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Sebastian van Strien

Publications and source records attributed to Sebastian van Strien.

At least 19 recordsLinked to original sources
arXiv2026

Intermingled basins: Kan's example on the Riemann sphere

Kan's discovery of dynamical systems with two attractors whose basins of attraction both have full support, featured specific examples of skew product systems of interval diffeomorphisms forced by expanding circle maps. The interval diffeomorphisms are polynomial maps and can hence be considered on the Riemann sphere. We consider the resulting skew product systems of holomorphic maps on the Riemann sphere forced by expanding circle maps, and establish the existence of three attractors whose basins of attraction all have full support and are thus intermingled.

math.DS↗
arXiv2026

Transversality for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we prove a transversality theorem for a family of dynamically defined vector subspaces that encode the dynamics of a given ITM. As a consequence, we establish a perturbation result that gives a precise control of the first return dynamics to subintervals in $I$, while preserving the remaining global dynamics of the system. Beyond their independent interest, these results are a key technical ingredient in the proof of the Characterisation of Stability of ITMs in arXiv:2605.00190, and in the establishment of the topological version of the Boshernitzan--Kornfeld Conjecture in arXiv:2605.00186.

math.DS↗
arXiv2026

Topological Prevalence of Finite Type Interval Translation Maps

An interval translation map (ITM) is a map $T \colon I \to I$ defined as a piecewise translation on a finite partition of an interval $I$ into $r \ge 2$ subintervals. Unlike classical interval exchange transformations (IETs), the images of these subintervals are allowed to overlap, making ITMs a natural generalisation of IETs. An ITM $T$ is said to be \textit{of finite type} if its attractor $\bigcap_{n\ge 0} T^n(I)$ is a finite union of intervals; in this case, restricted to this invariant set, $T$ is bijective and hence behaves like an IET. Otherwise, $T$ is of infinite type. In this paper, for every $r \ge 2$, we prove that the set of finite type ITMs contains an open and dense subset in the space of all possible ITMs on $r$ subintervals. This confirms a topological version of a long-standing conjecture due to Boshernitzan and Kornfeld.

math.DS↗
arXiv2026

Characterisation of Stability for Interval Translation Maps

An interval translation map (ITM) is a piece-wise translation $T \colon I \to I$ defined on a finite partition $I_1, \ldots, I_r$ of an interval $I$ into $r \ge 2$ subintervals. In contrast to classical interval exchange transformations (IETs), we do not require that the images of these subintervals are disjoint; in particular, ITMs are not assumed to be bijective. Thus, ITMs provide a natural non-invertible generalisation of IETs. In this paper, we formulate an appropriate notion of stability for general interval translation mappings and prove a characterisation of stability in terms of two dynamically natural properties called the Absence of Critical Connections and Matching. This result can be viewed as the foundational step towards the stability theory of general ITMs.

math.DS↗
arXiv2024

Density of Stable Interval Translation Maps

Assume that the interval $I=[0,1)$ is partitioned into finitely many intervals $I_1,\dots,I_r$ and consider a map $T\colon I\to I$ so that $T_{\vert I_s}$ is a translation for each $1 \le s \le r$. We do not assume that the images of these intervals are disjoint. Such maps are called Interval Translation Maps. Let $ITM(r)$ be the space of all such transformations, where we fix $r$ but not the intervals $I_1,\dots,I_r$, nor the translations. The set $X(T):=\bigcap_{n\ge 0} T^n[0,1)$ can be a finite union of intervals (in which case the map is called of finite type), or is a disjoint union of finitely many intervals and a Cantor set (in which case the map is called of infinite type). In this paper we show that there exists an open and dense subset $\mathcal{S}(r)$ of $ITM(r)$ consisting of stable maps, i.e. each $T\in \mathcal{S}(r)$ is of finite type, the first return map to any component of $X(T)$ corresponds to a circle rotation and $\mathcal{S}(r) \ni T \mapsto X(T)$ is continuous in the Hausdorff topology.

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arXiv2023

Robust reconstruction of sparse network dynamics

Reconstruction of the network interaction structure from multivariate time series is an important problem in multiple fields of science. This problem is ill-posed for large networks leading to the reconstruction of false interactions. We put forward the Ergodic Basis Pursuit (EBP) method that uses the network dynamics' statistical properties to ensure the exact reconstruction of sparse networks when a minimum length of time series is attained. We show that this minimum time series length scales quadratically with the node degree being probed and logarithmic with the network size. Our approach is robust against noise and allows us to treat the noise level as a parameter. We show the reconstruction power of the EBP in experimental multivariate time series from optoelectronic networks.

physics.data-an↗
arXiv2023

Holomorphic motions, natural families of entire maps, and multiplier-like objects for wandering domains

Structural stability of holomorphic functions has been the subject of much research in the last fifty years. Due to various technicalities, however, most of that work has focused on so-called finite-type functions (functions whose set of singular values has finite cardinality). Recent developments in the field go beyond this setting. In this paper we extend Eremenko and Lyubich's result on natural families of entire maps to the case where the set of singular values is not the entire complex plane, showing under this assumption that the set $M_f$ of entire functions quasiconformally equivalent to $f$ admits the structure of a complex manifold (of possibly infinite dimension). Moreover, we will consider functions with wandering domains -- another hot topic of research in complex dynamics. Given an entire function $f$ with a simply connected wandering domain $U$, we construct an analogue of the multiplier of a periodic orbit, called a distortion sequence, and show that, under some hypotheses, the distortion sequence moves analytically as $f$ moves within appropriate parameter families.

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arXiv2023

Dennis Sullivan's Work on Dynamics

In this expository paper, we provide the readers with an overview of Dennis Sullivan's major contributions to the area of Dynamical Systems.

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arXiv2023

Conjugacy classes of real analytic one-dimensional maps are analytic connected manifolds

An important question is to describe topological conjugacy classes of dynamical systems. Here we show that within the space of real analytic one-dimensional maps with critical points of prescribed order, the conjugacy class of a map is a real analytic manifold. This extends results of Avila-Lyubich-de Melo \cite{ALM} for the quasi-quadratic unimodal case and of Clark \cite{C} for the more general unimodal case. Their methods fail in the case where there are several critical points, and for this reason we introduce the new notions of {\em pruned Julia set} of a real analytic map, and associate to a real analytic map an {\em external map} of the circle {\em with discontinuities} and {\em a pruned polynomial-like} complex extension of the real analytic map. Using this we are also able to show that topological conjugacy classes are connected (something which was not even known in the general unimodal setting). Even more, this space is contractible. In a companion paper, further applications of this paper will be given. It will be shown that within any real analytic family of real analytic one-dimensional maps, hyperbolic parameters form an open and dense subset.

math.DS↗
arXiv2023

Transversality in the setting of hyperbolic and parabolic maps

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in \cite{LSvS1} to treat unfolding of critical relations can also be used to deal with cases where the critical orbit converges to a hyperbolic attracting or a parabolic periodic orbit. As before this result applies to rather general families of maps, such as polynomial-like mappings, provided some lifting property holds. Our Main Theorem states that either the multiplier of a hyperbolic attracting periodic orbit depends univalently on the parameter and bifurcations at parabolic periodic points are generic, or one has persistency of periodic orbits with a fixed multiplier.

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arXiv2022

The dynamics of complex box mappings

In holomorphic dynamics, complex box mappings arise as first return maps to well-chosen 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. -Many results in holomorphic dynamics rely on an interplay between combinatorial and analytic techniques: (*)the Enhanced Nest by Kozlovski-Shen-van Strien; (*)the Covering Lemma by Kahn-Lyubich; (*)the QC-Criterion, the Spreading Principle. 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 of the following results by Kozlovski-van Strien for non-renormalizable dynamically natural box mappings: (*)puzzle pieces shrink to points; (*)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 Lattes maps in this setting. -We prove a version of Mane's Theorem for complex box mappings concerning expansion along orbits of points that avoid a neighborhood of the set of critical points.

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arXiv2021

Revealing dynamics, communities and criticality from data

Complex systems such as ecological communities and neuron networks are essential parts of our everyday lives. These systems are composed of units which interact through intricate networks. The ability to predict sudden changes in the dynamics of these networks, known as critical transitions, from data is important to avert disastrous consequences of major disruptions. Predicting such changes is a major challenge as it requires forecasting the behaviour for parameter ranges for which no data on the system is available. We address this issue for networks with weak individual interactions and chaotic local dynamics. We do this by building a model network, termed an {\em effective network}, consisting of the underlying local dynamics and a statistical description of their interactions. We show that behaviour of such networks can be decomposed in terms of an emergent deterministic component and a {\em fluctuation} term. Traditionally, such fluctuations are filtered out. However, as we show, they are key to accessing the interaction structure.

nlin.AO↗
arXiv2020

Asymmetric unimodal maps with non-universal period-doubling scaling laws

We consider a family of strongly-asymmetric unimodal maps $\{f_t\}_{t\in [0,1]}$ of the form $f_t=t\cdot f$ where $f\colon [0,1]\to [0,1]$ is unimodal, $f(0)=f(1)=0$, $f(c)=1$ is of the form and $$f(x)=\left\{ \begin{array}{ll} 1-K_-|x-c|+o(|x-c|)& \mbox{ for }x c, \end{array}\right. $$ where we assume that $β>1$. We show that such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the $2^\infty$ map turn out to be super-exponential and non-universal (i.e. to depend on the map) and the scaling-law is different for odd and even steps of the renormalization. The conjugacy between the attracting Cantor sets of two such maps is smooth if and only if some invariant is satisfied. We also show that the Feigenbaum-Coullet-Tresser map does not have wandering intervals, but surprisingly we were only able to prove this using our rather detailed scaling results.

math.DS↗
arXiv2020

Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps

In this paper we will develop a general approach which shows that generalized "critical relations" of families of locally defined holomorphic maps on the complex plane unfold transversally. The main idea is to define a transfer operator, which is a local analogue of the Thurston pullback operator, using holomorphic motions. Assuming a so-called lifting property is satisfied, we obtain information about the spectrum of this transfer operator and thus about transversality. An important new feature of our method is that it is not global: the maps we consider are only required to be defined and holomorphic on a neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide class of one-parameter families of interval and circle maps, for example for maps with flat critical points, but also for maps with complex analytic extensions such as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1}, for real maps we obtain {\em positive} transversality (where $>0$ holds instead of just $\ne 0$), and thus monotonicity of entropy for these families, and also (as an easy application) for the real quadratic family. This method additionally gives results for unimodal families of the form $x\mapsto |x|^\ell+c$ for $\ell>1$ not necessarily an even integer and $c$ real.

math.DS↗
arXiv2019

Effective networks: a model to predict network structure and critical transitions from datasets

Real-world complex systems such as ecological communities and neuron networks are essential parts of our everyday lives. These systems are composed of units which interact through intricate networks. The ability to predict sudden changes in network behaviour, known as critical transitions, from data is important to avert disastrous consequences of major disruptions. Predicting such changes is a major challenge as it requires forecasting the behaviour for parameter ranges for which no data on the system is available. In this paper, we address this issue for networks with weak individual interactions and chaotic local dynamics. We do this by building a model network, termed an effective network, consisting of the underlying local dynamics at each node and a statistical description of their interactions. We illustrate this approach by reconstructing the dynamics and structure of realistic neuronal interaction networks of the cat cerebral cortex. We reconstruct the community structure by analysing the stochastic fluctuations generated by the network and predict critical transitions for coupling parameters outside the observed range.

nlin.AO↗
arXiv2018

Quasisymmetric rigidity in one-dimensional dynamics

In the late 1980's Sullivan initiated a programme to prove quasisymmetric rigidity in one-dimensional dynamics: interval or circle maps that are topologically conjugate are quasisymmetrically conjugate (provided some obvious necessary assumptions are satisfied). The aim of this paper is to conclude this programme in a natural class of $C^3$ mappings. Examples of such rigidity were established previously, but not, for example, for real polynomials with non-real critical points. Our results are also new for analytic mappings. The main new ingredients of the proof in the real analytic case are (i) the existence of infinitely many (complex) domains associated to its complex analytic extension so that these domains and their ranges are compatible, (ii) a methodology for showing that combinatorially equivalent complex box mappings are qc conjugate, (iii) a methodology for constructing qc conjugacies in the presence of parabolic periodic points. For a $C^3$ mapping, the dilatation of a high iterate of any complex extension of the real map will in general be unbounded. To deal with this, we introduce dynamically defined $qc\backslash bg$ $partitions$, where the appropriate mapping has bounded quasiconformal dilatation, except on sets with "bounded geometry". To obtain such a partition we prove that we have very good geometric control for infinitely many dynamically defined domains. Some of these results are new even for real polynomials, and in fact an important sequence of domains turn out to be quasidiscs. This technology also gives a new method for dealing with the infinitely renormalizable case. We will briefly also discuss why quasisymmetric rigidity is such a useful property in one-dimensional dynamics.

math.DS↗
arXiv2018

Dynamics of asymptotically holomorphic polynomial-like maps

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally as deep renormalizations of asymptotically holomorphic extensions of $C^r$ ($r>3$) unimodal maps that are infinitely renormalizable of bounded type. Here we prove a version of the Fatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. In particular, these maps do not have wandering domains and their Julia sets are locally connected.

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