Periodic Reporting for period 1 - CODAG (Complexity in Dynamical systems, Algebra, and Geometry)
Okres sprawozdawczy: 2023-09-01 do 2025-08-31
In the last decades, the field of complex dynamical systems has established itself not merely as a stand-alone subject, but as a source of deeply influential examples and paradigms for many other areas of mathematics, including topology, conformal geometry, and geometric group theory. Even more, the tools and methods developed within this field have provided new approaches to long-standing problems in other fields, such as statistical physics.
One of the fundamental objects of study in all these areas is that of fractal spaces, i.e. spaces exhibiting an infinitely complex self-similar structure (a property also observed in natural phenomena such as crystals, coastlines, or proteins). In the mathematical settings, such fractal spaces are often encoded or approximated by combinatorial data (such as finite graphs). One of the central research questions is then how the global and local structure of these spaces depends on this combinatorial data, and conversely, how this data influences the properties of the associated dynamical systems, groups, and related objects. In this vein, the overall goal of the project was to study the intricate relations between different measures of complexity of dynamical systems, fractal sets, groups, and graphs. The project was organized around three complementary research directions, each strengthening a conceptual bridge between dynamical systems and another mathematical domain.
A. Decomposition and approximation theory
Two natural ways to study an object with a complicated structure are decomposition and approximation. These strategies have proven very fruitful in complex dynamics and geometric group theory. In this subproject, we bridge these two subjects by developing a parallel decomposition theory and a unified approximation framework, thereby providing a toolbox for subsequent studies of more intricate properties of the underlying dynamical systems and spaces.
B. Iterated monodromy groups (IMGs)
The theory of IMGs forms a prominent bridge between dynamical systems and geometric group theory. IMGs are groups with an intrinsic self-similar structure naturally associated with specific dynamical systems. Over the past two decades, IMGs have attracted significant attention, as they provide an algebraic, and computationally efficient tool for addressing dynamical problems, while also yielding examples of groups that exhibit “exotic” behavior from the perspective of classical group theory. In this subproject, we systematically studied the algebraic properties of the IMGs associated with particular classes of polynomial maps. We particularly focused on characterizing the structure of these groups in terms of the dynamical properties of the corresponding maps.
C. Partition functions and spectral graph theory
The study of partition functions in mathematics is inspired by their fundamental role in statistical physics, where they encode the equilibrium properties of physical systems. In this subproject, we focused specifically on the independence polynomial—a partition function modeling “hard-core” gases, in which each particle occupies an exclusive region of space. The location of the complex roots of this polynomial is intimately connected to phase transitions in the underlying system. In a similar spirit, the study of the Laplacian spectrum and spectral measures is related to diffusion processes in physics and plays a central role in geometry and probability theory. In this subproject, we employed methods from (multidimensional) complex dynamics to investigate the structure of the zero-loci of independence and Laplacian polynomials for recursive graph sequences converging to fractal spaces.
One of the fundamental objects of study in all these areas is that of fractal spaces, i.e. spaces exhibiting an infinitely complex self-similar structure (a property also observed in natural phenomena such as crystals, coastlines, or proteins). In the mathematical settings, such fractal spaces are often encoded or approximated by combinatorial data (such as finite graphs). One of the central research questions is then how the global and local structure of these spaces depends on this combinatorial data, and conversely, how this data influences the properties of the associated dynamical systems, groups, and related objects. In this vein, the overall goal of the project was to study the intricate relations between different measures of complexity of dynamical systems, fractal sets, groups, and graphs. The project was organized around three complementary research directions, each strengthening a conceptual bridge between dynamical systems and another mathematical domain.
A. Decomposition and approximation theory
Two natural ways to study an object with a complicated structure are decomposition and approximation. These strategies have proven very fruitful in complex dynamics and geometric group theory. In this subproject, we bridge these two subjects by developing a parallel decomposition theory and a unified approximation framework, thereby providing a toolbox for subsequent studies of more intricate properties of the underlying dynamical systems and spaces.
B. Iterated monodromy groups (IMGs)
The theory of IMGs forms a prominent bridge between dynamical systems and geometric group theory. IMGs are groups with an intrinsic self-similar structure naturally associated with specific dynamical systems. Over the past two decades, IMGs have attracted significant attention, as they provide an algebraic, and computationally efficient tool for addressing dynamical problems, while also yielding examples of groups that exhibit “exotic” behavior from the perspective of classical group theory. In this subproject, we systematically studied the algebraic properties of the IMGs associated with particular classes of polynomial maps. We particularly focused on characterizing the structure of these groups in terms of the dynamical properties of the corresponding maps.
C. Partition functions and spectral graph theory
The study of partition functions in mathematics is inspired by their fundamental role in statistical physics, where they encode the equilibrium properties of physical systems. In this subproject, we focused specifically on the independence polynomial—a partition function modeling “hard-core” gases, in which each particle occupies an exclusive region of space. The location of the complex roots of this polynomial is intimately connected to phase transitions in the underlying system. In a similar spirit, the study of the Laplacian spectrum and spectral measures is related to diffusion processes in physics and plays a central role in geometry and probability theory. In this subproject, we employed methods from (multidimensional) complex dynamics to investigate the structure of the zero-loci of independence and Laplacian polynomials for recursive graph sequences converging to fractal spaces.
The PI carried out research across all three subprojects, achieving the following main results:
A. Decomposition and approximation theory
A1. Developed a decomposition theory for critically fixed branched covers of the 2-sphere (joint with L. Geyer; preprint available), and a corresponding parallel decomposition theory for special pared 3-manifolds (joint with L. Geyer, R. Lodge, Y. Luo, and S. Mukherjee; manuscript in preparation).
A2. Developed a framework of “quasi-visual approximations” for metric spaces, in connection with the theory of Gromov hyperbolic spaces. Such approximations were obtained for the Julia sets of semihyperbolic rational maps (joint with M. Bonk and D. Meyer; manuscript in preparation).
A3. Using decomposition ideas, developed a novel tiling approach for addressing the Global Curve Attractor Conjecture in complex dynamics (joint with M. Bonk and R. Lodge; preprint available; to appear in the Proceedings of the AMS).
B. Iterated monodromy groups (IMGs)
B1. Characterized the structure of the (profinite geometric) IMGs associated with bicritical cubic polynomials satisfying a natural non-degeneracy condition on their postcritical points (joint with O. Lukina and D. Wardell; preprint available). The main result demonstrates that these groups are determined by the isomorphism class of the ramification portrait of the corresponding polynomial. Further results were obtained on branch and torsion properties of these groups.
B2. Characterized the structure of the (profinite geometric) IMGs associated with critically fixed polynomials and anti-polynomials (joint with F. van Berkel; manuscript in preparation). It was shown that these groups are always regular branch over a specific characteristic subgroup.
C. Partition functions and spectral graph theory
C1. Developed a unified framework for studying the zero-loci of the independence polynomials for recursive graph sequences (joint with H. Peters). The main results show that, under natural conditions on the recursion, the zeroes are uniformly bounded (preprint available) and avoid a neighborhood of the positive real axis (manuscript in preparation). The latter implies that no phase transitions occur for the hard-core model on these graph sequences.
C2. Computed the Laplacian spectra for the Schreier graph sequences induced by the IMGs of specific critically fixed rational maps whose Julia sets exhibit rotational symmetry (manuscript in preparation).
The results of the project have been disseminated and promoted internationally through numerous conferences, workshops, and seminars.
During the fellowship, the PI has significantly broadened his research landscape, establishing new connections with such fields as Kleinian and braid groups (Subproject A), arithmetic dynamics and Galois theory (Subproject B), and statistical mechanics and algebraic geometry (Subproject C). Besides new collaborations within the project’s main topics, new research projects were initiated with graph theorists (F. Bencs, P. Buys, and J. van den Heuvel).
The PI actively contributed to the training of early-stage researchers by supervising one MSc and one BSc student on topics within Subprojects B and C. The MSc thesis results were presented at an international workshop and are currently being prepared for publication. In addition, the PI continued to co-supervise two PhD students who successfully completed their doctoral studies during the fellowship period.
The PI co-designed and taught one local Master’s class at the University of Amsterdam (in complex dynamical systems), as well as a nationwide Dutch Master’s class (on Dynamical Systems and Group Theory). The PI also participated as a lecturer at the Summer School on Teichmüller Theory, held at the University of Liverpool, which brought together the researchers from dynamical systems, geometry, and probability theory.
Finally, the PI continued to co-organize the “Quasiworld Seminar”—an international virtual seminar focused on quasiconformal geometry, complex dynamics, and analysis on metric spaces. Each session attracts an international audience of 50–100 participants (including additional views via the seminar’s YouTube channel), thereby fostering global collaboration and knowledge exchange.
A. Decomposition and approximation theory
A1. Developed a decomposition theory for critically fixed branched covers of the 2-sphere (joint with L. Geyer; preprint available), and a corresponding parallel decomposition theory for special pared 3-manifolds (joint with L. Geyer, R. Lodge, Y. Luo, and S. Mukherjee; manuscript in preparation).
A2. Developed a framework of “quasi-visual approximations” for metric spaces, in connection with the theory of Gromov hyperbolic spaces. Such approximations were obtained for the Julia sets of semihyperbolic rational maps (joint with M. Bonk and D. Meyer; manuscript in preparation).
A3. Using decomposition ideas, developed a novel tiling approach for addressing the Global Curve Attractor Conjecture in complex dynamics (joint with M. Bonk and R. Lodge; preprint available; to appear in the Proceedings of the AMS).
B. Iterated monodromy groups (IMGs)
B1. Characterized the structure of the (profinite geometric) IMGs associated with bicritical cubic polynomials satisfying a natural non-degeneracy condition on their postcritical points (joint with O. Lukina and D. Wardell; preprint available). The main result demonstrates that these groups are determined by the isomorphism class of the ramification portrait of the corresponding polynomial. Further results were obtained on branch and torsion properties of these groups.
B2. Characterized the structure of the (profinite geometric) IMGs associated with critically fixed polynomials and anti-polynomials (joint with F. van Berkel; manuscript in preparation). It was shown that these groups are always regular branch over a specific characteristic subgroup.
C. Partition functions and spectral graph theory
C1. Developed a unified framework for studying the zero-loci of the independence polynomials for recursive graph sequences (joint with H. Peters). The main results show that, under natural conditions on the recursion, the zeroes are uniformly bounded (preprint available) and avoid a neighborhood of the positive real axis (manuscript in preparation). The latter implies that no phase transitions occur for the hard-core model on these graph sequences.
C2. Computed the Laplacian spectra for the Schreier graph sequences induced by the IMGs of specific critically fixed rational maps whose Julia sets exhibit rotational symmetry (manuscript in preparation).
The results of the project have been disseminated and promoted internationally through numerous conferences, workshops, and seminars.
During the fellowship, the PI has significantly broadened his research landscape, establishing new connections with such fields as Kleinian and braid groups (Subproject A), arithmetic dynamics and Galois theory (Subproject B), and statistical mechanics and algebraic geometry (Subproject C). Besides new collaborations within the project’s main topics, new research projects were initiated with graph theorists (F. Bencs, P. Buys, and J. van den Heuvel).
The PI actively contributed to the training of early-stage researchers by supervising one MSc and one BSc student on topics within Subprojects B and C. The MSc thesis results were presented at an international workshop and are currently being prepared for publication. In addition, the PI continued to co-supervise two PhD students who successfully completed their doctoral studies during the fellowship period.
The PI co-designed and taught one local Master’s class at the University of Amsterdam (in complex dynamical systems), as well as a nationwide Dutch Master’s class (on Dynamical Systems and Group Theory). The PI also participated as a lecturer at the Summer School on Teichmüller Theory, held at the University of Liverpool, which brought together the researchers from dynamical systems, geometry, and probability theory.
Finally, the PI continued to co-organize the “Quasiworld Seminar”—an international virtual seminar focused on quasiconformal geometry, complex dynamics, and analysis on metric spaces. Each session attracts an international audience of 50–100 participants (including additional views via the seminar’s YouTube channel), thereby fostering global collaboration and knowledge exchange.
A. The developed parallel decomposition theory for critically fixed branched covers and Schottky pared 3-manifolds provides a novel entry for "Sullivan's dictionary"—the conceptual framework linking the dynamics of rational maps and Kleinian groups. It opens the possibility of establishing further parallelism between the structures of the corresponding deformation spaces. In addition, this subproject introduced the theory of "quasi-visual approximations", which provides a unified framework for the geometric study of fractal spaces arising in diverse settings such as complex dynamics, geometric group theory, and iterated function systems. We aim to combine this framework with decomposition theory to investigate the conformal dimension of Julia sets of expanding dynamical systems.
B. While the structure of the (profinite geometric) IMGs of unicritical polynomials is now well understood, the case of general rational maps remains largely unexplored. The results obtained in this subproject provide the first substantial evidence supporting the conjecture that the geometric profinite IMGs should be completely determined by simple finite combinatorial data—the ramification portrait—associated with the corresponding maps.
C. The (multidimensional) dynamical framework developed in this subproject for studying the hard-core model on recursive graph sequences is expected be transferable to other models and partition functions. While the present work focused on identifying conditions on the recursion that guarantee the absence of phase transitions, the developed tools also provide a foundation for analysis in the reverse setting, although more advanced dynamical and analytical techniques are required.
B. While the structure of the (profinite geometric) IMGs of unicritical polynomials is now well understood, the case of general rational maps remains largely unexplored. The results obtained in this subproject provide the first substantial evidence supporting the conjecture that the geometric profinite IMGs should be completely determined by simple finite combinatorial data—the ramification portrait—associated with the corresponding maps.
C. The (multidimensional) dynamical framework developed in this subproject for studying the hard-core model on recursive graph sequences is expected be transferable to other models and partition functions. While the present work focused on identifying conditions on the recursion that guarantee the absence of phase transitions, the developed tools also provide a foundation for analysis in the reverse setting, although more advanced dynamical and analytical techniques are required.