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Roberto De Leo

Publications and source records attributed to Roberto De Leo.

At least 19 recordsLinked to original sources
arXiv2026

Free maps in critical dimension on $\mathbb{T}^m$

The critical dimension for free maps on m-manifolds is $q_m=m(m+3)/2$. We show that $Free^\infty(\mathbb{T}^m,\mathbb{R}^{q_m})\neq\emptyset$ for every $m\ge 1$. The main tool is a product construction: given free maps in critical dimension on $\mathbb{T}^a$ and on $\mathbb{T}^b$, together with a cross-free map on $\mathbb{T}^a\times \mathbb{T}^b$, whose mixed Hessian is invertible everywhere, we obtain a free map in critical dimension on $\mathbb{T}^{a+b}$, the osculating matrix of the product being block-triangular. Cross-free maps are additive in each argument, and in the torus setting none exists when one factor is one-dimensional. We construct explicit cross-free maps on $\mathbb{T}^2\times \mathbb{T}^2$ and on $\mathbb{T}^3\times\mathbb{T}^4$ using quaternionic multiplication. Combined with the low-dimensional cases $m\leq 5$, constructed explicitly by the author in a previous publication, these yield the result by induction.

math.DG↗
arXiv2026

Recurrent and gradient dynamics of Iterated Function Systems

We study the qualitative dynamics of general Iterated Function Systems (IFSs) through chain recurrence and a decomposition into recurrent and gradient-like behavior. At the same time, we move the focus from the topological property of the phase space to the dynamical properties of the system and we say that an IFS has compact dynamics if it has an invariant compact set (global attractor) that attracts every compact subset of the phase space. For an IFS with compact dynamics, we introduce a distinction that has no counterpart for a single map: ordinary chain recurrence, which may be realized along a suitable itinerary, and orbitwise chain recurrence, which requires the whole semigroup orbit of a point to remain in its chain component. This leads naturally to two directed graphs encoding the recurrent components and the downstream relations between them. We prove that the chain structure of an IFS with compact dynamics over a locally compact space is completely determined by the restriction of the IFS to its compact global attractor. We then obtain a Conley-type trichotomy: every point of the global attractor has either orbitwise recurrent dynamics, mixed recurrent-gradient dynamics, or purely gradient dynamics. The three alternatives admit equivalent descriptions in terms of bitrajectories and strict Lyapunov functions. We also give conditions guaranteeing connectedness of the chain graph and of the orbitwise chain graph. Finally, we compare the point dynamics of the IFS with two associated single-map systems and establish a localization principle extending several compact-space results from Conley theory for closed relations to IFSs whose global attractor has a compact neighborhood. This is a second and highly modified version of the original manuscript. Most of the examples present in the first version have been removed and will appear soon in a second manuscript.

math.DS↗
arXiv2026

The Chain Graph of a general Iterated Function System

Classical fractal geometry describes the metric and topological structure of attractors generated by contractive Iterated Function Systems (IFSs). Much less is known about their global qualitative dynamics once the contractive hypothesis is abandoned. In this article we focus on IFSs with "compact dynamics", namely those IFSs for which there exists a non-empty compact set invariant under the Hutchinson map that attracts every compact subset of the phase space. We call such a set the "global attractor" of the IFS. We introduce the "chain graph" of a general IFS as a directed graph encoding the qualitative dynamics features of the IFS. For an IFS with compact dynamics, the chain graph contains at least one node. Our main results are that the chain graph of an IFS with compact dynamics coincides with the chain graph of its restriction to its global attractor and that the chain graph of an IFS has at most as many nodes as the graph of its Hutchinson map.

math.DS↗
arXiv2026

Shadow chains and Conley chains for continuous-time semiflows

In a recent series of articles we introduced the concept of "stream of a semiflow. A stream is a closed and transitive binary relation which extends the relation "being on the orbit of" and allows to encode the qualitative behavior of a semiflow into a direct graph. The most important stream of a semiflow is its chain stream, based on Charles Conley's chains. In those previous works we omitted several details and proofs on continuous-time semiflows. In the present work we complement those articles as follows: (i) we provide a full proof of the closedness and transitivity of the chain stream for continuous-time semiflows; (ii) we introduce the concept of ``shadow chain'' for a continuous-time semiflow, based on the Anosov-Sinai-Bowen idea of pseudo-orbit. Shadow chains have the advantage that fit naturally with semiflows arising from differential equations. Our main result is that, although the shadow chain stream and the Conley chain stream are in general distinct as binary relations, they yield the same chain-recurrent set, the same nodes, and the same chain graph whenever the semiflow has strong compact dynamics. While doing this, we also introduce an equivalent definition of recurrent point of a stream in terms of forward-orbit equivalence, which simplifies several arguments below, and we strengthen the definition of s-uniform continuity of a semiflow, fixing a gap in the proof of some important results when the space is not locally compact. This is a radical revision of the first version posted to the arXiv.

math.DS↗
arXiv2026

Free maps in critical dimension on low-dimensional tori and closed surfaces

We present a method to build free immersions in critical dimension on $m$-tori for $m=2,3,4,5$ by using a factorization trick inspired by tori immersions in critical dimension. As an application, we show that the set of smooth free maps from a closed surface $M$ to \(\mathbb R^5\) is nonempty. In particular, every closed surface embeds freely in \(\mathbb R^5\).

math.GT↗
arXiv2025

What is the graph of a dynamical system?

Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our goal has been to define a meaningful concept of graph of any dynamical system. As a result, we base our definition of ``chain graph'' on ``epsilon-chains'', defining both nodes and edges of the graph in terms of chains. In particular, nodes are often maximal limit sets and there is an edge between two nodes if there is a trajectory whose forward limit set is in one node and its backward limit set is in the other. Our initial goal was to prove that every ``chain graph'' of a dynamical system is, in some sense, connected, and we prove connectedness under mild hypotheses.

math.DS↗
arXiv2025

Wandering Flows on the Plane

We study planar flows without non-wandering points and prove several properties of these flows in relation with their prolongational relation. The main results of this article are that a planar (regular) wandering flow has no generalized recurrence and has only two topological invariants: the space of its orbits and its prolongational relation (or, equivalently, its smallest stream). As a byproduct, our results show that, even in absence of any type of recurrence, the stream of a flow contains fundamental information on its behavior.

math.DS↗
arXiv2025

Streams, Graphs and Global Attractors of Dynamical Systems on Locally Compact Spaces

In a recent article, we introduced the concept of streams and graphs of a semiflow. An important related concept is the one of semiflow with {\em compact dynamics}, which we defined as a semiflow $F$ with a {\em compact global trapping region}. In this follow-up, we restrict to the important case where the phase space $X$ is locally compact and we move the focus on the concept of {\em global attractor}, a maximal compact set that attracts every compact subset of $X$. A semiflow $F$ can have many global trapping regions but, if it has a global attractor, this is unique. We modify here our original definition and we say that $F$ has compact dynamics if it has a global attractor $G$. We show that most of the qualitative properties of $F$ are inherited by the restriction $F_G$ of $F$ to $G$ and that, in case of Conley's chains stream of $F$, the qualitative behavior of $F$ and $F_G$ coincide. Moreover, if $F$ is a continuous-time semiflow, then its graph is identical to the graph of its time-1 map. Our main result is that, for each semiflow $F$ with compact dynamics over a locally compact space, the graphs of the prolongational relation of $F$ and of every stream of $F$ are connected if the global attractor is connected.

math.DS↗
arXiv2025

Streams and Graphs of Dynamical Systems

While studying gradient dynamical systems (DSs), Morse introduced the idea of encoding the qualitative behavior of a DS into a graph. Smale later refined Morse's idea and extended it to Axiom-A diffeomorphisms on manifolds. In Smale's vision, nodes are indecomposable closed invariant subsets of the non-wandering set with a dense orbit and there is an edge from node N to node M if the unstable manifold of N intersects the stable manifold of M. Since then, the decomposition of the non-wandering set was studied in many other settings, while the edges component of Smale's construction has been often overlooked. In the same years, more sophisticated generalizations of the non-wandering set were elaborated first by Auslander in 60s, by Conley in 70s and later by Easton and other authors. In our language, each of these generalizations involves the introduction of a closed and transitive extension of the non-wandering relation, that is closed but not transitive. In the present article, we develop a theory that generalizes at the same time both these lines of research. We study the general properties of closed transitive relations ("streams") containing the space of orbits of a discrete- or continuous-time semi-flow and we argue that these relations play a central role in the qualitative study of DSs. All most studied concepts of recurrence currently in literature can be defined in terms of our streams. Finally, we show how to associate to each stream a graph encoding its qualitative properties. The current revision fixes some proof, adds some missing one and adds some example and clarification.

math.DS↗
arXiv2023

Graph and backward asymptotics of the tent map

The tent map family is arguably the simplest 1-parametric family of maps with non-trivial dynamics and it is still an active subject of research. In recent works the second author, jointly with J. Yorke, studied the graph and backward limits of S-unimodal maps. In this article we generalize those results to tent-like unimodal maps. By tent-like here we mean maps that share fundamental properties that characterize tent maps, namely unimodal maps without wandering intervals nor attracting cycles and whose graph has a finite number of nodes.

math.DS↗
arXiv2022

Backward asymptotics in S-unimodal maps

While the forward trajectory of a point in a discrete dynamical system is always unique, in general a point can have infinitely many backward trajectories. The union of the limit points of all backward trajectories through $x$ was called by M.~Hero the "special $α$-limit" ($sα$-limit for short) of $x$. In this article we show that there is a hierarchy of $sα$-limits of points under iterations of a S-unimodal map: the size of the $sα$-limit of a point increases monotonically as the point gets closer and closer to the attractor. The $sα$-limit of any point of the attractor is the whole non-wandering set. This hierarchy reflects the structure of the graph of a S-unimodal map recently introduced jointly by Jim Yorke and the present author.

math.DS↗
arXiv2021

The graph of the logistic map is a tower

The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node $A$ to node $B$ if, using arbitrary small perturbations, a trajectory starting from any point of A can be steered to any point of B. In this article we describe the graph of the logistic map. Our main result is that the graph is always a tower, namely there is an edge connecting each pair of distinct nodes. Notice that these graphs never contain cycles. If there is an edge from node A to node B, the unstable manifold of some periodic orbit in A contains points that eventually map onto B. For special parameter values, this tower has infinitely many nodes.

math.DS↗
arXiv2021

Infinite towers in the graph of a dynamical system

Chaotic attractors, chaotic saddles and periodic orbits are examples of chain-recurrent sets. Using arbitrary small controls, a trajectory starting from any point in a chain-recurrent set can be steered to any other in that set. The qualitative behavior of a dynamical system can be encapsulated in a graph. Its nodes are chain-recurrent sets. There is an edge from node A to node B if, using arbitrary small controls, a trajectory starting from any point of A can be steered to any point of B. We discuss physical systems that have infinitely many disjoint coexisting nodes. Such infinite collections can occur for many carefully chosen parameter values. The logistic map is such a system, as we showed in arXiv:2008.08338. To illustrate these very common phenomena, we compare the Lorenz system and the logistic map and we show how extremely similar their bifurcation diagrams are in some parameter ranges.

nlin.CD↗
arXiv2019

Julia sets of Newton maps of real quadratic polynomial maps on the plane

We study numerically the $α$- and $ω$-limits of the Newton maps of two of the most elementary families of polynomial transformations on the plane: those with a linear component and those with both components of degree two. Our results are fully consistent with some conjectures we posed in a recent work about the dynamics of Newton maps.

math.DS↗
arXiv2018

"Simple Dynamics" conjectures for some real Newton maps on the plane

We collect from several sources some of the most important results on the forward and backward limits of points under real and complex rational functions, and in particular real and complex Newton maps, and we provide numerical evidence that a fundamental result by B.Barna on the dynamics of Newton's method on the real line extends to the real plane.

math.DS↗
arXiv2018

Quasiperiodic functions on the plane and electron transport phenomena

While quasiperiodic functions in one variable appeared in applications since Eighteen hundreds, for example in connection with the trajectories of mechanical systems with 2n degress of freedom having n commuting first integrals, the first applications of multivariable quasiperiodic functions were found only in Seventies, in connection with solitonic solutions of the KdV equation. Later, several other physical applications were found, especially in connection with Solid State Physics, in particular with the electron transport phenomena. In this article we reformulate, specifically in terms of the topology of level sets of quasiperiodic functions on the plane, some fundamental theoretical results found in Eighties and Nineties, then we review the physical models of electron transport and their connections with quasiperiodic functions and finally we present some old and new numerical results on the level sets of some specific family of quasiperiodic functions, some of which related to the magnetoresistance in normal metals.

math-ph↗
arXiv2017

A survey on quasiperiodic topology

This article is a survey of the Novikov problem of the structure of leaves of the foliations induced by a collection of closed 1-forms in a compact manifold $M$. Equivalently, this is to the study of the level sets of multivalued functions on $M$. To date, this problem was thoroughly investigated only for $M=\Bbb T^n$ and multivalued maps $F:\Bbb T^n\to\Bbb R^{n-1}$ in three different particular cases: when all components of $F$ but one are multivalued, started by Novikov in 1981, when all components of $F$ but one are singlevalued, started by Zorich in 1994, when none of the components is singlevalued, started by Arnold in 1991. The first two problems can be formulated as the study of the level sets of certain quasiperiodic functions, the last as level sets of pseudoperiodic functions. In this survey we present the main analytical and numerical results to date and some physical phenomena where they play a fundamental role.

math.GT↗