arXiv subjects

Davi Obata

Publications and source records attributed to Davi Obata.

17 recordsLinked to original sources

arXiv2025

Uniqueness of the measure of maximal entropy for geodesic flows on surfaces

We prove that if a geodesic flow on a closed orientable $C^\infty$ surface is transitive and has positive topological entropy, then it has a unique measure of maximal entropy. This covers all previous results of the literature on the uniqueness of the measure of maximal entropy in this context, as well as it applies to new examples such as the ones constructed by Donnay and Burns-Donnay. We also prove that, in the above context, there is at most one SRB measure.

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arXiv2025

Absolute continuity of stationary measures for random surface dynamics

We find conditions for stationary measures of random dynamical systems on surfaces having dissipative diffeomorphisms to be absolutely continuous. These conditions involve a uniformly expanding on average property in the future (UEF) and past (UEP). Our results can cover random dynamical systems generated by "very dissipative" diffeomorphisms and perturbations of volume preserving surface diffeomorphisms. For example, we can consider a random dynamical system on $\mathbb{T}^2$ generated by perturbations of a pair of non-commuting infinite order toral automorphisms with any arbitrary single diffeomorphism. In this case, we conclude that stationary measures are either atomic or absolutely continuous. We also obtain an orbit classification and equidistribution result.

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arXiv2025

Simple Lyapunov spectrum of partially hyperbolic diffeomorphisms

We study the simplicity of the Lyapunov spectrum of partially hyperbolic diffeomorphisms. We prove that a class of volume-preserving partially hyperbolic diffeomorphisms is $C^r$-accumulated by $C^2$-open sets with simple spectrum. Also we prove that a class of partially hyperbolic maps has simple spectrum generically for the measures of maximal entropy. In order to prove these results, we give a criterion for simplicity of the Lyapunov spectrum in terms of periodic points homoclinically related to the invariant measure.

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arXiv2024

Absolute continuity of stationary measures

Let $f$ and $g$ be two volume preserving, Anosov diffeomorphisms on $\mathbb{T}^2$, sharing common stable and unstable cones. In this paper, we find conditions for the existence of (dissipative) neighborhoods of $f$ and $g$, $\mathcal{U}_f$ and $\mathcal{U}_g$, with the following property: for any probability measure $\mu$, supported on the union of these neighborhoods, and verifying certain conditions, the unique $\mu$-stationary SRB measure is absolutely continuous with respect to the ambient Haar measure. Our proof is inspired in the work of Tsujii for partially hyperbolic endomorphisms [Tsu05]. We also obtain some equidistribution results using the main result of [BRH17].

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arXiv2024

Measures of maximal entropy for non-uniformly hyperbolic maps

For $C^{1+}$ maps, possibly non-invertible and with singularities, we prove that each homoclinic class of an ergodic adapted hyperbolic measure carries at most one adapted hyperbolic measure of maximal entropy. We then apply this to study the finiteness/uniqueness of such measures in several different settings: finite horizon dispersing billiards, codimension one partially hyperbolic endomorphisms with ``large'' entropy, robustly non-uniformly hyperbolic volume-preserving endomorphisms as in Andersson-Carrasco-Saghin (2025), and Viana maps (1997).

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arXiv2022

Uniqueness of $u$-Gibbs measures for hyperbolic skew products on $\mathbb{T}^4$

We study the $u$-Gibbs measures of a certain class of uniformly hyperbolic skew products on $\mathbb{T}^4$. These systems have a strong unstable and a weak unstable directions. We show that $C^r$-dense and $C^2$-open in this set every $u$-Gibbs measure is SRB, in particular, there is only one such measure. As an application of this, we can obtain the minimality of the strong unstable foliation.

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arXiv2022

Rigidity of $\mathbf{\textit{U}}$-Gibbs measures near conservative Anosov diffeomorphisms on $\mathbb{T}^3$

We show that within a $C^1$-neighbourhood $\mathcal{U}$ of the set of volume preserving Anosov diffeomorphisms on the three-torus $\mathbb{T}^3$ which are strongly partially hyperbolic with expanding center, any $f\in\mathcal{U}\cap\operatorname{Diff}^2(\mathbb{T}^3)$ satisfies the dichotomy: either the strong stable and unstable bundles $E^s$ and $E^u$ of $f$ are jointly integrable, or any fully supported $u$-Gibbs measure of $f$ is SRB.

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arXiv2021

Open sets of partially hyperbolic skew products having a unique SRB measure

In this paper we obtain $C^2$-open sets of dissipative, partially hyperbolic skew products having a unique SRB measure with full support and full basin. These partially hyperbolic systems have a two dimensional center bundle which presents both expansion and contraction but does not admit any further dominated splitting of the center. These systems are non conservative perturbations of an example introduced by Berger-Carrasco. To prove the existence of SRB measures for these perturbations, we obtain a measure rigidity result for $u$-Gibbs measures for partially hyperbolic skew products. This is an adaptation of a measure rigidity result by A. Brown and F. Rodriguez Hertz for stationary measures of random product of surface diffeomorphisms. In particular, we classify all the possible $u$-Gibbs measures that may appear in a neighborhood of the example. Using this classification, and ruling out some of the possibilities, we obtain open sets of systems, in a neighborhood of the example, having a unique $u$-Gibbs measure which is SRB.

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arXiv2020

New examples of stably ergodic diffeomorphisms in dimension 3

We prove that in the isotopy class of any volume preserving partially hyperbolic diffeomorphism in a $3$-dimensional manifold, there is a non-partially hyperbolic stably ergodic diffeomorphism. In particular, we provide new examples of stably ergodic diffeomorphisms in 3-dimensional manifolds with respect to a smooth volume measure.

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arXiv2020

Uniqueness of the measure of maximal entropy for the standard map

In this paper we prove that for sufficiently large parameters the standard map has a unique measure of maximal entropy (m.m.e.). Moreover, we prove: the m.m.e. is Bernoulli, and the periodic points with Lyapunov exponents bounded away from zero equidistribute with respect to the m.m.e. We prove some estimates regarding the Hausdorff dimension of the m.m.e. and about the density of the support of the measure on the manifold. For a generic large parameter, we prove that the support of the m.m.e. has Hausdorff dimension $2$. We also obtain the $C^2$-robustness of several of these properties.

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arXiv2019

A new example of robustly transitive diffeomorphism

We present an example of a $\mathcal{C}^1$-robustly transitive skew-product with non-trivial, non-hyperbolic action on homology. The example is conservative, ergodic, non-uniformly hyperbolic and its fiber directions cannot be decomposed into two dominated expanded/contracted bundles.

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arXiv2019

Symmetries of vector fields: the diffeomorphism centralizer

In this paper we study the diffeomorphism centralizer of a vector field: given a vector field it is the set of diffeomorphisms that commutes with the flow. Our main theorem states that for a $C^1$-generic diffeomorphism having at most finitely many sinks or sources, the diffeomorphism centralizer is quasi-trivial. In certain cases, we can promote the quasi-triviality to triviality. We also obtain a criterion for a diffeomorphism in the centralizer to be a reparametrization of the flow.

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arXiv2018

On the centralizer of vector fields: criteria of triviality and genericity results

In this paper, we investigate the question of whether a typical vector field on a compact connected Riemannian manifold $M^d$ has a `small' centralizer. In the $C^1$ case, we give two criteria, one of which is $C^1$-generic, which guarantees that the centralizer of a $C^1$-generic vector field is indeed small, namely \textit{collinear}. The other criterion states that a $C^1$ \textit{separating} flow has a collinear $C^1$-centralizer. When all the singularities are hyperbolic, we prove that the collinearity property can actually be promoted to a stronger one, refered as \textit{quasi-triviality}. In particular, the $C^1$-centralizer of a $C^1$-generic vector field is quasi-trivial. In certain cases, we obtain the triviality of the centralizer of a $C^1$-generic vector field, which includes $C^1$-generic Axiom A (or sectional Axiom A) vector fields and $C^1$-generic vector fields with countably many chain recurrent classes. For sufficiently regular vector fields, we also obtain various criteria which ensure that the centralizer is \textit{trivial} (as small as it can be), and we show that in higher regularity, collinearity and triviality of the $C^d$-centralizer are equivalent properties for a generic vector field in the $C^d$ topology. We also obtain that in the non-uniformly hyperbolic scenario, with regularity $C^2$, the $C^1$-centralizer is trivial.

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arXiv2018

Entropy, Lyapunov exponents, and rigidity of group actions

This text is an expanded series of lecture notes based on a 5-hour course given at the workshop entitled "Workshop for young researchers: Groups acting on manifolds" held in Teres\'opolis, Brazil in June 2016. The course introduced a number of classical tools in smooth ergodic theory -- particularly Lyapunov exponents and metric entropy -- as tools to study rigidity properties of group actions on manifolds. We do not present comprehensive treatment of group actions or general rigidity programs. Rather, we focus on two rigidity results in higher-rank dynamics: the measure rigidity theorem for affine Anosov abelian actions on tori due to A. Katok and R. Spatzier [Ergodic Theory Dynam. Systems 16, 1996] and recent the work of the main author with D. Fisher, S. Hurtado, F. Rodriguez Hertz, and Z. Wang on actions of lattices in higher-rank semisimple Lie groups on manifolds [arXiv:1608.04995; arXiv:1610.09997]. We give complete proofs of these results and present sufficient background in smooth ergodic theory needed for the proofs. A unifying theme in this text is the use of metric entropy and its relation to the geometry of conditional measures along foliations as a mechanism to verify invariance of measures.

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arXiv2018

On the genericity of positive exponents of conservative skew products with two-dimensional fibers

In this paper we study the existence of positive Lyapunov exponents for three different types of skew products, whose fibers are compact Riemannian surfaces and the action on the fibers are by volume preserving diffeomorphisms. These three types include skew products with a volume preserving Anosov diffeomorphism on the basis; or with a subshift of finite type on the basis preserving a measure with product structure; or locally constant skew products with Bernoulli shifts on the basis. We prove the $C^1$-density and $C^r$-openess of the existence of positive Lyapunov exponents on a set of positive measure in the space of such skew products.

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arXiv2018

On the stable ergodicity of diffeomorphisms with dominated splitting

In this paper we obtain two criteria of stable ergodicity outside the partially hyperbolic scenario. In both criteria, we use a weak form of hyperbolicity called chain-hyperbolicity. It is obtained one criterion for diffeomorphisms with dominated splitting and one criterion for weakly partially hyperbolic diffeomorphisms. As an application of one of these criteria, we obtain the $C^1$-density of stable ergodicity inside a certain class of weakly partially hyperbolic diffeomorphisms.

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arXiv2018

On the Stable Ergodicity of Berger-Carrasco's example

We prove the stable ergodicity of an example of a volume-preserving, partially hyperbolic diffeomorphism introduced by Pierre Berger and Pablo Carrasco. This example is robustly non-uniformly hyperbolic, with two dimensional center, almost every point has both positive and negative Lyapunov exponents along the center direction and does not admit a dominated splitting of the center direction. The main novelty of our proof is that we do not use accessibility.

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