arXiv subjects

F. Micena

Publications and source records attributed to F. Micena.

11 recordsLinked to original sources

arXiv2026

A generalized Liv\v{s}ic--Sinai Theorem for endomorphisms

The classical theorem of Liv\v{s}ic and Sinai states that a transitive $C^2$ Anosov diffeomorphism whose Jacobian along every periodic orbit equals one admits an invariant volume form. Recently, it was observed that the periodic Jacobian condition alone already implies transitivity, rendering the transitivity assumption unnecessary. We extend this rigidity phenomenon to the non-invertible setting. We prove that if a $C^2$ Anosov endomorphism satisfies the natural periodic Jacobian condition $J(f^n(p)) = \deg(f)^n$ for every periodic point $p,$ such that $f^n(p) = p,$ then the system is automatically transitive and preserves a $C^1$ volume form. As a key ingredient, we establish a $C^1$ version of the Liv\v{s}ic cohomological theorem for hyperbolic endomorphisms.

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arXiv2026

On Non-Wandering Sets with Non-empty Interior for Endomorphisms

In this paper, we study transitivity for endomorphisms. Our results are related to a conjecture of F. Abdenur, C. Bonatti and L. D\'iaz, concerning the relationship between transitivity and the existence of a non-wandering set with nonempty interior. We obtain transitivity under the assumption that there exists a hyperbolic non-wandering set with non-empty interior, and in the setting of accessible partially hyperbolic endomorphisms.

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arXiv2022

On Measurable Properties of Anosov Endomorphisms of Torus

We found a dichotomy involving the unstable Lyapunov exponent of a special Anosov endomorphism of the torus induced by the conjugacy with the linearization. In fact, either every unstable leaf meets on a set of zero measure the set for which is defined such unstable Lyapunov exponent or the endomorphims is smoothly conjugated with its linearization. Also we are able to characterize the absolute continuity of the intermediate foliation for a class of volume preserving special Anosov endomorphisms of $\mathbb{T}^3.$

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arXiv2022

Some sufficient conditions for transitivity of Anosov diffeomorphisms

Given a $C^2$- Anosov diffemorphism $f: M \rightarrow M,$ we prove that the jacobian condition $Jf^n(p) = 1,$ for every point $p$ such that $f^n(p) = p,$ implies transitivity. As application in the celebrated theory of Sinai-Ruelle-Bowen, this result allows us to state a classical theorem of Livsic-Sinai without directly assuming transitivity as a general hypothesis. A special consequence of our result is that every $C^2$-Anosov diffeomorphism, for which every point is regular, is indeed transitive.

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arXiv2022

A relation between entropy and transitivity of Anosov diffeomorphisms

It is known that transitive Anosov diffeomorphisms have a unique measure of maximal entropy (MME). Here we discuss the converse question. Under suitable hypothesis on Lyapunov exponents on the set of periodic points and the structure of the MME we get transitivity of $C^1-$Anosov diffeomorphisms.

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arXiv2021

Some Generic Properties of Partially Hyperbolic Endomorphisms

In this work, we deal with a notion of partially hyperbolic endomorphism. We explore topological properties of this definition and we obtain, among other results, obstructions to get center leaf conjugacy with the linear part, for a class of partially hyperbolic endomorphism $C^1-$sufficiently close to a hyperbolic linear endomorphism. Indeed such obstructions are related to the number of center directions of a point. We provide examples illustrating these obstructions. We show that for a manifold $M$ with dimension $n \geq 3,$ admitting a non-invertible partially hyperbolic endomorphisms, there is a $C^1$ open and dense subset $\mathcal{U}$ of all partially hyperbolic endomorphisms with degree $d \geq n,$ such that any $f \in \mathcal{U}$ is neither $c$ nor $u$ special.

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arXiv2019

Constant periodic data and rigidity

In this work we lead with expanding maps of the circle and Anosov diffeomorphisms on $\mathbb{T}^d, d \geq 2.$ We prove that, for these maps, \textit{constant periodic data} imply \textit{same periodic data of these maps and their linearizations}, so in particular we have smooth conjugacy. For expanding maps of the circle and Anosov diffeomorphism on $\mathbb{T}^d, d= 2, 3,$ we have global rigidity. In higher dimensions, $d \geq 4,$ we can establish a result of local rigidity, in several cases. The main tools of this work are celebrated results of rigidity involving same periodic data with linearization and results involving topological entropy of a diffeomorphism along an expanding invariant foliation.

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arXiv2018

A Note on Rigidity of Anosov diffeomorphisms of the Three Torus

We consider Anosov diffeomorphisms on $\mathbb{T}^3$ such that the tangent bundle splits into three subbundles $E^s_f \oplus E^{wu}_f \oplus E^{su}_f.$ We show that if $f$ is $C^r, r \geq 2,$ volume preserving, then $f$ is $C^1$ conjugated with its linear part $A$ if and only if the center foliation $\mathcal{F}^{wu}_f$ is absolutely continuous and the equality $λ^{wu}_f(x) = λ^{wu}_A,$ between center Lyapunov exponents of $f$ and $A,$ holds for $m$ a.e. $x \in \mathbb{T}^3.$ We also conclude rigidity of derived from Anosov diffeomorphism, assuming an strong absolute continuity property (Uniform bounded density property) of strong stable and strong unstable foliations.

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arXiv2017

Pathological center foliation with dimension greater than one

In this paper we are considering partially hyperbolic diffeomorphims of the torus, with $dim(E^c) > 1.$ We prove, under some conditions, that if the all center Lyapunov exponents of the linearization $A,$ of a \mbox{DA-diffeomorphism} $f,$ are positive and the center foliation of $f$ is absolutely continuous, then the sum of the center Lyapunov exponents of $f$ is bounded by the sum of the center Lyapunov exponents of $A.$ After, we construct a $C^1-$open class of volume preserving \mbox{DA-diffeomorphisms}, far from Anosov diffeomorphisms, with non compact pathological two dimensional center foliation. Indeed, each $f$ in this open set satisfies the previously established hypothesis, but the sum of the center Lyapunov exponents of $f$ is greater than the corresponding sum with respect to its linearization. It allows to conclude that the center foliation of $f$ is non absolutely continuous. We still build an example of a DA-diffeomorphism, such that the disintegration of volume along the two dimensional, non compact center foliation is neither Lebesgue nor atomic.

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arXiv2015

New Derived from Anosov Diffeomorphisms with pathological center foliation

In this paper we focused our study on Derived From Anosov diffeomorphisms (DA diffeomorphisms ) of the torus $\mathbb{T}^3,$ it is, an absolute partially hyperbolic diffeomorphism on $\mathbb{T}^3$ homotopic to an Anosov linear automorphism of the $\mathbb{T}^3.$ We can prove that if $f: \mathbb{T}^3 \rightarrow \mathbb{T}^3 $ is a volume preserving DA diffeomorphism homotopic to linear Anosov $A,$ such that the center Lyapunov exponent satisfies $λ^c_f(x) > λ^c_A > 0,$ with $x $ belongs to a positive volume set, then the center foliation of $f$ is non absolutely continuous. We construct a new open class $U$ of non Anosov and volume preserving DA diffeomorphisms, satisfying the property $λ^c_f(x) > λ^c_A > 0$ for $m-$almost everywhere $x \in \mathbb{T}^3.$ Particularly for every $f \in U,$ the center foliation of $f$ is non absolutely continuous.

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arXiv2014

On the Unstable Directions and Lyapunov Exponents of Anosov Endomorphisms

Despite the invertible setting, Anosov endomorphisms may have infinitely many unstable directions. Here we prove, under transitivity assumption, that an Anosov endomorphism on a closed manifold $M,$ is either special (that is, every $x \in M$ has only one unstable direction) or for a typical point in $M$ there are infinitely many unstable directions. Other result of this work is the semi rigidity of the unstable Lyapunov exponent of a $C^{1+α}$ codimension one Anosov endomorphism and $C^1$ close to a linear endomorphism of $\mathbb{T}^n$ for $(n \geq 2).$ In the appendix we give a proof for ergodicity of $C^{1+α}, α> 0,$ conservative Anosov endomorphism.

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