## Publications

91 results found

van Strien S, De Faria E, Clark T, 2022, Asymptotically holomorphic methods for infinitely renormalizable Cr unimodal maps, *Ergodic Theory and Dynamical Systems*, ISSN: 0143-3857

The purpose of this paper is to initiate a theory concerning the dynamics of asymptotically holomorphic polynomial-like maps. Our maps arise naturally asdeep renormalizations of asymptotically holomorphic extensions of Cr(r > 3) unimodalmaps that are infinitely renormalizable of bounded type. Here we prove a version of theFatou-Julia-Sullivan theorem and a topological straightening theorem in this setting. Inparticular, these maps do not have wandering domains and their Julia sets are locallyconnected.

Clark T, Drach K, Kozlovski O, et al., 2022, The dynamics of complex box mappings, Publisher: Springer

In holomorphic dynamics, complex box mappings arise as first return maps towell-chosen domains. They are a generalization of polynomial-like mapping,where the domain of the return map can have infinitely many components. Theyturned out to be extremely useful in tackling diverse problems. The purpose ofthis paper is: -To illustrate some pathologies that can occur when a complex box mapping isnot induced by a globally defined map and when its domain has infinitely manycomponents, and to give conditions to avoid these issues. -To show that once one has a box mapping for a rational map, these conditionscan be assumed to hold in a very natural setting. Thus we call such complex boxmappings dynamically natural. -Many results in holomorphic dynamics rely on an interplay betweencombinatorial and analytic techniques: (*)the Enhanced Nest byKozlovski-Shen-van Strien; (*)the Covering Lemma by Kahn-Lyubich; (*)theQC-Criterion, the Spreading Principle. The purpose of this paper is to makethese tools more accessible so that they can be used as a 'black box', so onedoes not have to redo the proofs in new settings. -To give an intuitive, but also rather detailed, outline of the proof of thefollowing results by Kozlovski-van Strien for non-renormalizable dynamicallynatural box mappings: (*)puzzle pieces shrink to points; (*)topologicallyconjugate non-renormalizable polynomials and box mappings are quasiconformallyconjugate. -We prove the fundamental ergodic properties for dynamically natural boxmappings. This leads to some necessary conditions for when such a box mappingsupports a measurable invariant line field on its filled Julia set. Thesemappings are the analogues of Lattes maps in this setting. -We prove a version of Mane's Theorem for complex box mappings concerningexpansion along orbits of points that avoid a neighborhood of the set ofcritical points.

Clark T, Drach K, Kozlovski O,
et al., 2022, Correction to: The Dynamics of Complex Box Mappings (Arnold Mathematical Journal, (2022), 8, 2, (319-410), 10.1007/s40598-022-00200-7), *Arnold Mathematical Journal*, ISSN: 2199-6792

Dear readers, The typesetters introduced inconsistencies in an earlier unpublished version of the paper “The Dynamics of Complex Box Mappings”. Unfortunately, this unpublished version appeared on the journal’s website. Although this version no longer is accessible, if you downloaded this earlier version and noticed some inconsistencies, then please download the current version of the paper. Yours sincerely, Trevor Clark, Kostiantyn Drach, Oleg Kozlovski and Sebastian van Strien.

Clark T, Drach K, Kozlovski O,
et al., 2022, Correction to: The Dynamics of Complex Box Mappings (Arnold Mathematical Journal, (2022), 8, 2, (319-410), 10.1007/s40598-022-00200-7), *Arnold Mathematical Journal*, ISSN: 2199-6792

Dear readers, Unfortunately, the original article was published without the author’s final correction requests. This has since been corrected. Please excuse this error. “The original article has been corrected”.

Kozlovski O, van Strien S, 2020, Asymmetric unimodal maps with non-universal period-doubling scaling laws, *Communications in Mathematical Physics*, Vol: 379, Pages: 103-143, ISSN: 0010-3616

We consider a family of strongly-asymmetric unimodal maps {ft}t∈[0,1] of the form ft=t⋅f where f:[0,1]→[0,1] is unimodal, f(0)=f(1)=0, f(c)=1 is of the form andf(x)={1−K−|x−c|+o(|x−c|) for x<c, 1−K+|x−c|β+o(|x−c|β) for x>c,where we assume that β>1. We show that such a family contains a Feigenbaum–Coullet–Tresser 2∞ map, and develop a renormalization theory for these maps. The scalings of the renormalization intervals of the 2∞ 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.

Levin G, Shen W, van Strien S, 2020, Transversality in the setting of hyperbolic and parabolic maps, *Journal d'Analyse Mathematique*, Vol: 141, Pages: 247-284, ISSN: 0021-7670

In this paper we consider families of holomorphic maps defined on subsets of the complex plane, and show that the technique developed in [24] 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.

Eroglu D, Tanzi M, van Strien S,
et al., 2020, Revealing dynamics, communities and criticality from data, *Physical Review X*, Vol: 10, ISSN: 2160-3308

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 {}, 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 {} term. Traditionally, such fluctuations are filtered out. However, as we show, they are key to accessing the interaction structure. { We illustrate this approach on synthetic time-series of realistic neuronal interaction networks of the cat cerebral cortex and on experimental multivariate data of optoelectronic oscillators. } 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.

van Strien S, Pereira T, Tanzi M, 2020, Heterogeneously coupled maps: hub dynamics and emergence across connectivity layers, *Journal of the European Mathematical Society*, Vol: 22, Pages: 2183-2252, ISSN: 1435-9855

The aim of this paper is to rigorously study the dynamics of Heterogeneously Coupled Maps (HCM). Such systems are determined by a network with heterogeneous degrees. Some nodes, called hubs, are very well connected while most nodes interact with few others. The local dynamics on each node is chaotic, coupled with other nodes according to the network structure. Such high-dimensional systems are hard to understand in full, nevertheless we are able to describe the system over exponentially large time scales. In particular, we show that the dynamics of hub nodes can be very well approximated by a low-dimensional system. This allows us to establish the emergence of macroscopic behaviour such as coherence of dynamics among hubs of the same connectivity layer (i.e. with the same number of connections), and chaotic behaviour of the poorly connected nodes. The HCM we study provide a paradigm to explain why and how the dynamics of the network can change across layers.

Levin G, Shen W, Strien SV, 2020, Positive Transversality via transfer operators and holomorphic motions with applications to monotonicity for interval maps, *Nonlinearity*, Vol: 33, Pages: 1-43, ISSN: 0951-7715

In this paper we will develop a general approach which shows that generalized"critical relations" of families of locally defined holomorphic maps on thecomplex plane unfold transversally. The main idea is to define a transferoperator, which is a local analogue of the Thurston pullback operator, usingholomorphic motions. Assuming a so-called lifting property is satisfied, weobtain information about the spectrum of this transfer operator and thus abouttransversality. An important new feature of our method is that it is notglobal: the maps we consider are only required to be defined and holomorphic ona neighbourhood of some finite set. We will illustrate this method by obtaining transversality for a wide classof one-parameter families of interval and circle maps, for example for mapswith flat critical points, but also for maps with complex analytic extensionssuch as certain polynomial-like maps. As in Tsujii's approach \cite{Tsu0,Tsu1},for real maps we obtain {\em positive} transversality (where $>0$ holds insteadof 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.

Levin G, Shen W, van Strien S, 2019, Transversality for critical relations of families of rational maps: an elementary proof, New Trends in One-Dimensional Dynamics 2016, Publisher: Springer Nature, Pages: 201-220, ISSN: 2194-1017

In this paper we will give a short and elementary proofthat critical relations unfold transversally in the space of rationalmaps.

Kozlovski O, Strien SV, 2019, 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 $$f(x)=\left\{ \begin{array}{ll}1-K_-|x-c|+o(|x-c|)& \mbox{ for }x<c, \\ 1-K_+|x-c|^\beta + o(|x-c|^\beta)&\mbox{ for }x>c, \end{array}\right. $$ where we assume that $\beta>1$. We showthat such a family contains a Feigenbaum-Coullet-Tresser $2^\infty$ map, anddevelop a renormalization theory for these maps. The scalings of therenormalization intervals of the $2^\infty$ map turn out to besuper-exponential and non-universal (i.e. to depend on the map) and thescaling-law is different for odd and even steps of the renormalization. Theconjugacy between the attracting Cantor sets of two such maps is smooth if andonly if some invariant is satisfied. We also show that theFeigenbaum-Coullet-Tresser map does not have wandering intervals, butsurprisingly we were only able to prove this using our rather detailed scalingresults.

Tanzi M, Pereira T, van Strien S, 2019, Robustness of ergodic properties of non-autonomous piecewise expanding maps, *Ergodic Theory and Dynamical Systems*, Vol: 39, Pages: 1121-1152, ISSN: 0143-3857

Recently, there has been an increasing interest in non-autonomous composition of perturbed hyperbolic systems: composing perturbations of a given hyperbolic map results in statistical behaviour close to that of . We show this fact in the case of piecewise regular expanding maps. In particular, we impose conditions on perturbations of this class of maps that include situations slightly more general than what has been considered so far, and prove that these are stochastically stable in the usual sense. We then prove that the evolution of a given distribution of mass under composition of time-dependent perturbations (arbitrarily—rather than randomly—chosen at each step) close to a given map remains close to the invariant mass distribution of . Moreover, for almost every point, Birkhoff averages along trajectories do not fluctuate wildly. This result complements recent results on memory loss for non-autonomous dynamical systems.

Levin G, Shen W, Strien SV, 2019, Transversality in the setting of hyperbolic and parabolic maps

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

Clark T, van Strien S, Trejo S, 2017, Complex box bounds for real maps, *Communications in Mathematical Physics*, Vol: 355, Pages: 1001-1119, ISSN: 0010-3616

In this paper we prove complex bounds, also referred to as a priori bounds,for real analytic (and even C3) interval maps. This means that we associate tosuch a map a complex box mapping (which provides a kind of Markov structure),together with universal geometric bounds on the shape of the domains. Suchbounds show that the first return maps to these domains are well-controlled,and consequently form one of the corner stones in many recent results onone-dimensional dynamics: renormalisation theory, rigidity results, density ofhyperbolicity, local connectivity.

Hanssmann H, Hoveijn I, van Strien S,
et al., 2016, Preface, *Indagationes Mathematicae*, Vol: 27, Pages: 1029-1029, ISSN: 1872-6100

Grines VZ, Pochinka OV, Van Strien S, 2016, On 2-Diffeomorphisms with One-Dimensional Basic Sets and a Finite Number of Moduli, *MOSCOW MATHEMATICAL JOURNAL*, Vol: 16, Pages: 727-749, ISSN: 1609-3321

This paper is a step towards the complete topological classification of Ω-stable diffeomorphisms on an orientable closed surface, aiming to give necessary and sufficient conditions for two such diffeomorphisms to be topologically conjugate without assuming that the diffeomorphisms are necessarily close to each other. In this paper we will establish such a classification within a certain class Ψ of Ω-stable diffeomorphisms defined below. To determine whether two diffeomorphisms from this class Ψ are topologically conjugate, we give (i) an algebraic description of the dynamics on their non-trivial basic sets, (ii) a geometric description of how invariant manifolds intersect, and (iii) define numerical invariants, called moduli, associated to orbits of tangency of stable and unstable manifolds of saddle periodic orbits. This description determines the scheme of a diffeomorphism, and we will show that two diffeomorphisms from Ψ are topologically conjugate if and only if their schemes agree.

Rempe L, van Strien S, 2015, Density of hyperbolicity for classes of real transcendental entire functions and circle maps, *Duke Mathematical Journal*, Vol: 164, Pages: 1079-1137, ISSN: 0012-7094

We prove density of hyperbolicity in spaces of (i) real transcendental entirefunctions, bounded on the real line, whose singular set is finite and real and(ii) transcendental self-maps of the punctured plane which preserve the circleand whose singular set (apart from zero and infinity) is contained in thecircle. In particular, we prove density of hyperbolicity in the famous Arnol'dfamily of circle maps and its generalizations, and solve a number of other openproblems for these functions, including three conjectures by de Melo, Salomaoand Vargas. We also prove density of (real) hyperbolicity for certain families as in (i)but without the boundedness condition; in particular our results apply when thefunction in question has only finitely many critical points and asymptoticsingularities.

Bruin H, van Strien S, 2015, Monotonicity of entropy for real multimodal maps, *Journal of the American Mathematical Society*, Vol: 28, Pages: 1-61, ISSN: 0894-0347

In 1992, Milnor posed the Monotonicity Conjecture that within a family of real multimodal polynomial interval maps with only real critical points, the isentropes, i.e., the sets of parameters for which the topological entropy is constant, are connected. This conjecture was already proved in the mid-1980s for quadratic maps by a number of different methods, see A. Douady (1993, 1995), A. Douady and J.H. Hubbard (1984, 1985), W. de Melo and S. van Strein (1993), J. Milnor and W. Thurston (1986, 1988), and M. Tsujii (2000). In 2000, Milnor and Tresser, provided a proof for the case of cubic maps. In this paper we will prove the general case of this 20 year old conjecture.

Ostrovski G, van Strien S, 2014, Payoff Performance of Fictitious Play, *Journal of Dynamics and Games*, ISSN: 2164-6066

We investigate how well continuous-time fictitious play in two-player gamesperforms in terms of average payoff, particularly compared to Nash equilibriumpayoff. We show that in many games, fictitious play outperforms Nashequilibrium on average or even at all times, and moreover that any game islinearly equivalent to one in which this is the case. Conversely, we provideconditions under which Nash equilibrium payoff dominates fictitious playpayoff. A key step in our analysis is to show that fictitious play dynamicsasymptotically converges the set of coarse correlated equilibria (a fact whichis implicit in the literature).

van Strien S, 2014, Milnor’s Conjecture on Monotonicity of Topological Entropy: results and questions, Frontiers in Complex Dynamics:In Celebration of John Milnor's 80th Birthday, Editors: Bonifant, Lyubich, Sutherland, Publisher: Princeton University Press, ISBN: 9780691159294

This note discusses Milnor’s conjecture on monotonicity of entropy and gives a short exposition of the ideas used in its proof which was obtained in joint work with Henk Bruin, see [BvS09]. At the end of this note we explore some related conjectures and questions.

Shen W, van Strien S, 2014, Recent developments in interval dynamics, International Congress of Mathematicians (ICM), Publisher: Kyung Moon SA Co Ltd, Pages: 699-719

Shen W, van Strien S, 2013, On stochastic stability of expanding circle maps with neutral fixed points, *DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL*, Vol: 28, Pages: 423-452, ISSN: 1468-9367

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- Citations: 11

Bruin H, van Strien S, 2013, On the structure of isentropes of polynomial maps, *DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL*, Vol: 28, Pages: 381-392, ISSN: 1468-9367

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- Citations: 7

Pereira T, van Strien S, Lamb JSW, 2013, Dynamics of Coupled Maps in Heterogeneous Random Networks

Eremenko A, van Strien S, 2011, RATIONAL MAPS WITH REAL MULTIPLIERS, *TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY*, Vol: 363, Pages: 6453-6463, ISSN: 0002-9947

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- Citations: 18

Broer H, van Strien S, 2011, Preface, *INDAGATIONES MATHEMATICAE-NEW SERIES*, Vol: 22, Pages: 135-135, ISSN: 0019-3577

Broer H, van Strien S, 2011, In Memoriam, Floris Takens 1940-2010, *INDAGATIONES MATHEMATICAE-NEW SERIES*, Vol: 22, Pages: 137-143, ISSN: 0019-3577

Eremenko A, van Strien S, 2011, Rational maps with real multipliers, *Transactions of the American Mathematical Society*, Vol: 363, Pages: 6453-6463, ISSN: 0002-9947

Let $ f$ be a rational function such that the multipliers of all repelling periodic points are real. We prove that the Julia set of such a function belongs to a circle. Combining this with a result of Fatou we conclude that whenever $ J(f)$ belongs to a smooth curve, it also belongs to a circle. Then we discuss rational functions whose Julia sets belong to a circle.

Strien SV, 2011, A new class of Hamiltonian flows with random-walk behavior originating from zero-sum games and Fictitious Play, *Nonlinearity*, Vol: 24, Pages: 1715-1742, ISSN: 0951-7715

In this paper we introduce Hamiltonian dynamics, inspired by zero-sum games (best response and fictitious play dynamics). The Hamiltonian functions we consider are continuous and piecewise affine (and of a very simple form). It follows that the corresponding Hamiltonian vector fields are discontinuous and multi-valued. Differential equations with discontinuities along a hyperplane are often called 'Filippov systems', and there is a large literature on such systems, see for example (di Bernardo et al 2008 Theory and applications Piecewise-Smooth Dynamical Systems (Applied Mathematical Sciences vol 163) (London: Springer); Kunze 2000 Non-Smooth Dynamical Systems (Lecture Notes in Mathematics vol 1744) (Berlin: Springer); Leine and Nijmeijer 2004 Dynamics and Bifurcations of Non-smooth Mechanical Systems (Lecture Notes in Applied and Computational Mechanics vol 18) (Berlin: Springer)). The special feature of the systems we consider here is that they have discontinuities along a large number of intersecting hyperplanes. Nevertheless, somewhat surprisingly, the flow corresponding to such a vector field exists, is unique and continuous. We believe that these vector fields deserve attention, because it turns out that the resulting dynamics are rather different from those found in more classically defined Hamiltonian dynamics. The vector field is extremely simple: outside codimension-one hyperplanes it is piecewise constant and so the flow phgrt piecewise a translation (without stationary points). Even so, the dynamics can be rather rich and complicated as a detailed study of specific examples show (see for example theorems 7.1 and 7.2 and also (Ostrovski and van Strien 2011 Regular Chaotic Dynf. 16 129–54)). In the last two sections of the paper we give some applications to game theory, and finish with posing a version of the Palis conjecture in the context of the class of non-smooth systems studied in this paper.

van Strien S, Sparrow C, 2011, Dynamics associated to games (fictitious play) with chaotic behavior, Dynamics, Games and Science I, Editors: Peixoto, Pinto, Rand, Publisher: Springer, ISBN: 9783642114557

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