Statistical and geometric properties of tree-shifts and fractals: a dynamical perspective
Thesis event information
Date and time of the thesis defence
Place of the thesis defence
L10, Linnanmaa
Topic of the dissertation
Statistical and geometric properties of tree-shifts and fractals: a dynamical perspective
Doctoral candidate
Master of Science Yu-Liang Wu
Faculty and unit
University of Oulu Graduate School, Faculty of Science, Research Unit of Mathematical Sciences
Subject of study
Mathematics
Opponent
Associate Professor Tuomas Sahlsten, University of Helsinki
Custos
Professor Ville Suomala, University of Oulu
The dynamic and random structures of fractals
Dynamical systems are ubiquitous in our lives, from the evolution of the universe to changes in weather and climate. Curiosity toward these systems is naturally driven by their practical importance: the need to predict changes in the short or long term has fostered the development of a wide range of scientific studies. One branch of mathematics that embodies this pursuit is fractal geometry. Fractals, as sets with intricate structures at all scales, inherently carry the dynamics of magnification: when observed more closely, they reveal new details that resemble the whole. The statistical patterns gathered through such magnification often uncover the hidden rules governing these complex structures.
In my doctoral research, I applied methods from dynamical systems to study the statistical and geometric properties of fractals. My aim was to investigate what these repeating structures can tell us about size, dimension, and regularity, and how they reveal deeper connections between geometry, probability, and dynamics.
The dissertation makes four principal contributions. First, it develops novel approaches to describe tree-lattice models in statistical physics and uncovers an unusual phenomenon absent in conventional grid-based models. Second, it quantifies recurrence in certain fractal sets—that is, how often points return close to their starting position under the system’s dynamics. Third, it proves that in some families of fractals, the numbers generated are “regular,” behaving much like well-distributed random numbers. Finally, it shows that for a broad class of fractal measures, the local statistics observed under magnification are independent of the chosen location, revealing a form of universal behavior.
Taken together, these results deepen our understanding of how complexity arises from simple rules. They demonstrate the power of dynamical systems to uncover order within apparent irregularity and offer new perspectives for the study of fractals, probability, and mathematical physics.
In my doctoral research, I applied methods from dynamical systems to study the statistical and geometric properties of fractals. My aim was to investigate what these repeating structures can tell us about size, dimension, and regularity, and how they reveal deeper connections between geometry, probability, and dynamics.
The dissertation makes four principal contributions. First, it develops novel approaches to describe tree-lattice models in statistical physics and uncovers an unusual phenomenon absent in conventional grid-based models. Second, it quantifies recurrence in certain fractal sets—that is, how often points return close to their starting position under the system’s dynamics. Third, it proves that in some families of fractals, the numbers generated are “regular,” behaving much like well-distributed random numbers. Finally, it shows that for a broad class of fractal measures, the local statistics observed under magnification are independent of the chosen location, revealing a form of universal behavior.
Taken together, these results deepen our understanding of how complexity arises from simple rules. They demonstrate the power of dynamical systems to uncover order within apparent irregularity and offer new perspectives for the study of fractals, probability, and mathematical physics.
Last updated: 6.10.2025