arXiv subjects

Gabriel Ponce

Publications and source records attributed to Gabriel Ponce.

12 recordsLinked to original sources

arXiv2026

Infinite-Piecewise Expanding Maps: Chaos, Ergodicity and Invariant-Set Complexity

In this paper, we study a class of one-dimensional piecewise maps defined by infinitely many smooth expanding branches. This class arises naturally in the context of non-smooth dynamical systems and includes the first-return maps locally defined near sliding Shilnikov connections. By means of the theory of conformal iterated function systems (CIFS), we investigate several dynamical properties of these maps as well as the topological complexity of their invariant sets. In particular, we show that the dynamics restricted to the invariant set is topologically conjugate to the shift on $\mathbb{N}^{\mathbb{N}}$. We also establish the existence of a unique conformal measure that is invariant and ergodic under the map.

math.DS↗

arXiv2025

Homoclinic classes for flows: ergodicity and SRB measures

In this work we intend to study homoclinic classes for some classes of flows. To this end we obtain analogous results those obtained by Hertz-Hertz-Tahzibi-Ures in the flow setting. Namely we prove that if the Lesbegue measure gives positive measure to both stable and unstable homoclinic classes of a periodic hyperbolic orbit, then their intersection constitute an ergodic component. Futhermore, with similar techiniques we state several results concerning regular SRB measures.

math.DS↗

arXiv2023

On the Hausdorff dimension and Cantor set structure of sliding Shilnikov invariant sets

The concept of sliding Shilnikov connection has been recently introduced and represents an important notion in Filippov systems, because its existence implies chaotic behavior on an invariant subset of the system. The investigation of its properties has just begun, and understanding the topology and complexity of its invariant set is of interest. In this paper, we conduct a local analysis on the first return map associated to a Shilnikov sliding connection, which reveals a conformal iterated function system (CIFS) structure. By using the theory of CIFS, we estimate the Hausdorff dimension of the local invariant set of the first return map, showing, in particular, that it is strictly greater than $0$ and strictly less than $1$, and its one-dimensional Lebesgue measure is 0. Moreover, we prove that the closure of the local invariant set is a Cantor set and has the same Hausdorff dimension and Lebesgue measure of the original invariant set. Furthermore, it is given by the invariant set adjoined with the set of all pre-images of the regular-fold point.

math.DS↗

arXiv2019

Ergodic properties of partially hyperbolic diffeomorphisms with topological neutral center

In this work we obtain some metric and ergodic properties of $C^{1+}$ partially hyperbolic diffeomorphisms with one-dimensional topological neutral center, mainly regarding the behavior of its center foliation. Based on a trichotomy for the center conditional measures of any invariant ergodic measure, we show that if these conditionals have full support, then the center foliation is leafwise absolutely continuous, the diffeomorphism is Bernoulli in the $C^{1+}$ case, and an invariance principle occurs in the sense that M may be covered by a finite number of open sets where the system of center conditionals is continuous and su-invariant. Using this invariance principle we show that if a local accessibility hypothesis occurs then the center foliation must be as regular as the partially hyperbolic dynamics.

math.DS↗

arXiv2018

Classification of Conditional Measures Along Certain Invariant One-Dimensional Foliations

Let $f:M\to M$ be a homeomorphism over a compact Riemannian manifold, ergodic with respect to a measure $\mu$ defined on the completion of the Borel $\sigma$-algebra and $\mathcal F$ a $f$-invariant one dimensional continuous foliation of $M$ by $C^1$-leaves. Then, if $f$ preserves a continuous $\mathcal{F}$-arc length system, then we only have three possibilities for the conditional measures of $\mu$ along $\mathcal F$, namely: - they are atomic for almost every leaf, or - for almost every leaf they are equivalent to the measure $\lambda_x$ induced by the invariant arc-length system over $\mathcal F$, or - for almost every leaf their support is a nowhere dense, perfect subset of the leaf. Furthermore, we show that restricted to ergodic partially hyperbolic diffeomorphism with one-dimensional topological neutral center direction, we are able to eliminate the third case obtaining a dichotomy.

math.DS↗

arXiv2018

Unstable entropy of partially hyperbolic diffeomorphisms along non-compact subsets

Given a partially hyperbolic diffeomorphism $f:M \rightarrow M$ defined on a compact Riemannian manifold $M$, in this paper we define the concept of unstable topological entropy of $f$ on a set $Y \subset M$ not necessarily compact and we extend a theorem of R. Bowen proving that, for an ergodic $f$-invariant measure $\mu$, the unstable measure theoretical entropy of $f$ is upper bounded by the unstable topological entropy of $f$ on any set of full $\mu$-measure. We also define a notion of unstable topological entropy of $f$ using a Hausdorff dimension like characterization.

math.DS↗

arXiv2017

Measure Rigidity and Disintegration: Time-one map of flows

An invariant measure for a flow is, of course, an invariant measure for any of its time-t maps. But the converse is far from being true. Hence, one may naturally ask: What is the obstruction for an invariant measure for the time-one map to be invariant for the flow itself? We give an answer in terms of measure disintegration. Surprisingly all it takes is the measure not to be "too much pathological in the orbits". We prove the following rigidity result. If $\mu$ is an ergodic probability for the time-one map of a flow, then it is either highly pathological in the orbits, or it is highly regular (i.e invariant for the flow). In particular this measure rigidity result is also true for measurable flows by the classical Ambrose-Kakutani's representation theorem for measurable flows.

math.DS↗

arXiv2016

Chaos induced by sliding phenomena in Filippov systems

In this paper we provide a full topological and ergodic description of the dynamics of Filippov systems nearby a sliding Shilnikov orbit. More specifically we prove that the first return map, defined nearby this orbit, is topologically conjugate to a Bernoulli shift with infinite topological entropy. In particular, we see that for each natural number m it has infinitely many periodic points with period m.

math.DS↗

arXiv2016

Generating positive geometric entropy from recurrent leaves

In this paper we introduce a new $C^r-$perturbation procedure, with respect to the $C^r$-Epstein topology, for $C^r$-foliations by surfaces. Using this perturbation procedure we show how one can use the existence of recurrent leaves of certain $C^r-$foliation $\mathcal F$ to obtain a foliation $\mathcal G$, $C^r-$close to $\mathcal F$ in the $C^r-$Epstein topology, which has a resilient leaf. In particular, one can take advantage of recurrence property to construct examples of $C^r-$foliations by surfaces, $C^r-$close to each other and such that one of them has a resilient leaf while the other is Riemannian (therefore has trivial dynamics).

math.DS↗

arXiv2016

On the Bernoulli property for certain partially hyperbolic diffeomorphisms

We address the classical problem of equivalence between Kolmogorov and Bernoulli property of smooth dynamical systems. In a natural class of volume preserving partially hyperbolic diffeomorphisms homotopic to Anosov ("derived from Anosov") on 3-torus, we prove that Kolmogorov and Bernoulli properties are equivalent. In our approach, we propose to study the conditional measures of volume along central foliation to recover fine ergodic properties for partially hyperbolic diffeomorphisms. As an important consequence we obtain that there exists an almost everywhere conjugacy between any volume preserving derived from Anosov diffeomorphism of 3-torus and its linearization. Our results also hold in higher dimensional case when central bundle is one dimensional and stable and unstable foliations are quasi-isometric.

math.DS↗

arXiv2013

Mono-atomic disintegration and Lyapunov exponents for derived from Anosov diffeomorphisms

In this paper we mainly address the problem of disintegration of Lebesgue measure and measure of maximal entropy along the central foliation of (conservative) Derived from Anosov (DA) diffeomorphisms. We prove that for accessible DA diffeomorphisms of $\mathbb{T}^3$, atomic disintegration has the peculiarity of being mono-atomic (one atom per leaf). We further provide open and non-empty condition for the existence of atomic disintegration. Finally, we prove some new relations between Lyapunov exponents of DA diffeomorphisms and their linearization.

math.DS↗

arXiv2012

Central Lyapunov exponent of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$

In this paper we construct some "pathological" volume preserving partially hyperbolic diffeomorphisms on $\toro{3}$ such that their behaviour in small scales in the central direction (Lyapunov exponent) is opposite to the behavior of their linearization. These examples are isotopic to Anosov. We also get partially hyperbolic diffeomorphisms isotopic to Anosov (consequently with non-compact central leaves) with zero central Lyapunov exponent at almost every point.

math.DS↗