KTH Dynamics Seminar

TBA

Collaborators?

TBA

Collaborators?

Starting with the pioneering work of Furstenberg and Kesten in the sixties, the quest for non-commutative analogues of classical limit theorems in probability theory has gathered continuous attention. We will discuss aspects of the study of random walks driven by actions of discrete countable groups by isometries of metric spaces, with a focus on those possessing certain hyperbolic-like features, which are most clearly on display for actions on Gromov-hyperbolic spaces and for the dynamics of the mapping class group on the Teichmüller space of a closed oriented surface. The emphasis is on large deviations for the process of renormalized distances from the origin: we prove existence of a large deviation principle, with identification of the rate function as an appropriate Fenchel-Legendre transform, for a broad class of isometric group actions encompassing the two aforementioned examples, as well as many others of relevance in geometric group theory and geometric topology. This at once extends recent work of Boulanger, Mathieu, Sert and Sisto, and provides a more streamlined argument for their result. If time permits, quantitative refinements of the main result, afforded by thermodynamic formalism on symbolic spaces in the non-additive regime, will also be presented.

We consider the direct product of two Hamiltonian diffeomorphisms, one of which exhibits a basic set and the other one a non-degenerate elliptic equilibrium. Under a local domination condition we show that a typical real-analytic perturbation of this system displays a large normally hyperbolic lamination on which the natural invariant measure is ergodic.

In 1972, Livsic proved his famous theorem for Anosov systems that when a Holder continuous real valued function has zero averages around all closed orbits then the function necessarily takes a special form (i.e., it is a coboundary). This is a simple, but striking, example of where local information (on closed orbits) gives a global conclusion (the function is a coboundary). Recently, A. Gogolev and F. Rodriguez Hertz showed that knowing this property for a suitable smaller collection of closed orbits still gives that the Holder function takes a special form (The Abelian Livsic Theorem). We will show a generalization of this result.

This is joint work with Richard Sharp (Warwick).

In 1984, Grebogi et al. showed the occurrence of strange non-chaotic attractors in a specific class of quasi-periodically forced systems. Since then, various numerical studies have indicated the prevalence of geometrically intricate invariant graphs in quasi-periodically forced systems. This has continued to motivate theoretical results that can uncover the underlying mechanisms that generate such invariant structures.

In this talk, we discuss quasi-periodically forced circle diffeomorphisms, with a focus on circle diffeomorphisms which have exactly two attracting and two repelling fixed points. Under a sufficiently weak forcing, one can prove that the corresponding skew-product map is uniformly hyperbolic. Under a strong quasi-periodic forcing, the dynamics become notably more complicated. In such cases, for certain weak conditions of asymmetry on the fixed points, we establish that the corresponding map $F_{\omega}$ exhibits synchronisation along the fibre direction for a set of frequencies $\omega$ of positive measure. Further, we show that the hyperbolic behaviour in the system can depend sensitively on the arithmetic properties of the frequency.

How fast can a quantum particle spread through a crystal material? Dispersion estimates measure this spreading through the decay of quantum probability amplitudes over time. I will discuss joint work with Gaétan Leclerc and Mostafa Sabri constructing one-dimensional crystals with long-range hopping and arbitrarily fast polynomial dispersion. The link is through Fourier decay for images of Lebesgue measure under Weierstrass functions. Our main tool is an adaptation of Dolgopyat’s method from hyperbolic dynamics, using the self-similarity of these functions to obtain a version of van der Corput’s lemma for non-smooth phases. This also answers a question of Geman and Horowitz from 1980 on the existence of local times for classical Weierstrass functions with sufficiently large lacunarity.

Given a finitely generated group acting on the circle by diffeomorphisms, we study the dimension of its mininal sets. In this talk, I will present a notion of dynamical critical exponent. Combined with other tools such as random walks, conformal measures and variational principle, this gives some characterisation of the Hausdorff and box dimensions of the minimal sets.

Based on joint work with Yuxiang Jiao and Disheng Xu.

We will present a measure rigidity theorem for quotients of $SL_2(R)^r$, focusing on the newer ideas in the proof.

This is joint work with Elon Lindenstrauss. (Extensions of this theorem are in the works jointly with Shem-Tov and Wieser.)

We discuss a recent example of a matrix cocycle exhibiting a freezing phase transition in the negative-temperature regime. The cocycle is generated by two simple parabolic matrices and admits a complete description of its equilibrium measures. A phase transition occurs at $t=-2$: for $t>-2$ there is a unique fully supported equilibrium measure, while for $t\leq -2$ the system freezes and only two atomic equilibrium measures remain. The example is strongly irreducible and proximal, showing that non-uniqueness of equilibrium measures can occur for negative parameters in the matrix cocycle setting.

This is joint work with Anthony Quas (University of Victoria, Canada).

It is a classical result that topological entropy and kneading sequences depend in a monotone way monotone on parameters for the quadratic family. We prove this for the more general family $f_{a}(x) = a - \vert x\vert^r$, $r$ real and $ > 1$. The proof uses a modification of the Milnor-Thurston method, which is based on a Thurston pullback map and a Teichmüller space contraction

This is joint work with Ana Rodrigues.

In 1985, Odoni introduced a novel approach to study the density of primes dividing iterates of a given polynomial via the fixed-point proportion of the associated iterated Galois group. His approach has found many further applications in arithmetic dynamics. In 2016 Juul, Kurlberg, Madhu and Tucker obtained several results on the proportion of periodic points of rational functions over finite fields via this approach. In that paper, they further asked for a classification of the rational functions for which the limit inferior of this proportion becomes $0$.

In this talk, we give the sought classification for all polynomials. In fact, this is deduced as a result of a long-awaited classification of the fixed-point proportion of geometric iterated Galois groups of polynomials. Since the first examples of zero fixed-point proportion were obtained by Odoni in 1985, more and more evidence has been found on the fixed-point proportion being zero unless the polynomial under consideration is linearly conjugate up to a sign to a Chebyshev polynomial. We confirm this well-known conjecture. Our original approach is based on a new ergodic theory for self-similar groups and it further uses classical techniques from groups acting on rooted trees, martingale theory and complex dynamics.

This is joint work with Santiago Radi.

The understanding of universality in transitions to chaos achieved in the 1970ies in the sense of physics, and later as rigorous mathematics, is based on renormalization. In the abstract renormalization is a change of the representation of some object from a change of resolution of observation. As applied to dynamics at the transition to chaos the change is from observing the system at every (discrete) time point to more widely spaced time points. The paradigmatic case is period-doubling where the sequence is every time point, every second time point, every fourth time point, and so on.

From the fixed point of renormalization one can compute metric properties, of which one of the most famous is Feigenbaum's quantity $\alpha$ in period-doubling. Wikipedia says that it is $2.502907875095892822283902873218$ (I personally never computed it with so many digits).

This $\alpha$ is however just one out of many metric properties which can also be computed from the renormalization fixed point. For instance, it is well known that $2^k$ almost returns of the critical point (conventionally $x=0$) to itself at the accumulation point of period doublings scale as $(1/-\alpha)^k$, the renormalization transformation in fact encodes this regularity. However, it is then obvious that $2^k$ almost returns of one iteration of the critical point (conventionally $f(0)$) must scale as $(1/\alpha)^{2k}$. Hence $(1/-\alpha)$ and $(1/\alpha^2)$ both appear as metric regularities, and these numbers can be extended to a real function $\sigma(t)$ on $[0,1/2]$ which Feigenbaum first called `scaling function' and later `presentation function'.

A bit less known is the fact that the presentation function can itself be computed as a fixed point, without direct reference to the standard renormalization, or its fixed point.

I will present how this is done, with no pretence of mathematical rigor, and without showing my own numerical data on the problem, which I did at an earlier time in this seminar. I will also not consider the case of golden mean mode locking of circle maps, where a similar analysis has been carried out, but only focus on period-doubling.

The talk is based on M Feigenbaum, J Stat Phys vol 52 Nos 3/4 (1988) pp. 527-569, a paper which has been cited $121$ times according to Google scholar, but where the main idea, to my knowledge, has not been much developed further, nor has it been taken up by others.

I will review the information lattice, which we recently introduced as a way to decompose information in a quantum state into local information in position. Using this decomposition allows both universal characterisation of many-body states as well as an informed construction of efficient numerical algorithms for time evolution of many-body states to long times, way beyond the entanglement barrier that limits the time range of matrix product state time evolution. We apply this to the problem of simulating thermalising quantum many-body system to long times such that diffusion emerges.

In this talk I will describe new results establishing the existence of a strong spectral gap for any geometrically finite, $n$ dimensional, hyperbolic manifold with critical exponent greater than $(n-1)/2$. This settles a conjecture of Mohammadi and Oh from 2015.

This is based on joint work with Osama Kahlil and Pratyush Sarkar.

The discrete one-dimensional Schrödinger operator with a quasi-periodic potential has been extensively studied. Dynamical methods have shown powerful to analyze the associated Schrödinger equation. In this talk we will show how such type of methods can be used to obtain certain (asymptotic) estimates of the Lebesgue measure of (parts) of the spectrum of the operator.

A Fibonacci map is a piecewise defined map of a subset of an interval $I$ onto $I$ with a unique critical point of order $d$ whose orbit undergoes nearest returns at Fibonacci times. It has been shown by Bruin, Keller, Nowicki and van Strien that such maps exhibit "wild" attractors: Cantor sets of zero Lebesgue measure whose basin of attraction is meager but has positive Lebesgue measure. We will discuss real renormalization, and a trichotomy for Fibonacci maps, similar to the Avila–Lyubich trichotomy for Feigenbaum Julia sets, which, in particular, allows us to show that Fibonacci maps admit wild attractors for $d = 5.1$, and do not for $d = 3.9$ (and, conjecturally, for $2 < d \leq 3.9$).

This is a joint work with Artem Dudko of IMPAN, Warsaw.

We will discuss the speed of mixing of extensions of chaotic (Anosov) flows to possibly non-compact principal bundles. For example, this includes the frame flow on the bundle of orthonormal frames over a negatively curved Riemannian manifold, and our result in this setting guarantees that ergodicity of the frame flow implies its rapid mixing, that is, mixing faster than $C_N t^{-N}$ for any $N > 0$ (here $t > 0$ denotes time and $C_N > 0$). In the second situation of interest, we will consider extensions to $\mathbb{Z}^d$ covers; in this case, we will prove the optimal speed of mixing $Ct^{-d/2}$, and moreover we will provide the full asymptotic expansion in decaying integer powers of the time $t$. To prove these results we use the Fourier expansion in vertical fibers, suitably constructed anisotropic Sobolev spaces tailored to the dynamics, and a novel semiclassical calculus on principal $G$-bundles that we call the Borel–Weil calculus, where the semiclassical parameters correspond to the highest roots parametrizing irreducible representations of $G$. The calculus can be shown to have further applications to hypoellipticity and quantum ergodicity of horizontal (sub-)Laplacians.

Joint works with Thibault Lefeuvre and Sebastián Muñoz Thon.

The quantum nature of black holes has remained a key problem in fundamental physics since the discovery of Hawking radiation, now more than half a century ago. It was early on pointed out by Bekenstein and Mukhanov that if black holes are quantum objects with a density of states corresponding to Bekenstein–Hawking entropy, then the density of states of those objects must grow as $e^{Const. E^2}$. An entropy growing quadratically with $E$ is a concave and not a convex function, and the canonical distribution is not defined for any finite temperature (the partition function is infinite). This already says that a quantum black hole, if such a thing exists, must be a thing with rather exotic properties.

While natural objects with sub-exponential or exponential spectral growth, i.e. $e^{Const. E^a}$ with $a\leq 1$, are known in both physics and mathematics, if mathematicians also know of natural objects with super-exponential spectral growth, that knowledge has not reached outsiders.

I will present a simple physical toy model which does have super-exponential growth, though it obviously cannot be a reasonable model of a quantum black hole as all states are very delocalized. I will then use this observation as starting point of a discussion of why it is hard to construct models of compact objects with super-exponential spectral growth, and what this may perhaps tell us about quantum black holes.

The talk is based on joint work with Satya N. Majumdar Physical Review Research 7: 043165 (2025) [arXiv:2504.06623].

We will discuss Sinai's principle for recurrence in the unicritical dynamics from the point of view of harmonic measure. In particular we show that for almost all $c$ with respect to the harmonic measure on the boundary of the Mandelbrot set, \[ \vert f_c^n(c) - c \vert \leq \frac{1}{n^\alpha} \quad \text{infinitely often} \] if and only if $\alpha \leq 1$.

This talk is based on joint work with Graczyk and Świątek.

In chaotic systems, the mixing property is known for the fast decay of correlation. It is called rapid mixing if the correlation function decays super-polynomially. The mixing mechanism for hyperbolic systems and its compact group extensions were studied by Dolgopyat in a series of his papers in the late 90s'.

In this talk, we prove rapid mixing for almost all random walks generated by $m\geq 2$ translations on an arbitrary nilmanifold. For several classical classes of nilmanifolds, we show $m = 2$ suffices. This provides a partial answer to the question raised in the work of Dolgopyat ('02) about the prevalence of rapid mixing for random walks on homogeneous spaces.

This is joint work with Dmitry Dolgopyat and Spencer Durham.

In this talk, we will introduce a novel approach linking fractal geometry to partially hyperbolic dynamics, revealing several new phenomena related to regularity jumps and rigidity. One key result demonstrates a sharp phase transition for partially hyperbolic diffeomorphisms $f$ with a contracting center direction on $T^3$: $f$ is $C^\infty$-rigid if and only if both $E^s$ and $E^c$ exhibit Hölder exponents exceeding the expected threshold. These and related results originate from a general non-fractal invariance principle: for a skew product $F$ over a partially hyperbolic system $f$, if $F$ expands fibers more weakly than $f$ along $W^u_f$ in the base, then for any $F$-invariant section $\Phi$, if $\Phi$ has no fractal graph, then it is smooth along $W^u_f$ and holonomy-invariant.

This is based on a joint work with Disheng Xu.

Let $D>1$ be a square-free integer. Marklof and Welsh established limit laws for the fine-scale distribution of roots of the quadratic congruence $\mu^2\equiv D \pmod m$ with $D \not\equiv 1 \pmod 4$. This is achieved by considering the convergence of certain geodesic random line processes in the hyperbolic plane. Strikingly, they derived an explicit formula for the pair correlation density of the roots. In this talk, I will introduce their work and our recent result without the restriction $D \not\equiv 1 \pmod 4$.

This is joint work with Matthew Welsh.

Abstract: The classical shrinking target problem involves a sequence of `target' subsets $A_n \subset X$ and a dynamic $T: X \to X$. The focus is on the set of all points $R \subset X$ whose $n$-th iterate hits the $n$-th target, $T^n(x) \in A_n$, infinitely often. Numerous variants of this problem exist, including a `dynamical covering problem', which can be viewed as a dynamical analogue of the Dvoretzky covering problem. Another version of these `limsup set' problems involves replacing them with `liminf sets'. There are few results on these latter two sets. I will provide an overview of such problems and present a recent result that examines the size of coverings for `self-similar' dynamical systems.

Joint with Balázs Bárány (BME Budapest) and Henna Koivusalo (Bristol).

There is a natural upper bound for the dimension of self-similar sets, usually coined the similarity dimension. On the real line there is a folklore conjecture that the only way that the actual dimension drops below the similarity dimension is if exact overlaps are present. In a breakthrough paper, Hochman confirmed the conjecture assuming the exponential separation condition (ESC). This was recently extended by Rapaport to analytic self-conformal sets on the real line. Together with Balázs Bárány and Sascha Troscheit, we show that the ESC is a topologically typical property, moreover, we give an explicit sufficient condition to verify the ESC by introducing a novel concept that we call the dual iterated function system. I will give an overview of all necessary background before presenting our new results.

The Brjuno and Wilton functions bear a striking resemblance, despite their very different origins; while the Brjuno function $B(x)$ is a fundamental tool in one-dimensional holomorphic dynamics, the Wilton function $W(x)$ stems from the study of divisor sums and self-correlation functions in analytic number theory. I will discuss how these two functions are related to each other also by the means of a generalization of the Brjuno function – the so-called semi-Brjuno function $B_0(x)$ – built using the by-excess continued fraction map. Namely, $B(x)$ and $W(x)$ can be expressed in terms of the even $B_0^+$ and odd $B_0^-$ parts of $B_0(x)$, respectively, up to a bounded defect. Moreover we study the regularity of the differences $\Delta^+(x) = B^+(x) -ˆ’ 2B_0^+(x)$ and $\Delta^-ˆ’(x) = W^-ˆ’(x) -ˆ’ 2B_0^-ˆ’(x)$, the first of which is Hölder continuous whereas the second exhibits discontinuities at rationals, behaving similarly to the classical popcorn function.

This is joint work with Claire Burrin (University of Zurich) and Seul Bee Lee (Seoul National University): see https://arxiv.org/abs/2503.08206.

Recently, Fisher and Melnick classified $\operatorname{SL}(n,R)$-actions on $n$-dimensional manifolds for $n \geq 3$. In this talk, we generalize this result by classifying smooth or real-analytic $\operatorname{SL}(n,R)$-actions on $m$-dimensional manifolds for $3 \leq n \leq m \leq 2n-3$. This work is motivated by the Zimmer program and is central to it, as Lie group actions restrict to their lattice actions. This classification relies on the linearization of $\operatorname{SL}(n,R)$-actions when there is a global fixed point. The analytic case was proved by Guillemin–Sternberg and Kushinirenko.

We discuss the smooth case which is ongoing joint work with Insung Park.

The statistical characteristics of interval maps having indifferent fixed points have been extensively investigated. A method to examine a comparable issue in higher dimensions is through the skew product system, which is a coupling of two systems non-autonomously. This presentation will discuss a joint work with Stefano Luzzatto, where we analyzed a skew product system by coupling a uniformly expanding circle map with a double intermittent interval map exhibiting unbounded derivatives. By utilizing the inducing technique, we have demonstrated that this system possesses a unique ergodic absolutely continuous invariant probability (acip) with a polynomial decay of correlations.

The set of real numbers which are badly approximable by rationals admits an exhaustion by sets $\operatorname{Bad}(c)$, whose dimension converges to $1$ as $c$ goes to zero. D. Hensley computed the asymptotic for the dimension up to the first order in $c$, via an analogous estimate for the set of real numbers whose continued fraction has all entries uniformly bounded. We consider diophantine approximations by parabolic fixed points of any non-uniform lattice in $\operatorname{PSL}(2,R)$ and a geometric notion of $c$-badly approximable points. We compute the dimension of the set of such points up to the first order in $c$, via the thermodynamic method of Ruelle and Bowen. Geometric good approximations are related to a notion of bounded partial quotients for the Bowen-Series expansion. This gives a family of Cantor sets and associated quasi-compact transfer operators, with simple and positive maximal eigenvalue. Perturbative analysis of spectra applies.

In this talk, we will discuss about smooth random dynamical systems and group actions on surfaces. When a group action on torus is given by toral automorphisms, Benoist–Quint classified orbit closures (and stationary measures), using random dynamics. In this talk, we will study group actions that are given by diffeomorphisms, using random dynamics. We will discuss about absolutely continuity of stationary measures, classification of orbit closure, and exact dimensionality of stationary measures.

This talk will be mostly about an ongoing work with Aaron Brown, Davi Obata, and Yuping Ruan.

I plan to review problems and recent results regarding smooth curves in the moduli space of translation surfaces and Teichmüller positive semi-orbits starting from such curves. The main part of the talk is applications that motivate abstract results. Two main applications relate to billiards on nibbled ellipses and Hamiltonian systems with reflections.

Motivated by a hyperbolic analog of the Lester–Wigman "vanishing area correlations"-conjecture for euclidean lattice points we investigate the dynamical properties of a natural choice of a Poincaré section, associated with $\operatorname{H}/\operatorname{SL}(2,Z)$, and the horocycle flow on the upper half plane $H$. Since the horocycle *flow* is mixing, one might hope for an easy proof of vanishing area correlations by showing that the Poincaré map is mixing. However, not only is the Poincaré map non-mixing; even equidistribution/ergodicity breaks down badly due to escape of mass. Amusingly, we can still show vanishing of area correlations (but "for the wrong reason".)

The Collet–Eckmann condition is one, quite strong, condition which is used to exhibit chaotic behaviour. In this talk I will present a recent result (joint work with M. Bylund and W. Cui) about perturbations of CE-maps in the unicritical family $f_c(z) = z^d + c$, where $c$ is a parameter in the complex plane. This is a well studied family of maps and it has a long history. Let $\mathcal{M}_d$ be the connectedness locus, or simply, the Mandelbrot set, i.e. parameters $c$ for which the Julia set is connected. Around the millenia shift, J. Rivera-Letelier proved that critically non-recurrent maps (Misiurewicz maps) in this family are Lebesgue density points of the complement of $\mathcal{M}_d$ (I proved a corresponding result for rational maps in 2009). In a series of quite recent papers, J. Graczyk and G. Świątek proved, among other things, that typical CE-parameters w.r.t. harmonic measure are Lebesgue density points of the complement of $\mathcal{M}_d$. For these maps, in particular, the critical point is allowed to be slowly recurrent. Moreover, in 2011, A. Avila, M. Lyubich and W. Shen proved that, in particular, CE-maps cannot be density points of $\mathcal{M}_d$. The main result I will present is that for each CE-map $f_c$ in the unicritical family, $c$ is a Lebesgue density point of the complement of the Mandelbrot set. It also generalizes earlier results by M. Bylund, W. Cui and myself for slowly recurrent rational maps.

A fibrewise Anosov flow is an Anosov flow on a fibre bundle whose flow direction is tangent to the base. We are particularly interested in those examples that fibre over $3$-dimensional Anosov flows, where the non-algebraic examples are rich, in order to obtain non-algebraic examples in higher dimensions. We review Tomter's algebraic example that is over a geodesic flow and discuss a construction over the Bonatti Langevin example.

This is an ongoing project joint with Neige Paulet.

We will discuss some surprising rigidity phenomena for Anosov flows in dimension $3$. For example, in the context of generic transitive $3$-dimensional Anosov flows, any continuous conjugacy is either smooth or reverses the positive and negative SRB measures.

This is joint work with Martin Leguil and Federico Rodriguez Hertz.

In recent years there has been an explosion of interest and progress on the Fourier decay problem for fractal measures. In this talk I will discuss some of these results, focusing on how ideas and methods from random walks and spectral theory of transfer operators play a key role in their proof.

Based on joint works with Federico Rodriguez Hertz and Zhiren Wang.

In this talk I will introduce the (planetary) $N$-body problem and briefly discuss its historical background: From Ptolemy, Copernicus, Kepler and Tycho Brahe to Newton; from Laplace, Euler and Lagrange, to Mittag-Leffler and Poincaré, ending up with Kolmogorov, Arnold and Moser. Then I will move to discuss the stability problem of our Solar system and, in particular, the Sun-Jupiter-Saturn. Is it stable? Do the planets orbit around the Sun in a (quasi)periodic fashion?

Let $A$ be a subset of the positive integers. For a dynamical system $(T,X)$ we are interested in ergodic averages along $A$, i.e. for $x\in X$ and $f\in C(X)$ we look at the limiting behavior of $\frac{1}{|A_N|}\sum_{k\in A_N}f(T^k x)$, where $A_N=\{k\in A: k\leq N\}$. We will focus on the case where $A$ is the set of primes or values of polynomials with integer coefficients. We will recall some classical results, mention some recent developments and highlight some interesting open problems.