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Marco Martens

Publications and source records attributed to Marco Martens.

At least 19 recordsLinked to original sources
arXiv2026

Equation free data-driven modelling of chaotic processes

We introduce a method for constructing predictive models of non cyclic physical processes directly from time-series data, without assuming an underlying differential equation. The observations define a discrete evolution rule whose recurrent behaviour captures the essential dynamics of the process. Analysing this behaviour across multiple geometric scales leads to probabilistic models in the form of Markov chains. Hyperbolicity criteria identify when these models provide a consistent statistical description of the data. The method is inspired by, and illustrated through, the analysis of a biological imaging data set referred to as the Cell Process.

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arXiv2025

Non-Hyperbolic Chaotic Dynamics: Renormalization and More

In two-dimensional unfoldings of homoclinic tangencies, the parameter space contains codimension-1 laminations whose leaves consist of maps with invariant non-hyperbolic Cantor sets. These sets are wild and unstable in the sense of Newhouse and contain Collet-Eckmann points with dense orbits. Thus, wild and non-hyperbolic chaotic dynamics can coexist on a single invariant set, while persisting along codimension-1 manifolds. Even more strikingly, each leaf of the lamination contains a map with infinitely many sinks accumulating on the invariant Cantor set carrying the Collet-Eckmann dynamics. In particular, the occurrence of infinitely many sinks is compatible with this type of non-hyperbolic chaotic behavior, with both phenomena organized around the same invariant Cantor set. To analyze these phenomena, we introduce a generalized renormalization scheme for two-dimensional systems, which describes the dynamics at successive scales and reveals a finite-scale hyperbolic structure within these non-hyperbolic regimes.

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arXiv2020

Newhouse Laminations of polynomials on $\mathbb{C}^2$

It has been recently discovered that in smooth unfoldings of maps with a rank-one homoclinic tangency there are codimension two laminations of maps with infinitely many sinks. Indeed, these laminations, called Newhouse laminations, occur also in the holomorphic context. In the space of polynomials of $\mathbb{C}^2$, with bounded degree, there are Newhouse laminations.

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arXiv2019

Newhouse Laminations

Newhouse laminations occur in unfoldings of rank-one homoclinic tangencies. Namely, in these unfoldings, there exist codimension $2$ laminations of maps with infinitely many sinks which move simultaneously along the leaves. As consequence, in the space of real polynomial maps, there are examples of: Hénon maps, in any dimension, with infinitely many sinks, quadratic Hénon-like maps with infinitely many sinks and a period doubling attractor, quadratic Hénon-like maps with infinitely many sinks and a strange attractor, non trivial analytic families of polynomial maps with infinitely many sinks.

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arXiv2017

Infinitely Many Moduli of Stability at the Dissipative Boundary of Chaos

In the family of area-contracting Hénon-like maps with zero topological entropy we show that there are maps with infinitely many moduli of stability. Thus one cannot find all the possible topological types for non-chaotic area-contracting Hénon-like maps in a family with finitely many parameters. A similar result, but for the chaotic maps in the family, became part of the folklore a short time after Hénon used such maps to produce what was soon conjectured to be the first non-hyperbolic strange attractor in $\mathbb{R}^2$. Our proof uses recent results about infinitely renormalisable area-contracting Hénon-like maps; it suggests that the number of parameters needed to represent all possible topological types for area-contracting Hénon-like maps whose sets of periods of their periodic orbits are finite (and in particular are equal to $\{1,\, 2,\dots,\,2^{n-1}\}$ or an initial segment of this $n$-tuple) increases with the number of periods. In comparison, among $C^k$-embeddings of the 2-disk with $k\geq 1$, the maximal moduli number for non-chaotic but non area-contracting maps in the interior of the set of zero-entropy is infinite.

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arXiv2017

Instability of Renormalization

In the theory of renormalization for classical dynamical systems, e.g. unimodal maps and critical circle maps, topological conjugacy classes are stable manifolds of renormalization. Physically more realistic systems on the other hand may exhibit instability of renormalization within a topological class. This instability gives rise to new phenomena and opens up directions of inquiry that go beyond the classical theory. In phase space it leads to the coexistence phenomenon, i.e. there are systems whose attractor has bounded geometry but which are topologically conjugate to systems whose attractor has degenerate geometry; in parameter space it causes dimensional discrepancy, i.e. a topologically full family has too few dimensions to realize all possible geometric behavior.

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arXiv2017

The Rigidity Conjecture

A central question in dynamics is whether the topology of a system determines its geometry. This is known as rigidity. Under mild topological conditions rigidity holds for many classical cases, including: Kleinian groups, circle diffeomorphisms, unimodal interval maps, critical circle maps, and circle maps with a break point. More recent developments show that under similar topological conditions, rigidity does not hold for slightly more general systems. In this paper we state a conjecture which describes how topological classes are organized into rigidity classes.

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arXiv2015

Rigidity of critical circle maps

We prove that any two $C^4$ critical circle maps with the same irrational rotation number and the same odd criticality are conjugate to each other by a $C^1$ circle diffeomorphism. The conjugacy is $C^{1+α}$ for Lebesgue almost every rotation number.

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arXiv2014

Physical Measures for Infinitely Renormalizable Lorenz Maps

A physical measure on the attractor of a system describes the statistical behavior of typical orbits. An example occurs in unimodal dynamics. Namely, all infinitely renormalizable unimodal maps have a physical measure. For Lorenz dynamics, even in the simple case of infinitely renormalizable systems, the existence of physical measures is more delicate. In this article we construct examples of infinitely renormalizable Lorenz maps which do not have a physical measure. A priori bounds on the geometry play a crucial role in (unimodal) dynamics. There are infinitely renormalizable Lorenz maps which do not have a priori bounds. This phenomenon is related to the position of the critical point of the consecutive renormalizations. The crucial technical ingredient used to obtain these examples without a physical measure, is the control of the position of these critical points.

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arXiv2014

Rigidity for infinitely renormalizable area-preserving maps

The period doubling Cantor sets of strongly dissipative Henon-like maps with different average Jacobian are not smoothly conjugated. The Jacobian Rigidity Conjecture says that the period doubling Cantor sets of two-dimensional Henon-like maps with the same average Jacobian are smoothly conjugated. This conjecture is true for average Jacobian zero, e.g. the one-dimensional case. The other extreme case is when the maps preserve area, e.g. the average Jacobian is one. Indeed, the period doubling Cantor set of area-preserving maps in the universality class of the Eckmann-Koch-Wittwer renormalization fixed point are smoothly conjugated.

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arXiv2012

On the Hyperbolicity of Lorenz Renormalization

We consider infinitely renormalizable Lorenz maps with real critical exponent $α>1$ and combinatorial type which is monotone and satisfies a long return condition. For these combinatorial types we prove the existence of periodic points of the renormalization operator, and that each map in the limit set of renormalization has an associated unstable manifold. An unstable manifold defines a family of Lorenz maps and we prove that each infinitely renormalizable combinatorial type (satisfying the above conditions) has a unique representative within such a family. We also prove that each infinitely renormalizable map has no wandering intervals and that the closure of the forward orbits of its critical values is a Cantor attractor of measure zero.

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arXiv2011

Focal Rigidity of Flat Tori

Given a closed Riemannian manifold (M, g), there is a partition Σ_i of its tangent bundle TM called the focal decomposition. The sets Σ_i are closely associated to focusing of geodesics of (M, g), i.e. to the situation where there are exactly i geodesic arcs of the same length joining points p and q in M. In this note, we study the topological structure of the focal decomposition of a closed Riemannian manifold and its relation with the metric structure of the manifold. Our main result is that the flat n-tori are focally rigid, in the sense that if two flat tori are focally equivalent, then the tori are isometric up to rescaling.

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arXiv2011

Probabilistic Universality in two-dimensional Dynamics

In this paper we continue to explore infinitely renormalizable Hénon maps with small Jacobian. It was shown in [CLM] that contrary to the one-dimensional intuition, the Cantor attractor of such a map is non-rigid and the conjugacy with the one-dimensional Cantor attractor is at most 1/2-Hölder. Another formulation of this phenomenon is that the scaling structure of the Hénon Cantor attractor differs from its one-dimensional counterpart. However, in this paper we prove that the weight assigned by the canonical invariant measure to these bad spots tends to zero on microscopic scales. This phenomenon is called {\it Probabilistic Universality}. It implies, in particular, that the Hausdorff dimension of the canonical measure is universal. In this way, universality and rigidity phenomena of one-dimensional dynamics assume a probabilistic nature in the two-dimensional world.

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arXiv2010

Renormalisable Henon-like Maps and Unbounded Geometry

We show that given a one parameter family $F_b$ of strongly dissipative infinitely renormalisable Hénon-like maps, parametrised by a quantity called the `average Jacobian' $b$, the set of all parameters $b$ such that $F_b$ has a Cantor set with unbounded geometry has full Lebesgue measure.

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arXiv2009

Geometry of Polynomials and Root-Finding via Path-Lifting

Using the interplay between topological, combinatorial, and geometric properties of polynomials and analytic results (primarily the covering structure and distortion estimates), we analyze a path-lifting method for finding approximate zeros, similar to those studied by Smale, Shub, Kim, and others. Given any polynomial, this simple algorithm always converges to a root, except on a finite set of initial points lying on a circle of a given radius. Specifically, the algorithm we analyze consists of iterating $$z - \frac{f(z)-t_kf(z_0)}{f'(z)}$$ where the $t_k$ form a decreasing sequence of real numbers and $z_0$ is chosen on a circle containing all the roots. We show that the number of iterates required to locate an approximate zero of a polynomial $f$ depends only on $\log|f(z_0)/\rho_\zeta|$ (where $\rho_\zeta$ is the radius of convergence of the branch of $f^{-1}$ taking $0$ to a root $\zeta$) and the logarithm of the angle between $f(z_0)$ and certain critical values. Previous complexity results for related algorithms depend linearly on the reciprocals of these angles. Note that the complexity of the algorithm does not depend directly on the degree of $f$, but only on the geometry of the critical values. Furthermore, for any polynomial $f$ with distinct roots, the average number of steps required over all starting points taken on a circle containing all the roots is bounded by a constant times the average of $\log(1/\rho_\zeta)$. The average of $\log(1/\rho_\zeta)$ over all polynomials $f$ with $d$ roots in the unit disk is ${\mathcal{O}}({d})$. This algorithm readily generalizes to finding all roots of a polynomial (without deflation); doing so increases the complexity by a factor of at most $d$.

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arXiv2008

Renormalization in the Hénon family, II: The heteroclinic web

We study highly dissipative Hénon maps $$ F_{c,b}: (x,y) \mapsto (c-x^2-by, x) $$ with zero entropy. They form a region $Π$ in the parameter plane bounded on the left by the curve $W$ of infinitely renormalizable maps. We prove that Morse-Smale maps are dense in $Π$, but there exist infinitely many different topological types of such maps (even away from $W$). We also prove that in the infinitely renormalizable case, the average Jacobian $b_F$ on the attracting Cantor set $\OO_F$ is a topological invariant. These results come from the analysis of the heteroclinic web of the saddle periodic points based on the renormalization theory. Along these lines, we show that the unstable manifolds of the periodic points form a lamination outside $\OO_F$ if and only if there are no heteroclinic tangencies.

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arXiv1998

Invariant measures for typical quadratic maps

A sufficient geometrical condition for the existence of absolutely continuous invariant probability measures for S-unimodal maps will be discussed. The Lebesgue typical existence of such measures in the quadratic family will be a consequence.

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