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Jairo Bochi

Publications and source records attributed to Jairo Bochi.

At least 19 recordsLinked to original sources
arXiv2026

A Poincar\'e-Bendixson theorem for Bebutov shifts and applications to switched systems

We prove a version of the Poincar\'e-Bendixson theorem for certain classes of curves on the 2-sphere which are not required to be the trajectories of an underlying flow or semiflow on the sphere itself. Using this result we extend the Poincar\'e-Bendixson theorem to the context of continuous semiflows on compact subsets of the 2-sphere and the projective plane, give new sufficient conditions for the existence of periodic trajectories of certain low-dimensional affine control systems, and give a new criterion for the global uniform exponential stability of switched systems of homogeneous ODEs in dimension three. We prove in particular that periodic asymptotic stability implies global uniform exponential stability for real linear switched systems of dimension three and complex linear switched systems of dimension two. In combination with a recent result of the second author, this resolves a question of R. Shorten, F. Wirth, O. Mason, K. Wulff and C. King and resolves a natural analogue of the Lagarias-Wang finiteness conjecture in continuous time.

math.DS↗
arXiv2025

Removing quasiconformal orbits

We show that generic continuous linear cocycles over shifts and other zero-dimensional systems admit no quasiconformal orbits, thus providing a partial answer to a question of Nassiri, Rajabzadeh, and Reshadat. The proof relies on a new result about towers for homeomorphisms of zero-dimensional spaces, which may be of independent interest.

math.DS↗
arXiv2025

An isolated Lyapunov exponent

We construct a continuous linear cocycle over an expanding base dynamics for which the Lyapunov exponents of all ergodic invariant probability measures are small, except for one measure whose Lyapunov exponents are away from zero. The support of this distinguished measure is not a periodic orbit. In particular, our example violates the periodic approximation property.

math.DS↗
arXiv2025

On the distribution of the angle between Oseledets spaces

We study the distribution of the angles between Oseledets subspaces and their log-integrability, focusing on dimension $2$. For random i.i.d. products of matrices, we construct examples of probability measures on $\mathrm{GL}_2(\mathbb{R})$ with finite first moment where the Oseledets angle is not log-integrable. We also show that for probability measures with finite second moment the angle is always log-integrable. We then consider general measurable $\mathrm{GL}_2(\mathbb{R})$-cocycles over an arbitrary ergodic automorphism of a non-atomic Lebesgue space, proving that no integrability condition on the matrix distribution ensures log-integrability of the angle. In fact, the joint distribution of the Oseledets spaces can be chosen arbitrarily. A similar flexibility result for bounded cocycles holds under an unavoidable technical restriction.

math.DS↗
arXiv2024

Complete regularity of linear cocycles and the Baire category of the set of Lyapunov-Perron regular points

Given a continuous linear cocycle A over a homeomorphism f of a compact metric space X, we investigate its set R of Lyapunov-Perron regular points, that is, the collection of trajectories of f that obey the conclusions of the Multiplicative Ergodic Theorem. We obtain results roughly saying that the set R is of first Baire category (i.e., meager) in X, unless some rigid structure is present. In some settings, this rigid structure forces the Lyapunov exponents to be defined everywhere and to be independent of the point; that is what we call complete regularity.

math.DS↗
arXiv2024

Monochromatic nonuniform hyperbolicity

We construct examples of continuous $\mathrm{GL}(2,\mathbb{R})$-cocycles which are not uniformly hyperbolic despite having the same non-zero Lyapunov exponents with respect to all invariant measures. The base dynamics can be any non-trivial subshift of finite type. According to a theorem of DeWitt--Gogolev and Guysinsky, such cocycles cannot be H\"older-continuous. Our construction uses the nonuniformly hyperbolic cocycles discovered by Walters in 1984.

math.DS↗
arXiv2023

Spectrum maximizing products are not generically unique

It is widely believed that typical finite families of $d \times d$ matrices admit finite products that attain the joint spectral radius. This conjecture is supported by computational experiments and it naturally leads to the following question: are these spectrum maximizing products typically unique, up to cyclic permutations and powers? We answer this question negatively. As discovered by Horowitz around fifty years ago, there are products of matrices that always have the same spectral radius despite not being cyclic permutations of one another. We show that the simplest Horowitz products can be spectrum maximizing in a robust way; more precisely, we exhibit a small but nonempty open subset of pairs of $2 \times 2$ matrices $(A,B)$ for which the products $A^2 B A B^2$ and $B^2 A B A^2$ are both spectrum maximizing.

math.OC↗
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.

math.DS↗
arXiv2021

Flexibility of Lyapunov exponents

We outline the flexibility program in smooth dynamics, focusing on flexibility of Lyapunov exponents for volume-preserving diffeomorphisms. We prove flexibility results for Anosov diffeomorphisms admitting dominated splittings into one-dimensional bundles.

math.DS↗
arXiv2021

The Hal\'asz-Sz\'ekely Barycenter

We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Hal\'asz and Sz\'ekely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Hal\'asz-Sz\'ekely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Hal\'asz-Sz\'ekely barycenter of the corresponding distribution.

math.PR↗
arXiv2020

Anosov representations and dominated splittings

We provide a link between Anosov representations introduced by Labourie and dominated splitting of linear cocycles. This allows us to obtain equivalent characterizations for Anosov representations and to recover recent results due to Guéritaud-Guichard-Kassel-Wienhard and Kapovich-Leeb-Porti by different methods. We also give characterizations in terms of multicones and cone-types inspired in the work of Avila-Bochi-Yoccoz and Bochi-Gourmelon. Finally we provide a new proof of the higher rank Morse Lemma of Kapovich-Leeb-Porti.

math.GR↗
arXiv2019

Extremal Norms for Fiber Bunched Cocycles

In traditional Ergodic Optimization, one seeks to maximize Birkhoff averages. The most useful tool in this area is the celebrated Mañé Lemma, in its various forms. In this paper, we prove a non-commutative Mañé Lemma, suited to the problem of maximization of Lyapunov exponents of linear cocycles or, more generally, vector bundle automorphisms. More precisely, we provide conditions that ensure the existence of an extremal norm, that is, a Finsler norm with respect to which no vector can be expanded in a single iterate by a factor bigger than the maximal asymptotic expansion rate. These conditions are essentially irreducibility and sufficiently strong fiber bunching. Therefore we extend the classic concept of Barabanov norm, which is used in the study of the joint spectral radius. We obtain several consequences, including sufficient conditions for the existence of Lyapunov maximizing sets.

math.DS↗
arXiv2018

On the approximation of convex bodies by ellipses with respect to the symmetric difference metric

Given a centrally symmetric convex body $K \subset \mathbb{R}^d$ and a positive number $λ$, we consider, among all ellipsoids $E \subset \mathbb{R}^d$ of volume $λ$, those that best approximate $K$ with respect to the symmetric difference metric, or equivalently that maximize the volume of $E\cap K$: these are the maximal intersection (MI) ellipsoids introduced by Artstein-Avidan and Katzin. The question of uniqueness of MI ellipsoids (under the obviously necessary assumption that $λ$ is between the volumes of the John and the Loewner ellipsoids of $K$) is open in general. We provide a positive answer to this question in dimension $d=2$. Therefore we obtain a continuous $1$-parameter family of ellipses interpolating between the John and the Loewner ellipses of $K$. In order to prove uniqueness, we show that the area $I_K(E)$ of the intersection $K \cap E$ is a strictly quasiconcave function of the ellipse $E$, with respect to the natural affine structure on the set of ellipses of area $λ$. The proof relies on smoothening $K$, putting it in general position, and obtaining uniform estimates for certain derivatives of the function $I_K(.)$. Finally, we provide a characterization of maximal intersection positions, that is, the situation where the MI ellipse of $K$ is the unit disk, under the assumption that the two boundaries are transverse.

math.MG↗
arXiv2018

Equilibrium states of generalised singular value potentials and applications to affine iterated function systems

We completely describe the equilibrium states of a class of potentials over the full shift which includes Falconer's singular value function for affine iterated function systems with invertible affinities. We show that the number of distinct ergodic equilibrium states of such a potential is bounded by a number depending only on the dimension, answering a question of A. Käenmäki. We prove that all such equilibrium states are fully supported and satisfy a Gibbs inequality with respect to a suitable subadditive potential. We apply these results to demonstrate that the affinity dimension of an iterated function system with invertible affinities is always strictly reduced when any one of the maps is removed, resolving a folklore open problem in the dimension theory of self-affine fractals. We deduce a natural criterion under which the Hausdorff dimension of the attractor has the same strict reduction property.

math.DS↗
arXiv2017

Dominated Pesin theory: convex sum of hyperbolic measures

In the uniformly hyperbolic setting it is well known that the set of all measures supported on periodic orbits is dense in the convex space of all invariant measures. In this paper we consider the converse question, in the non-uniformly hyperbolic setting: assuming that some ergodic measure converges to a convex combination of hyperbolic ergodic measures, what can we deduce about the initial measures? To every hyperbolic measure $μ$ whose stable/unstable Oseledets splitting is dominated we associate canonically a unique class $H(μ)$ of periodic orbits for the homoclinic relation, called its \emph{intersection class}. In a dominated setting, we prove that a measure for which almost every measure in its ergodic decomposition is hyperbolic with the same index such as the dominated splitting is accumulated by ergodic measures if, and only if, almost all such ergodic measures have a common intersection class. We provide examples which indicate the importance of the domination assumption.

math.DS↗
arXiv2016

Perfect Matchings in Inhomogeneous Random Bipartite Graphs in Random Environment

In this note we study inhomogeneous random bipartite graphs in random environment. These graphs can be thought of as an extension of the classical Erdös-Rényi random graphs in a random environment. We show that the expected number of perfect matchings obeys a precise quenched asymptotic and that it can be approximated using an iterative process that converges exponentially fast.

math.CO↗