaine Quackenbush, Jordan Rowley, Tony Samuel and Matt West. Continuity of entropy of Lorenz maps. Indag. Math. (N.S.) 31(1), 96–105 (2020).
• Marc Kesseböhmer, Tony Samuel and Karenina Sender. The Sierpiński gasket as the Martin boundary of a non–isotropic Markov chain. J. Fractal Geom. 7(2), 113–136 (2020).
• Marc Kesseböhmer, Tony Samuel and Hendrik Weyer. Measure-geometric Laplacians for partially atomic measures. Comment. Math. Univ. Carolin. 61(3), 313–335 (2020).
• Blaine Quackenbush, Tony Samuel and Matt West. Periodic intermediate β-expansions of Pisot numbers. Mathematics 8(903), 1–16 (2020).
• Maik Gröger, Marc Kesseböhmer, Arne Mosbach, Tony Samuel and Malte Steffens. A classification of aperiodic order via spectral metrics and Jarnìk sets. Ergod. Dyn. Sys. 39(11), 3031–3065 (2019).
• Marc Kesseböhmer, Arne Mosbach, Tony Samuel and Malte Steffens. Diffraction of return time measures. J. Stat. Phys. 174, 519–535 (2019).
• Marc Kesseböhmer, Tony Samuel and Hendrik Weyer. Measure–geometric Laplacians for discrete distributions. Contemp. Math. 731, 133–142 (2019).
• Sabrina Kombrink and Tony Samuel. Fractal geometry and dynamics. Lond. Math. Soc. Newsl. 481, 24–29 (2019).
• Bing Li, Tuomas Sahlsten, Tony Samuel and Wolfgang Steiner. Denseness of intermediate β-shifts of finite type. Proc. Amer. Math. Soc. 147, 2045–2055 (2019).
• Robert Niemeyer, Erin Pearse, John Rock and Tony Samuel. Preface: Horizons of fractal geometry and complex dimensions. Contemp. Math. 731, vii-ix (2019).
• Fabian Dreher, Marc Kesseböhmer, Arne Mosbach, Tony Samuel and Malte Steffens. Regularity of aperiodic minimal subshifts. Bull. Math. Sci. 8, 413–434 (2018).
• Kurt Falk, Tobias Oertel-Jäger, Marc Kesseböhmer, Jens Rademacher and Tony Samuel. Preface: Diffusion on fractals and non–linear dynamics. Discrete Contin. Dyn. Syst. Ser. S 10(2), i–iv (2017).
• Johannes Kautzsch, Marc Kesseböhmer and Tony Samuel. On the convergence to equilibrium of unbounded observables under a family of intermittent interval maps. Ann. Henri Poincaré 17, 2585–2621 (2016).
• Marc Kesseböhmer, Tony Samuel and Hendrik Weyer. A note on measure–geometric Laplacians. Monatsh. Math. 181, 643–655 (2016).
• Bing Li, Tuomas Sahlsten and Tony Samuel. Intermediate β-shifts of finite type. Discrete Contin. Dyn. Syst. 36(1), 323–344 (2016).
• Johannes Kautzsch, Marc Kesseböhmer, Tony Samuel and Bernd Stratmann. On the asymptotics of the α–Farey transfer operator. Nonlinearity 28(1), 143–166 (2015).
• Fabian Dreher and Tony Samuel. Continuous images of Cantor’s ternary set. Amer. Math. Monthly 121(7), 640–643 (2014). Translated into Mandarin: Shu Xue Yi Lin 35(4), 381–384 (2016).
• Tony Samuel, Nina Snigireva and Andrew Vince. Embedding the symbolic dynamics of Lorenz maps. Math. Proc. Camb. Phil. Soc. 156(3), 505–519 (2014).
• Marc Kesseböhmer and Tony Samuel. Spectral metric spaces for Gibbs measures. J. Funct. Anal. 265(9), 1801–1828 (2013).
• Tony Samuel. A simple proof of Vitali’s theorem for signed measures. Amer. Math. Monthly 120(7), 654–659 (2013).
• Kenneth Falconer and Tony Samuel. Dixmier traces and coarse multifractal analysis. Ergod. Dyn. Sys. 31(2), 369–381 (2011).

Dissertations

• Tony Samuel. A commutative noncommutative fractal geometry. University of St Andrews. PhD Thesis (2011).
• Tony Samuel. Space filling curves and invariance of domain. University of St Andrews. BSc Thesis (2005).