@phdthesis{22255, abstract = {This thesis studies spectral rigidity and nonrigidity phenomena in dynamical systems. The central question is whether a dynamical system can be determined, up to a natural conjugacy, from its spectrum. We consider three related spectra: the length spectrum, the action spectrum, and the Lyapunov spectrum. The first part of the thesis concerns Liouville metrics on the two-dimensional torus. It is a long-standing folklore conjecture that Liouville metrics are the only integrable metrics on the torus. We prove a length-spectral rigidity result for linear conformal deformations of Liouville metrics by exploiting the dynamical properties of the rational tori -- analogues of the resonant convex caustics in billiards. We also establish a complementary classification result showing that marked-length-isospectral Liouville metrics are characterized by rearrangements of the one-dimensional functions appearing in their conformal factors, generalizing a theorem of Abbondandolo-Mazzucchelli. In particular, the second result gives nonrigidity examples within the class of Liouville metrics. The second part of the thesis studies the standard map from the viewpoint of action and Lyapunov spectra. We construct nontrivial deformations of the standard map which preserve the symplectic actions (respectively, the Lyapunov exponents) of infinitely many periodic orbits accumulating on an invariant curve. The proof combines a resonant normal form construction with Picard iteration schemes to obtain a sequence of periodic orbits accumulating on an invariant curve with a Liouville rotation number. Within the resonant normal forms we capture the dependence of these periodic orbits on the resonant Fourier coefficients of the dynamics on the invariant curve and, using the contraction mapping principle, obtain a suitable deformation achieving the prescribed spectral data associated with this sequence of orbits. The result can be viewed as a symplectic twist-map analogue of a length-spectral nonrigidity phenomenon for Riemannian manifolds and convex billiards, and it motivates the existence problem for similar 'partially length-isospectral' deformations of strictly convex billiard tables. }, author = {Li, Yunzhe}, issn = {2663-337X}, pages = {131}, publisher = {Institute of Science and Technology Austria}, title = {{Spectral rigidity and nonrigidity of dynamical systems}}, doi = {10.15479/AT-ISTA-22255}, year = {2026}, } @article{14278, abstract = {The Birkhoff conjecture says that the boundary of a strictly convex integrable billiard table is necessarily an ellipse. In this article, we consider a stronger notion of integrability, namely, integrability close to the boundary, and prove a local version of this conjecture: a small perturbation of almost every ellipse that preserves integrability near the boundary, is itself an ellipse. We apply this result to study local spectral uniqueness of ellipses using the connection between the wave trace of the Laplacian and the dynamics near the boundary and establish local uniqueness for almost all of them.}, author = {Koval, Illya}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {221--298}, publisher = {Springer Nature}, title = {{Local strong Birkhoff conjecture and local spectral rigidity of almost every ellipse}}, doi = {10.1007/s00222-025-01397-y}, volume = {244}, year = {2026}, } @article{20839, abstract = {For every couple of Hausdorff functions ψ and φ verifying some mild assumptions, there exists a compact subset K of the Baire space such that the φ-Hausdorff measure and the ψ-packing measure on K are both finite and positive. Such examples are then embedded in any infinite dimensional Banach space to answer positively a question of Fan on the existence of metric spaces with arbitrary scales.}, author = {Helfter, Mathieu}, issn = {2308-1317}, journal = {Journal of Fractal Geometry}, publisher = {EMS Press}, title = {{Sets with arbitrary Hausdorff and packing scales in infinite dimensional Banach spaces}}, doi = {10.4171/jfg/177}, year = {2025}, } @article{19496, abstract = {We introduce the notions of scale for sets and measures on metric space by generalizing the usual notions of dimension. Several versions of scales are introduced such as Hausdorff, packing, box, local and quantization. They are defined for different growth, allowing a refined study of infinite dimensional spaces. We prove general theorems comparing the different versions of scales. They are applied to describe geometries of ergodic decompositions, of the Wiener measure and from functional spaces. The first application solves a problem of Berger on the notions of emergence (2020); the second lies in the geometry of the Wiener measure and extends the work of Dereich–Lifshits (2005); the last refines Kolmogorov–Tikhomirov (1958) study on finitely differentiable functions.}, author = {Helfter, Mathieu}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, publisher = {Springer Nature}, title = {{Scales}}, doi = {10.1007/s00209-025-03719-5}, volume = {310}, year = {2025}, } @article{20185, abstract = {We show a local rigidity result for the integrability of symplectic billiards. We prove that any domain which is close to an ellipse, and for which the symplectic billiard map is rationally integrable must be an ellipse as well. This is in spirit of the result of [2] for Birkhoff billiards.}, author = {Tsodikovich, Daniel}, issn = {1050-6926}, journal = {Journal of Geometric Analysis}, number = {10}, publisher = {Springer Nature}, title = {{Local rigidity for symplectic billiards}}, doi = {10.1007/s12220-025-02148-4}, volume = {35}, year = {2025}, } @unpublished{22340, abstract = {We show that Laplace isospectral deformations within a conformal class of generic Liouville metrics on the two-dimensional torus that are linear in the deformation parameter are necessarily trivial. Two of the main ingredients in our proof are a noncancellation result for the wave trace and an analysis of the second order variational formula for the energy functional associated to closed geodesics. Noncancellation allows us to detect parts of the length spectrum from the Laplace spectrum and conclude rational integrability for the deformed geodesic flow (Liouville metrics are folklorically conjectured to be the only Riemannian metrics with integrable geodesic flow on the torus). We then use the second variational formula to show how the preservation of a single rational torus is sufficient to conclude triviality of the deformation, assuming linearity. We also present some evidence that our hypothesis of linearity may indeed be necessary.}, author = {Henheik, Sven Joscha and Kaloshin, Vadim and Li, Yunzhe and Vig, Amir}, booktitle = {arXiv}, keywords = {Differential Geometry (math.DG), Mathematical Physics (math-ph), Dynamical Systems (math.DS), Spectral Theory (math.SP), FOS: Mathematics, FOS: Mathematics, FOS: Physical sciences, FOS: Physical sciences, 58J42, 37J35, 37J35, 35P20, 58J40, 58J50, 37D40}, title = {{Spectral rigidity of Liouville tori}}, doi = {10.48550/ARXIV.2511.10398}, year = {2025}, } @unpublished{22341, abstract = {We construct nontrivial deformations of the standard map which preserve the symplectic actions, respectively the Lyapunov exponents, of infinitely many periodic orbits accumulating to an invariant curve. The proof uses a resonant normal-form construction to obtain a sequence of periodic orbits accumulating on an invariant curve with a Liouville rotation number. Within these normal forms we capture the dependence of these periodic orbits on the resonant Fourier coefficients of the dynamics on the invariant curve and, using the contraction mapping principle, obtain a suitable deformation achieving the prescribed spectral data associated with this sequence of orbits. The result can be viewed as a symplectic twist-map analogue of a length spectral nonrigidity phenomenon for Riemannian manifolds and convex billiards, and it motivates the existence problem for similar 'partially length-isospectral' deformations of strictly convex billiard tables.}, author = {Li, Yunzhe}, booktitle = {arXiv}, keywords = {Dynamical Systems (math.DS), FOS: Mathematics, FOS: Mathematics}, title = {{Deformations of the standard map with prescribed actions and Lyapunov exponents}}, doi = {10.48550/ARXIV.2512.03865}, year = {2025}, } @article{18483, abstract = {In this paper we prove a perturbative version of a remarkable Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) result. They prove non perturbative Birkhoff conjecture for centrally-symmetric convex domains, namely, a centrally-symmetric convex domain with integrable billiard is ellipse. We combine techniques from Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) with a local result by Kaloshin–Sorrentino (Ann. Math. 188(1):315–380, 2018) and show that a domain close enough to a centrally symmetric one with integrable billiard is ellipse. To combine these results we derive a slight extension of Bialy–Mironov (Ann. Math. 196(1):389–413, 2022) by proving that a notion of rational integrability is equivalent to the C0-integrability condition used in their paper.}, author = {Kaloshin, Vadim and Koudjinan, Edmond and Zhang, Ke}, issn = {1420-8970}, journal = {Geometric and Functional Analysis}, pages = {1973--2007}, publisher = {Springer Nature}, title = {{Birkhoff conjecture for nearly centrally symmetric domains}}, doi = {10.1007/s00039-024-00695-6}, volume = {34}, year = {2024}, } @article{18586, abstract = {We prove the Central Limit Theorem and superpolynomial mixing for environment viewed from the particle process in quasi periodic Diophantine random environment. The main ingredients are smoothness estimates for the solution of the Poisson equation and local limit asymptotics for certain accelerated walks.}, author = {Czudek, Klaudiusz S and Dolgopyat, Dmitry}, issn = {1980-0436}, journal = {Alea}, number = {2}, pages = {1853--1865}, publisher = {Instituto Nacional de Matematica Pura e Aplicada}, title = {{The central limit theorem and rate of mixing for simple random walks on the circle}}, doi = {10.30757/ALEA.V21-70}, volume = {21}, year = {2024}, } @article{17475, abstract = {As a discrete analogue of Kac’s celebrated question on ‘hearing the shape of a drum’ and towards a practical graph isomorphism test, it is of interest to understand which graphs are determined up to isomorphism by their spectrum (of their adjacency matrix). A striking conjecture in this area, due to van Dam and Haemers, is that ‘almost all graphs are determined by their spectrum’, meaning that the fraction of unlabelled n-vertex graphs which are determined by their spectrum converges to 1 as n → ∞. In this paper, we make a step towards this conjecture, showing that there are exponentially many n-vertex graphs which are determined by their spectrum. This improves on previous bounds (of shape e c √ n ). We also propose a number of further directions of research. }, author = {Koval, Illya and Kwan, Matthew Alan}, issn = {1464-3847}, journal = {Quarterly Journal of Mathematics}, number = {3}, pages = {869--899}, publisher = {Oxford University Press}, title = {{Exponentially many graphs are determined by their spectrum}}, doi = {10.1093/qmath/haae030}, volume = {75}, year = {2024}, } @article{18065, abstract = {We establish a close connection between acceleration and dynamical degree for one-frequency quasi-periodic compact cocycles, by showing that two vectors derived separately from each coincide. Based on this, we provide a dynamical classification of one-frequency quasi-periodic SO(3, R)-cocycles.}, author = {Hou, Xuanji and Pan, Yi and Zhou, Qi}, issn = {1090-2082}, journal = {Advances in Mathematics}, publisher = {Elsevier}, title = {{Dynamical classification of analytic one-frequency quasi-periodic SO(3,R)-cocycles}}, doi = {10.1016/j.aim.2024.109943}, volume = {457}, year = {2024}, } @article{17231, abstract = {In the class of projective billiards, which contains the usual billiards, we exhibit counter-examples to Ivrii's conjecture, which states that in any planar billiard with smooth boundary the set of periodic orbits has zero measure. The counter-examples are polygons admitting a 2-parameters family of n-periodic orbits, with n being either 3 or any even integer greater than 4.}, author = {Fiorebe, Corentin}, issn = {1553-5231}, journal = {Discrete and Continuous Dynamical Systems- Series A}, number = {11}, pages = {3287--3301}, publisher = {AIMS}, title = {{Examples of projective billiards with open sets of periodic orbits}}, doi = {10.3934/dcds.2024059}, volume = {44}, year = {2024}, } @article{12877, abstract = {We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.}, author = {De Simoi, Jacopo and Kaloshin, Vadim and Leguil, Martin}, issn = {1432-1297}, journal = {Inventiones Mathematicae}, pages = {829--901}, publisher = {Springer Nature}, title = {{Marked Length Spectral determination of analytic chaotic billiards with axial symmetries}}, doi = {10.1007/s00222-023-01191-8}, volume = {233}, year = {2023}, } @article{14427, abstract = {In the paper, we establish Squash Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich squash-type stadia whose convex arcs are homothetic. We also establish Stadium Rigidity Theorem—the dynamical spectral rigidity for piecewise analytic Bunimovich stadia whose flat boundaries are a priori fixed. In addition, for smooth Bunimovich squash-type stadia we compute the Lyapunov exponents along the maximal period two orbit, as well as the value of the Peierls’ Barrier function from the maximal marked length spectrum associated to the rotation number 2n/4n+1.}, author = {Chen, Jianyu and Kaloshin, Vadim and Zhang, Hong Kun}, issn = {1432-0916}, journal = {Communications in Mathematical Physics}, pages = {1--50}, publisher = {Springer Nature}, title = {{Length spectrum rigidity for piecewise analytic Bunimovich billiards}}, doi = {10.1007/s00220-023-04837-z}, volume = {404}, year = {2023}, } @article{18959, abstract = {This workshop continues a series of workshops whose current format originated in 1981 under then-organizers Moser and Zehnder, and whose latest iteration took place in July 2023. The general goal of this series of workshops is to discuss the latest developments in the field of dynamical systems, broadly construed, and its connections with neighboring areas of mathematics such as differential geometry, partial differential equations, and more recently contact and symplectic geometry. We continued this tradition, bringing in new participants working in areas of dynamical systems and its connections with other areas of mathematics that are currently highly active and/or showing great promise for future development. Key focus areas for the 2023 workshop include spectral rigidity for planar domains, chaotic and oscillatory motions in celestial mechanics, conformal symplectic dynamics, and relations between dynamics.he workshop by the grant DMS-2230648, “US Junior Oberwolfach Fellows”.}, author = {Arnaud, Marie-Claude and Hutchings, Michael and Kaloshin, Vadim}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {3}, pages = {1671--1730}, publisher = {EMS Press}, title = {{Dynamische Systeme}}, doi = {10.4171/owr/2023/30}, volume = {20}, year = {2023}, } @article{11553, abstract = {In holomorphic dynamics, complex box mappings arise as first return maps to wellchosen domains. They are a generalization of polynomial-like mapping, where the domain of the return map can have infinitely many components. They turned out to be extremely useful in tackling diverse problems. The purpose of this paper is: • To illustrate some pathologies that can occur when a complex box mapping is not induced by a globally defined map and when its domain has infinitely many components, and to give conditions to avoid these issues. • To show that once one has a box mapping for a rational map, these conditions can be assumed to hold in a very natural setting. Thus, we call such complex box mappings dynamically natural. Having such box mappings is the first step in tackling many problems in one-dimensional dynamics. • Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques. In this setting, some of these tools are: • the Enhanced Nest (a nest of puzzle pieces around critical points) from Kozlovski, Shen, van Strien (AnnMath 165:749–841, 2007), referred to below as KSS; • the Covering Lemma (which controls the moduli of pullbacks of annuli) from Kahn and Lyubich (Ann Math 169(2):561–593, 2009); • the QC-Criterion and the Spreading Principle from KSS. The purpose of this paper is to make these tools more accessible so that they can be used as a ‘black box’, so one does not have to redo the proofs in new settings. • To give an intuitive, but also rather detailed, outline of the proof from KSS and Kozlovski and van Strien (Proc Lond Math Soc (3) 99:275–296, 2009) of the following results for non-renormalizable dynamically natural complex box mappings: • puzzle pieces shrink to points, • (under some assumptions) topologically conjugate non-renormalizable polynomials and box mappings are quasiconformally conjugate. • We prove the fundamental ergodic properties for dynamically natural box mappings. This leads to some necessary conditions for when such a box mapping supports a measurable invariant line field on its filled Julia set. These mappings are the analogues of Lattès maps in this setting. • We prove a version of Mañé’s Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.}, author = {Clark, Trevor and Drach, Kostiantyn and Kozlovski, Oleg and Strien, Sebastian Van}, issn = {2199-6806}, journal = {Arnold Mathematical Journal}, number = {2}, pages = {319--410}, publisher = {Springer Nature}, title = {{The dynamics of complex box mappings}}, doi = {10.1007/s40598-022-00200-7}, volume = {8}, year = {2022}, } @article{11717, abstract = {We study rigidity of rational maps that come from Newton's root finding method for polynomials of arbitrary degrees. We establish dynamical rigidity of these maps: each point in the Julia set of a Newton map is either rigid (i.e. its orbit can be distinguished in combinatorial terms from all other orbits), or the orbit of this point eventually lands in the filled-in Julia set of a polynomial-like restriction of the original map. As a corollary, we show that the Julia sets of Newton maps in many non-trivial cases are locally connected; in particular, every cubic Newton map without Siegel points has locally connected Julia set. In the parameter space of Newton maps of arbitrary degree we obtain the following rigidity result: any two combinatorially equivalent Newton maps are quasiconformally conjugate in a neighborhood of their Julia sets provided that they either non-renormalizable, or they are both renormalizable “in the same way”. Our main tool is a generalized renormalization concept called “complex box mappings” for which we extend a dynamical rigidity result by Kozlovski and van Strien so as to include irrationally indifferent and renormalizable situations.}, author = {Drach, Kostiantyn and Schleicher, Dierk}, issn = {0001-8708}, journal = {Advances in Mathematics}, keywords = {General Mathematics}, number = {Part A}, publisher = {Elsevier}, title = {{Rigidity of Newton dynamics}}, doi = {10.1016/j.aim.2022.108591}, volume = {408}, year = {2022}, } @article{12145, abstract = {In the class of strictly convex smooth boundaries each of which has no strip around its boundary foliated by invariant curves, we prove that the Taylor coefficients of the “normalized” Mather’s β-function are invariant under C∞-conjugacies. In contrast, we prove that any two elliptic billiard maps are C0-conjugate near their respective boundaries, and C∞-conjugate, near the boundary and away from a line passing through the center of the underlying ellipse. We also prove that, if the billiard maps corresponding to two ellipses are topologically conjugate, then the two ellipses are similar.}, author = {Koudjinan, Edmond and Kaloshin, Vadim}, issn = {1468-4845}, journal = {Regular and Chaotic Dynamics}, keywords = {Mechanical Engineering, Applied Mathematics, Mathematical Physics, Modeling and Simulation, Statistical and Nonlinear Physics, Mathematics (miscellaneous)}, number = {6}, pages = {525--537}, publisher = {Springer Nature}, title = {{On some invariants of Birkhoff billiards under conjugacy}}, doi = {10.1134/S1560354722050021}, volume = {27}, year = {2022}, } @article{10706, abstract = {This is a collection of problems composed by some participants of the workshop “Differential Geometry, Billiards, and Geometric Optics” that took place at CIRM on October 4–8, 2021.}, author = {Bialy, Misha and Fiorebe, Corentin and Glutsyuk, Alexey and Levi, Mark and Plakhov, Alexander and Tabachnikov, Serge}, issn = {2199-6806}, journal = {Arnold Mathematical Journal}, location = {Hybrid}, pages = {411--422}, publisher = {Springer Nature}, title = {{Open problems on billiards and geometric optics}}, doi = {10.1007/s40598-022-00198-y}, volume = {8}, year = {2022}, } @article{17063, abstract = {This workshop continued a biannual series of workshops at Oberwolfach on dynamical systems that started with a meeting organized by Moser and Zehnder in 1981. Workshops in this series focus on new results and developments in dynamical systems and related areas of mathematics, with symplectic geometry playing an important role in recent years in connection with Hamiltonian dynamics. In this year special emphasis was placed on various kinds of spectra (in contact geometry, in Riemannian geometry, in dynamical systems and in symplectic topology) and their applications to dynamics.}, author = {Arnaud, Marie-Claude and Hofer, Helmut W. and Hutchings, Michael and Kaloshin, Vadim}, issn = {1660-8941}, journal = {Oberwolfach Reports}, number = {3}, pages = {1735--1803}, publisher = {EMS Press}, title = {{Dynamische Systeme}}, doi = {10.4171/owr/2021/33}, volume = {18}, year = {2022}, }