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Pierre Berger

Publications and source records attributed to Pierre Berger.

At least 19 recordsLinked to original sources
arXiv2025

On Kolmogorov-typical properties of symplectic dynamics

We propose a general framework, within which we prove that several properties, such as the fast growth of the number of periodic points, the universality, and the high emergence, hold true for every parameter value for a generic finite-parameter family of symplectic diffeomorphisms displaying an elliptic point.

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arXiv2024

Analytic pseudo-rotations II: a principle for spheres, disks and annuli

We construct analytic surface symplectomorphisms with unstable elliptic fixed points; this solves a problem of Birkhoff (1927). More precisely, we construct analytic symplectomorphisms of the sphere and of the disk which are transitive, with respectively only 2 and 1 periodic points. This also solves problems of proposed by Herman (1998), Fayad-Katok (2004) and Fayad-Krikorian (2018). To establish these results, we introduce a principle that enables to realize, by an analytic symplectomorphism, properties which are $C^0$-realizable by the approximation by conjugacy method of Anosov-Katok.

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arXiv2022

Generators of groups of Hamitonian maps

We prove that analytic Hamiltonian dynamics on tori, annuli, or Euclidean space can be approximated by a composition of nonlinear shear maps where each of the shears depends only on the position or only on the momentum.

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arXiv2022

Every diffeomorphism is a total renormalization of a close to identity map

For any $1\le r\le \infty$, we show that every diffeomorphism of a manifold of the form $\mathbb{R}/\mathbb{Z} \times M$ is a total renormalization of a $C^r$-close to identity map. In other words, for every diffeomorphism $f$ of $\mathbb{R}/\mathbb{Z} \times M$, there exists a map $g$ arbitrarily close to identity such that the first return map of $g$ to a domain is conjugate to $f$ and moreover the orbit of this domain is equal to $\mathbb{R}/\mathbb{Z} \times M$. This enables us to localize nearby the identity the existence of many properties in dynamical systems, such as being Bernoulli for a smooth volume form.

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arXiv2022

Parametric linearization of skew products

We establish a linearization criterion for skew products of contractions in any dimension. We prove their smooth or holomorphic parameter dependence. In the smooth setting, we use the language of tame Fréchet spaces. We apply our result to the linearization of totally projectively expanding Cantor sets.

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arXiv2022

Analytic pseudo-rotations

We construct analytic symplectomorphisms of the cylinder or the sphere with zero or exactly two periodic points and which are not conjugated to a rotation. In the case of the cylinder, we show that these symplectomorphisms can be chosen ergodic or to the contrary with local emergence of maximal order. In particular, this disproves a conjecture of Birkhoff (1941) and solve a problem of Herman (1998). One aspect of the proof provides a new approximation theorem, it enables in particular to implement the Anosov-Katok scheme in new analytic settings.

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arXiv2022

Emergence of wandering stable components

We prove the existence of a locally dense set of real polynomial automorphisms of C 2 displaying a wandering Fatou component; in particular this solves the problem of their existence, reported by Bedford and Smillie in 1991. These Fatou components have non-empty real trace and their statistical behavior is historical with high emergence. The proof is based on a geometric model for parameter families of surface real mappings. At a dense set of parameters, we show that the dynamics of the model displays a historical, high emergent, stable domain. We show that this model can be embedded into families of H{é}non maps of explicit degree and also in an open and dense set of 5-parameter C r-families of surface diffeomorphisms in the Newhouse domain, for every 2 $\le$ r $\le$ $\infty$ and r = $ω$. This implies a complement of the work of Kiriki and Soma (2017), a proof of the last Taken's problem in the C $\infty$ and C $ω$-case. The main difficulty is that here perturbations are done only along finite-dimensional parameter families. The proof is based on the multi-renormalization introduced in [Ber18].

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arXiv2022

Steady Euler flows on $\mathbb{R}^3$ with wild and universal dynamics

Understanding complexity in fluid mechanics is a major problem that has attracted the attention of physicists and mathematicians during the last decades. Using the concept of renormalization in dynamics, we show the existence of a locally dense set $\mathscr G$ of stationary solutions to the Euler equations in $\mathbb R^3$ such that each vector field $X\in \mathscr G$ is universal in the sense that any area preserving diffeomorphism of the disk can be approximated (with arbitrary precision) by the Poincar\'e map of $X$ at some transverse section. We remark that this universality is approximate but occurs at all scales. In particular, our results establish that a steady Euler flow may exhibit any conservative finite codimensional dynamical phenomenon; this includes the existence of horseshoes accumulated by elliptic islands, increasing union of horseshoes of Hausdorff dimension $3$ or homoclinic tangencies of arbitrarily high multiplicity. The steady solutions we construct are Beltrami fields with sharp decay at infinity. To prove these results we introduce new perturbation methods in the context of Beltrami fields that allow us to import deep techniques from bifurcation theory: the Gonchenko-Shilnikov-Turaev universality theory and the Newhouse and Duarte theorems on the geometry of wild hyperbolic sets. These perturbation methods rely on two tools from linear PDEs: global approximation and Cauchy-Kovalevskaya theorems. These results imply a strong version of V.I. Arnold's vision on the complexity of Beltrami fields in Euclidean space.

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arXiv2021

On Emergence and Complexity of Ergodic Decompositions

A concept of emergence was recently introduced in the paper [Berger] in order to quantify the richness of possible statistical behaviors of orbits of a given dynamical system. In this paper, we develop this concept and provide several new definitions, results, and examples. We introduce the notion of topological emergence of a dynamical system, which essentially evaluates how big the set of all its ergodic probability measures is. On the other hand, the metric emergence of a particular reference measure (usually Lebesgue) quantifies how non-ergodic this measure is. We prove fundamental properties of these two emergences, relating them with classical concepts such as Kolmogorov's $ε$-entropy of metric spaces and quantization of measures. We also relate the two types of emergences by means of a variational principle. Furthermore, we provide several examples of dynamics with high emergence. First, we show that the topological emergence of some standard classes of hyperbolic dynamical systems is essentially the maximal one allowed by the ambient. Secondly, we construct examples of smooth area-preserving diffeomorphisms that are extremely non-ergodic in the sense that the metric emergence of the Lebesgue measure is essentially maximal. These examples confirm that super-polynomial emergence indeed exists, as conjectured in the paper [Berger]. Finally, we prove that such examples are locally generic among smooth diffeomorphisms.

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arXiv2021

Germ-typicality of the coexistence of infinitely many sinks

In the spirit of Kolmogorov typicality, we introduce the notion of germ-typicality: in a space of dynamics, it encompass all these phenomena that occur for a dense and open subset of parameters of any generic parametrized family of systems. For any $2\le r<\infty$, we prove that the Newhouse phenomenon (the coexistence of infinitely many sinks) is locally $C^r$-germ-typical, nearby a dissipative bicycle: a dissipative homoclinic tangency linked to a special heterodimensional cycle. During the proof we show a result of independent interest: the stabilization of some heterodimensional cycles for any regularity class $r\in \{1, \dots, \infty\}\cup \{ω\}$ by introducing a new renormalization scheme. We also continue the study of the paradynamics done in [Be15,Be17,BCP16] and prove that parablenders appear by unfolding some heterodimensional cycles.

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arXiv2021

Generic family displaying robustly a fast growth of the number of periodic points

For any $ 2 \le r \le \infty$, $n\ge2$, we prove the existence of an open set $U$ of $C^r$-self-mappings of any $n$-manifold so that a generic map $f$ in $U$ displays a fast growth of the number of periodic points: the number of its $n$-periodic points grows as fast as asked. This complements the works of Martens-de Melo-van Strien, Kaloshin, Bonatti-D\' iaz-Fisher and Turaev, to give a full answer to questions asked by Smale in 1967, Bowen in 1978 and Arnold in 1989, for any manifold of any dimension and for any smoothness. Furthermore for any $1\le r<\infty$ and any $k\ge 0$, we prove the existence of an open set $\hat U$ of $k$-parameter families in $U$ so that for a generic $(f_p)_p\in \hat U$, for every $\|p\|\le 1$, the map $f_p$ displays a fast growth of the number of periodic points. This gives a negative answer to a problem asked by Arnold in 1992 in the finitely smooth case.

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arXiv2019

Complexities of differentiable dynamical systems

We define the notion of localizable property for a dynamical system. Then we survey three properties of complexity and relate how they are known to be typical among differentiable dynamical systems. These notions are the fast growth of the number of periodic points, the positive entropy and the high emergence. We finally propose a dictionary between the previously explained theory on entropy and the ongoing one on emergence.

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arXiv2019

Strong regularity

This is an introduction of a book called "strong regularity", to appear at Astérisque, containing: 1) Yoccoz' proof of Jakobson theorem www.college-de-france.fr/media/jean-christophe-yoccoz/UPL7416254474776698194_Jakobson_jcy.pdf 2) Berger's proof of the abundance of non-uniformly hyperbolic Hénon like endomorphisms arxiv.org/abs/0903.1473 It gives an overview of the main examples and conjectures of non-uniformly hyperbolic set for low dimensional dynamical systems. It compares the proofs of parameter selections based on the concept of binding with those based on the one of strong regularity.

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arXiv2018

Abundance of non-uniformly hyperbolic Hénon like endomorphisms

For every $C^2$-small function $B$, we prove that the map $(x,y)\mapsto (x^2+a,0)+B(x,y,a)$ leaves invariant a physical, SRB probability measure, for a set of parameters $a$ of positive Lebesgue measure. When the perturbation $B$ is zero, this is the Jakobson Theorem; when the perturbation is a small constant times $(0,x)$, this is the celebrated Benedicks-Carleson Theorem. In particular, a new proof of the last theorem is given, based on devellopment of the combinatorial formalism of the Yoccoz puzzles. By adding new geometrical and combinatorial ingredients, and restructuring classic analytical ideas, we are able to carry out our proof in the $C^2$-topology, even when the underlying dynamics are given by endomorphisms.

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arXiv2018

Zoology in the Hénon family: twin babies and Milnor's swallows

We study $C^{d,r}$-Hénon-like families $(f_{a\, b})_{a\, b}$ with two parameters $(a,b)\in \mathbb R^2$. We show the existence of an open set of parameters $(a,b)\in \mathcal D$, so that a renormalization chart conjugates an iterate of $f_{a\, b}$ to a perturbation of $(x,y)\mapsto ((x^2+c_1)^2+c_2,0)$. We prove that the map $(a,b)\in \mathcal D\mapsto (c_1,c_2)$ is a $C^d$-diffeomorphism; as first numerically conjectured by Milnor in 1992. Furthermore, we show the existence of an open set of parameters $(a,b)$ so that $f_{a\, b}$ displays exactly two different renormalized Hénon-like maps whose basins union attracts Lebesgue a.e. point with bounded forward orbit. A great freedom in the choice of the renormalized parameters enables us to deduce in particular the existence of a (unperturbed) Hénon map with exactly $2$ attracting cycles (an answer to a Question by Lyubich). The proof is based on a generalization of puzzle pieces for Hénon-like maps, and on a generalization of both the affine-like formalism of Palis-Yoccoz and the cross map of Shilnikov. The distortion bounds enable us to define (for the first time) $C^{r}$ and $C^{d,r}$-renormalizations and multi-renormalizations with bounds on all the derivatives.

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