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Alejandro Passeggi

Publications and source records attributed to Alejandro Passeggi.

15 recordsLinked to original sources
arXiv2026

Annular Chaos for Non-Wandering Homeomorphisms

We study topological conditions ensuring the presence of rotational chaos for non-wandering or area-preserving annular homeomorphisms. Compared to previous criteria, our main result provides a simpler alternative that avoids the need to locate periodic points, requiring only knowledge of the behavior of certain open sets. This feature is crucial for enabling concrete applications to Poincar\'e return maps arising in Hamiltonian systems with two degrees of freedom. The resulting topological criterion admits straightforward numerical implementation: a computer can verify all the required conditions using a simple algorithm that relies solely on basic data from the map. We illustrate this approach with the so-called driven pendulum.

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arXiv2023

Area preserving homeomorphisms of surfaces with rational rotational direction

Let $S$ be a closed surface of genus $g\geq 2$, furnished with a Borel probability measure $λ$ with total support. We show that if $f$ is a $λ$-preserving homeomorphism isotopic to the identity such that the rotation vector $\mathrm{rot}_f(λ)\in H_1(S,\mathbb R)$ is a multiple of an element of $H_1(S,\mathbb Z)$, then $f$ has infinitely many periodic orbits. Moreover, these periodic orbits can be supposed to have their rotation vectors arbitrarily close to the rotation vector of any fixed ergodic Borel probability measure.

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arXiv2023

Conditions implying annular chaos

This work investigates topological chaos for homeomorphisms of the open annulus, introducing a new set of sufficient conditions based on points with distinct rotation numbers and their topological relation to invariant continua. These conditions allow us to formulate classic methods for verifying annular chaos in a finitely verifiable version supported on basic properties of the map. The results pave the way for simple computer-assisted proofs of chaos in a wide range of annular maps, including many well known examples, and we present these proofs for some analytic families, demonstrating the effectiveness of the method. On the theoretical side, one of the consequences of the established conditions permits the proof of a folkloric conjecture about the relation between topological entropy and rotation sets.

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arXiv2019

The bifurcation set as a topological invariant for one-dimensional dynamics

For a continuous map on the unit interval or circle, we define the bifurcation set to be the collection of those interval holes whose surviving set is sensitive to arbitrarily small changes of their position. By assuming a global perspective and focusing on the geometric and topological properties of this collection rather than the surviving sets of individual holes, we obtain a novel topological invariant for one-dimensional dynamics. We provide a detailed description of this invariant in the realm of transitive maps and observe that it carries fundamental dynamical information. In particular, for transitive non-minimal piecewise monotone maps, the bifurcation set encodes the topological entropy and strongly depends on the behavior of the critical points.

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arXiv2018

A Poincaré-Bendixson theorem for translation lines and applications to prime ends

For an orientation-preserving homeomorphism of the sphere, we prove that if a translation line does not accumulate in a fixed point, then it necessarily spirals towards a topological attractor. This is in analogy with the description of flow lines given by Poincaré-Bendixson theorem. We then apply this result to the study of invariant continua without fixed points, in particular to circloids and boundaries of simply connected open sets. Among the applications, we show that if the prime ends rotation number of such an open set $U$ vanishes, then either there is a fixed point in the boundary, or the boundary of $U$ is contained in the basin of a finite family of topological "rotational" attractors. This description strongly improves a previous result by Cartwright and Littlewood, by passing from the prime ends compactification to the ambient space. Moreover, the dynamics in a neighborhood of the boundary is semiconjugate to a very simple model dynamics on a planar graph. Other applications involve the decomposability of invariant continua, and realization of rotation numbers by periodic points on circloids.

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arXiv2018

Deviations in the Franks-Misiurewicz conjecture

We show that if there exists a counter example for the rational case of the Franks-Misiurewicz conjecture, then it must exhibit unbounded deviations in the complementary direction of its rotation set.

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arXiv2017

Rotation intervals and entropy on attracting annular continua

We show that if $f$ is an annular homeomorphism admitting an attractor which is an irreducible annular continua with two different rotation numbers, then the entropy of $f$ is positive. Further, the entropy is shown to be associated to a $C^0$-robust rotational horseshoe. On the other hand, we construct examples of annular homeomorphisms with such attractors so that the rotation interval is uniformly large but the entropy approaches zero as much as desired. The developed techniques allow us to obtain similar results in the context of Birkhoff attractors.

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arXiv2016

The Franks-Misiurewicz conjecture for extensions of irrational rotations

We show that a toral homeomorphism which is homotopic to the identity and topologically semiconjugate to an irrational rotation of the circle is always a pseudo-rotation (i.e. its rotation set is a single point). In combination with recent results, this allows us to complete the study of the Franks-Misiurewicz conjecture in the minimal case.

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arXiv2014

Rotation sets and almost periodic sequences

We study the rotational behaviour on minimal sets of torus homeomorphisms and show that the associated rotation sets can be any type of line segments as well as non-convex and even plane-separating continua. This shows that restrictions holding for rotation sets on the whole torus are not valid on minimal sets. The proof uses a construction of rotational horseshoes by Kwapisz to transfer the problem to a symbolic level, where the desired rotational behaviour is implemented by means of suitable irregular Toeplitz sequences.

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arXiv2014

A Classification of Minimal Sets of Torus Homeomorphisms

We provide a classification of minimal sets of homeomorphisms of the two-torus, in terms of the structure of their complement. We show that this structure is exactly one of the following types: (1) a disjoint union of topological disks, or (2) a disjoint union of essential annuli and topological disks, or (3) a disjoint union of one doubly essential component and bounded topological disks. Periodic bounded disks can only occur in type 3. This result provides a framework for more detailed investigations, and additional information on the torus homeomorphism allows to draw further conclusions. In the non-wandering case, the classification can be significantly strengthened and we obtain that a minimal set other than the whole torus is either a periodic orbit, or the orbit of a periodic circloid, or the extension of a Cantor set. Further special cases are given by torus homeomorphisms homotopic to an Anosov, in which types 1 and 2 cannot occur, and the same holds for homeomorphisms homotopic to the identity with a rotation set which has non-empty interior. If a non-wandering torus homeomorphism has a unique and totally irrational rotation vector, then any minimal set other than the whole torus has to be the extension of a Cantor set.

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arXiv2013

On Torus Homeomorphisms Semiconjugate to irrational Rotations

In the context of the Franks-Misiurewicz Conjecture, we study homeomorphisms of the two-torus semiconjugate to an irrational rotation of the circle. As a special case, this conjecture asserts uniqueness of the rotation vector in this class of systems. We first characterise these maps by the existence of an invariant foliation by essential annular continua (essential subcontinua of the torus whose complement is an open annulus) which are permuted with irrational combinatorics. This result places the considered class close to skew products over irrational rotations. Generalising a well-known result of M. Herman on forced circle homeomorphisms, we provide a criterion, in terms of topological properties of the annular continua, for the uniqueness of the rotation vector. As a byproduct, we obtain a simple proof for the uniqueness of the rotation vector on decomposable invariant annular continua with empty interior. In addition, we collect a number of observations on the topology and rotation intervals of invariant annular continua with empty interior.

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arXiv2012

A classification of minimal sets for surface homeomorphisms

We classify minimal sets of (closed and oriented) hyperbolic surface homeomorphisms by studying the connected components of their complement. This extends the classification given by F. Kwakkel, T.Jäger and A. Passeggi in the torus. The given classification is studied in the non-wandering setting and in light of the Nielsen-Thurston Theory.

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arXiv2012

Rational Polygons as Rotation Sets of Generic Homeomorphisms of the Two-Torus

We prove the existence of an open and dense set D\subset? Homeo0(T2) (set of toral homeomorphisms homotopic to the identity) such that the rotation set of any element in D is a rational polygon. We also extend this result to the set of axiom A dif- feomorphisms in Homeo0(T2). Further we observe the existence of minimal sets whose rotation set is a non-trivial segment, for an open set in Homeo0(T2).

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arXiv2012

Examples of Minimal Diffeomorphisms on $t^{2}$ Semiconjugated to an Ergodic Translation

We prove that for every $ε>0$ there exists a minimal diffeomorphism $f:\T^{2}\rightarrow\T^{2}$ of class $C^{3-ε}$ and semiconjugate to an ergodic traslation, and have the following properties: zero entropy, sensitivity with respect to initial conditions and Li-Yorke chaos. These examples are obtained through the holonomy of the unstable foliation of Mañé's example of derived from Anosov diffeomorphism on $\T^3.$

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