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Pablo G. Barrientos

Publications and source records attributed to Pablo G. Barrientos.

At least 19 recordsLinked to original sources
arXiv2026

(co)Quasi-irreducible and (co)expanding random maps

We study quasi-irreducibility, expansion, and their cotangent duals for random $C^1$ (local) diffeomorphisms on compact invariant sets of a Riemannian manifold. Quasi-irreducibility is defined through stationary lifts to the projective tangent bundle: every lift must realize the Furstenberg--Kifer formula for the top Lyapunov exponent. We prove that, for ergodic stationary measures, this is equivalent to the absence of equators, namely non-random invariant subbundles on which the top exponent drops. Under simplicity of the first Lyapunov exponent for every stationary measure, quasi-irreducibility is also equivalent to vertical mostly contraction, contraction on average, a vertical spectral gap, and uniqueness of stationary projective lifts. We then apply these criteria to continuity of the top Lyapunov exponent and to expansion on average. In particular, expansion is characterized by positivity of the top Lyapunov exponent on all non-random invariant subbundles; under quasi-irreducibility, it is equivalent to positivity of the top Lyapunov exponent for every stationary measure. Dual statements hold for coquasi-irreducibility, coequators, coexpansion, and continuity of the bottom Lyapunov exponent. We also describe the interplay between expansion and coexpansion and extend the formalism to Grassmannian bundles, obtaining higher-dimensional versions controlling intermediate sums of Lyapunov exponents.

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arXiv2026

Integral representation of Lyapunov exponents

We introduce a new operator-theoretic construction of Lyapunov growth for Markov-driven systems. The construction is based on an abstract variational principle for asymptotic growth rates arising from a subadditive process generated by an intertwining pair of Markov operators on a measurable bundle with compact fibers: for each invariant base measure, the fiberwise-maximal growth rate equals the supremum of the fiber integral over invariant lifts, and this supremum is attained on an ergodic lift. For random linear bundle morphisms driven by Markovian place-dependent noise, this yields Lyapunov exponents defined intrinsically from the initial law on the state-phase space. We prove that these exponents coincide with the classical path-space skew-product Lyapunov spectrum, showing that the pointwise exponents depend only on the current noise state and initial position, not on the full noise realization. We also obtain new asymptotic formulas for the sums of Lyapunov exponents as conditional annealed growth rates along individual directions. As further consequences of this variational principle, we recover and extend to singular linear bundle morphisms the classical projective variational formulas.

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arXiv2025

Robust heterodimensional cycles of co-index two via split blending machines

We consider diffeomorphisms $f$ with heterodimensional cycles of co-index two, associated with saddles $P$ and $Q$ having unstable indices $\ell$ and $\ell+2$, respectively. In a partially hyperbolic setting, where a two-dimensional center direction and strong invariant manifolds are defined, we introduce the class of \emph{non-escaping cycles}, where the strong stable manifold of $P$ and the strong unstable manifold of $Q$ are involved in the cycle. This configuration guarantees the existence of orbits that remain in a neighbourhood of the cycle. We show that such diffeomorphisms $f$ can be $C^1$ approximated by diffeomorphisms exhibiting simultaneously $C^1$ robust heterodimensional cycles of co-indices one and two, encompassing all possible combinations among hyperbolic sets of unstable indices $\ell$, $\ell+1$, and $\ell+2$. The proof relies on the construction of \emph{split blending machines}. This tool extends Asaoka's blending machines to a partially hyperbolic setting, providing a mechanisms to generate and control robust intersections within a two-dimensional central bundle. We also present simple dynamical settings where such cycles occur, namely skew product dynamics with surface fiber maps. Non-escaping cycles also appear in contexts such as Derived from Anosov diffeomorphisms and matrix cocycles on $\mathrm{GL}(3,\mathbb{R})$.

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arXiv2025

Historical behavior of skew products and arcsine laws

We study the occurrence of historical behavior for almost every point in the setting of skew products with one-dimensional fiber dynamics. Under suitable ergodic conditions, we establish that a weak form of the arcsine law leads to the non-convergence of Birkhoff averages along almost every orbit. As an application, we show that this phenomenon occurs for one-step skew product maps over a Bernoulli shift, where the stochastic process induced by the iterates of the fiber maps is conjugate to a random walk. Furthermore, we revisit known examples of skew products that exhibit historical behavior almost everywhere, verifying that they fulfill the required ergodic and probabilistic conditions. Consequently, our results provide a unified and generalized framework that connects such behaviors to the arcsine distribution of the orbits.

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arXiv2025

Finitude of physical measures for Markovian random maps

We study the finiteness of physical measures for skew-product transformations $F$ associated with discrete-time random dynamical systems driven by ergodic Markov chains. We develop a framework, using an independent and identically distributed (i.i.d.) representation of the Markov process, that facilitates transferring results from the well-studied Bernoulli (i.i.d.) setting to the Markovian context. Specifically, we establish conditions for the existence of finitely many ergodic, $F$-invariant measures, absolutely continuous with respect to a reference measure, such that their statistical basins of attraction for measurable bounded observables cover the phase space almost everywhere. Furthermore, we investigate a weaker notion, which demands finitely many physical measures (not necessarily absolutely continuous) whose weak$^*$ basins of attraction cover the phase space almost everywhere. We show that for random maps on compact metric spaces driven by Markov chains on finite state spaces, this property holds if the system is mostly contracting, i.e., if all the Markovian invariant measures have negative maximal Lyapunov exponents. This result is applied to random $C^1$ diffeomorphisms of the circle and the interval under conditions based on the absence of invariant probability measures or finite invariant sets, respectively. We also connect our result to the quasi-compactness of the Koopman operator on the space of Hölder continuous functions.

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arXiv2024

Finitude of physical measures for random maps

For random compositions of independent and identically distributed measurable maps on a Polish space, we study the existence and finitude of absolutely continuous ergodic stationary probability measures (which are, in particular, physical measures) whose basins of attraction cover the whole space almost everywhere. We characterize and hierarchize such random maps in terms of their associated Markov operators, as well as show the difference between classes in the hierarchy by plenty of examples, including additive noise, multiplicative noise, and iterated function systems. We also provide sufficient practical conditions for a random map to belong to these classes. For instance, we establish that any continuous random map on a compact Riemannian manifold with absolutely continuous transition probability has finitely many physical measures whose basins of attraction cover Lebesgue almost all the manifold.

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arXiv2024

Mostly contracting random maps

We study the long-term behavior of independent random iterations of Lipschitz transformations on a compact metric space. Such a random map is said to be mostly contracting if all Lyapunov exponents associated with stationary measures are negative. This requires introducing the notion of (maximal) Lyapunov exponent in this general context of Lipschitz transformations on compact metric spaces. We show that this class is open with respect to the appropriate topology and satisfies the strong law of large numbers for non-uniquely ergodic systems, the limit theorem for the law of random iterations, Palis' global conjecture, and that the associated annealed Koopman operator is quasi-compact. This implies many statistical properties such as central limit theorems, large deviations, statistical stability, and the continuity and H\"older continuity of Lyapunov exponents. Examples from this class of random maps include random products of circle diffeomorphisms, interval diffeomorphisms onto their images, and diffeomorphisms of a Cantor set on a line, all considered under the assumption of no common invariant measure. This class also includes projective actions of locally constant linear cocycles under the assumptions of simplicity of the top Lyapunov exponent and a suitable irreducibility condition. Finally, it encompasses the classical theory of finite-state Markov chains, viewed as random walks on a finite set. One of the main tools to prove the above results is the generalization of Kingman's subadditive ergodic theorem and the uniform Kingman's subadditive ergodic theorem for general Markov operators. Another key ingredient is the exponential local contraction theorem, which can be viewed as a non-smooth metric analogue of the contraction mechanism behind stable manifold theory. These results are of independent interest, as they may have broad applications in other contexts.

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arXiv2024

Zero Lyapunov Exponents in Transitive Skew-products of Iterated Function Systems

We study the class of transitive skew-products associated with iterated function systems of circle diffeomorphisms. We can approximate any transitive skew-product by maps in this class that have a robustly zero Lyapunov exponent. In particular, we prove the existence of non-hyperbolic ergodic measures for an open and dense subset of transitive skew-products. Moreover, these measures have full support and are the weak$^*$ limit of periodic measures.

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arXiv2024

Minimal Strong Foliations in Skew-products of Iterated Function Systems

We study locally constant skew-product maps over full shifts of finite symbols with arbitrary compact metric spaces as fiber spaces. We introduce a new criterion to determine the density of leaves of the strong unstable (and strong stable) foliation, that is, for its minimality. When the fiber space is a circle, we show that both strong foliations are minimal for an open and dense set of robust transitive skew-products. We provide examples where either one foliation is minimal or neither is minimal. Our approach involves investigating the dynamics of the associated iterated function system (IFS). We establish the asymptotic stability of the phase space of the IFS when it is a strict attractor of the system. We also show that any transitive IFS consisting of circle diffeomorphisms that preserve orientation can be approximated by a robust forward and backward minimal, expanding, and ergodic (with respect to Lebesgue) IFS. Lastly, we provide examples of smooth robust transitive IFSs where either the forward or the backward minimal fails, or both.

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arXiv2023

Ping-pong partitions and locally discrete groups of real-analytic circle diffeomorphisms, II: Applications

In the first part of this work we have established an efficient method to obtain a topological classification of locally discrete, finitely generated, virtually free subgroups of real-analytic circle diffeomorphisms. In this second part we describe several consequences, among which the solution (within this setting) to an old conjecture by P. R. Dippolito [Ann. Math. 107 (1978), 403-453] that actions with invariant Cantor sets must be semi-conjugate to piecewise linear actions. In addition, we exhibit examples of locally discrete, minimal actions which are not of Fuchsian type.

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arXiv2023

Typical coexistence of infinitely many strange attractors

We prove that the coexistence of infinitely many prevalent Hénon-like phenomena is Kolmogorov typical in sectional dissipative $C^{d,r}$-Berger domains of parameter families of diffeomorphisms of dimension $m\geq 3$ for $d<r-1$. Namely, we answer an old question posed by Colli in [Annales de l'Institut Henri Poincare-Nonlinear Analysis, 15, 539--580 (1998)] on typicality of the coexistence of infinitely many non-hyperbolic strange attractors for $3\leq d<r-1$.

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arXiv2023

Historic wandering domains near cycles

We explain how to obtain non-trivial historic contractive wandering domains for a dense set of diffeomorphisms in certain type of $C^r$-Newhouse domains of homoclinic tangencies in dimension $d\geq 3$ and $r\geq 1$. In particular, this gives for the first time a contribution to Takens' last problem in the $C^1$ topology and in dimension $d>2$. We show how these Newhouse domains can be obtained arbitrarily close to diffeomorphisms exhibiting heterodimensional cycles (in dimension $d=3$) or non-transverse equidimensional cycles (in any dimension $d\ge 3$) associated with periodic points with non-real complex leading multipliers.

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arXiv2021

Homoclinic tangencies leading to robust heterodimensional cycles

We consider $C^r$ ($r\geqslant 1$) diffeomorphisms $f$ defined on manifolds of dimension $\geqslant 3$ with homoclinic tangencies associated to saddles. Under generic properties, we show that if the saddle is homoclinically related to a blender then the diffeomorphism $f$ can be {$C^r$} approximated by diffeomorphisms with {$C^1$} robust heterodimensional cycles. As an application, we show that the classic Simon-Asaoka's examples of diffeomorphisms with $C^1$ robust homoclinic tangencies also display {$C^1$} robust heterodimensional cycles. In a second application, we consider homoclinic tangencies associated to hyperbolic sets. When the entropy of these sets is large enough we obtain $C^1$ robust cycles after $C^1$ perturbations.

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arXiv2021

Berger domains and Kolmogorov typicality of infinitely many invariant circles

Using the novel notion of parablender, P. Berger proved that the existence of finitely many attractors is not Kolmogorov typical in parametric families of diffeomorphisms. Here, motivated by the concept of Newhouse domains we define Berger domains for families of diffeomorphisms. As an application, we show that the coexistence of infinitely many attracting invariant smooth circles is Kolmogorov typical in certain non-sectionally dissipative Berger domains of parametric families in dimension three or greater.

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arXiv2021

Extremal exponents of random products of conservative diffeomorphisms

We show that for a $C^1$-open and $C^{r}$-dense subset of the set of ergodic iterated function systems of conservative diffeomorphisms of a finite-volume manifold of dimension $d\geq 2$, the extremal Lyapunov exponents do not vanish. In particular, the set of non-uniform hyperbolic systems contains a $C^1$-open and $C^r$-dense subset of ergodic random products of i.i.d. conservative surface diffeomorphisms.

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arXiv2021

Robust degenerate unfoldings of cycles and tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

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arXiv2019

Historic behavior in non-hyperbolic homoclinic classes

We show that $C^1$-generically for diffeomorphisms of manifolds of dimension $d\geq3$, a homoclinic class containing saddles of different indices has a residual subset where the orbit of any point has historic behavior.

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