arXiv subjects

Francesc Mañosas

Publications and source records attributed to Francesc Mañosas.

11 recordsLinked to original sources

arXiv2026

Entropy and ω-limit sets on invariant graphs for a class of planar piecewise linear maps

We consider the family of piecewise linear maps $F(x,y)=\left(|x| - y + a, x - |y| + b\right),$ where $(a,b)\in \mathbb{R}^2$. In previous work, we identified that certain maps of this class possess one-dimensional invariant sets, planar graphs, that capture the global dynamics of the system. Within these graphs, chaotic dynamics emerge for certain parameter values, leading to an intermediate dynamical regime between regular behavior and full-plane chaos. In the present study, we revisit this family and analyze in detail the topological entropy as a function of a bifurcation parameter, finding that transitions from positive to zero entropy are continuous for certain parameter values and discontinuous for others. We also provide a methodology for determining arbitrarily sharp rational bounds for the bifurcation values at which this transition occurs. Finally, motivated by the limitations of numerical simulations in detecting the complex dynamics within these graphs, we prove that for some parameter values, there exists a full-measure set in these graphs where orbits converge to at most three omega-limit sets, which, when the parameter values are rational, correspond to periodic orbits.

math.DS↗

arXiv2025

On two families of iterative methods without memory

We study two natural families of methods of order $n\ge 2$ that are useful for solving numerically one variable equations $f(x)=0.$ The first family consists on the methods that depend on $x,f(x)$ and its successive derivatives up to $f^{(n-1)}(x)$ and the second family comprises methods that depend on $x,g(x)$ until $g^{\circ n}(x),$ where $g^{\circ m}(x)=g(g^{\circ (m-1)}(x))$ and $g(x)=f(x)+x$. The first family includes the well-known Newton, Chebyshev, and Halley methods, while the second one contains the Steffensen method. Although the results for the first type of methods are well known and classical, we provide new, simple, detailed, and self-contained proofs.

math.NA↗

arXiv2025

Invariant graphs and dynamics of a family of continuous piecewise linear planar maps

We consider the family of piecewise linear maps $$F_{a,b}(x,y)=\left(|x| - y + a, x - |y| + b\right),$$ where $(a,b)\in \mathbb{R}^2$. This family belongs to a wider one that has deserved some interest in the recent years as it provides a framework for generalized Lozi-type maps. Among our results, we prove that for $a\ge 0$ all the orbits are eventually periodic and moreover that there are at most three different periodic behaviors formed by at most seven points. For $a<0$ we prove that for each $b\in\mathbb{R}$ there exists a compact graph $Γ,$ which is invariant under the map $F$, such that for each $(x,y)\in \mathbb{R}^2$ there exists $n\in\mathbb{N}$ (that may depend on $x$) such that $F_{a,b}^n(x,y)\in Γ.$ We give explicitly all these invariant graphs and we characterize the dynamics of the map restricted to the corresponding graph for all $(a,b)\in\mathbb{R}^2$ obtaining, among other results, a full characterization of when $F_{a,b}|_Γ$ has positive or zero entropy.

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arXiv2023

Characterization of the tree cycles with minimum positive entropy for any period

Consider, for any integer $n\ge3$, the set $\text{Pos}_n$ of all $n$-periodic tree patterns with positive topological entropy and the set $\text{Irr}_n\subset\text{Pos}_n$ of all $n$-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\text{Pos}_n$, $\text{Irr}_n$ and $\text{Pos}_n\setminus\text{Irr}_n$. Let $λ_n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty)$. We explicitly construct an irreducible $n$-periodic tree pattern $\mathcal{Q}_n$ whose entropy is $\log(λ_n)$. We prove that this entropy is minimum in $\text{Pos}_n$. Since the pattern $\mathcal{Q}_n$ is irreducible, $\mathcal{Q}_n$ also minimizes the entropy in the family $\text{Irr}_n$. We also prove that the minimum positive entropy in the set $\text{Pos}_n\setminus\text{Irr}_n$ (which is nonempty only for composite integers $n\ge6$) is $\log(λ_{n/p})/p$, where $p$ is the least prime factor of $n$.

math.DS↗

arXiv2023

On some rational piecewise linear rotations

We study the dynamics of the piecewise planar rotations $F_λ(z)=λ(z-H(z)), $ with $z\in\C$, $H(z)=1$ if $\mathrm{Im}(z)\ge0,$ $H(z)=-1$ if $\mathrm{Im}(z)<0,$ and $λ=\mathrm{e}^{i α} \in\C$, being $α$ a rational multiple of $π$. Our main results establish the dynamics in the so called regular set, which is the complementary of the closure of the set formed by the preimages of the discontinuity line. We prove that any connected component of this set is open, bounded and periodic under the action of $F_λ$, with a period $\ell,$ that depends on the connected component. Furthermore, $F_λ^\ell $ restricted to each component acts as a rotation with a period which also depends on the connected component. As a consequence, any point in the regular set is periodic. Among other results, we also prove that for any connected component of the regular set, its boundary is a convex polygon with certain maximum number of sides.

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arXiv2021

Pointwise periodic maps with quantized first integrals

We describe the global dynamics of some pointwise periodic piecewise linear maps in the plane that exhibit interesting dynamic features. For each of these maps we find a first integral. For these integrals the set of values are discrete, thus quantized. Furthermore, the level sets are bounded sets whose interior is formed by a finite number of open tiles of certain regular or uniform tessellations. The action of the maps on each invariant set of tiles is described geometrically.

math.DS↗

arXiv2016

A quasiperiodically forced skew-product on the cylinder without fixed-curves

In [FJJK] the Sharkovskiĭ Theorem was extended to periodic orbits of strips of quasiperiodic skew products in the cylinder. In this paper we deal with the following natural question that arises in this setting: Does Sharkovskiĭ Theorem holds when restricted to curves instead of general strips?mWe answer this question in the negative by constructing a counterexample: We construct a map having a periodic orbit of period 2 of curves (which is, in fact, the upper and lower circles of the cylinder) and without any invariant curve. In particular this shows that there exist quasiperiodic skew products in the cylinder without invariant curves.

math.DS↗

arXiv2016

On the number of limit cycles for perturbed pendulum equations

We consider perturbed pendulum-like equations on the cylinder of the form $ \ddot x+\sin(x)= \varepsilon \sum_{s=0}^{m}{Q_{n,s} (x)\, \dot x^{s}}$ where $Q_{n,s}$ are trigonometric polynomials of degree $n$, and study the number of limit cycles that bifurcate from the periodic orbits of the unperturbed case $\varepsilon=0$ in terms of $m$ and $n$. Our first result gives upper bounds on the number of zeros of its associated first order Melnikov function, in both the oscillatory and the rotary regions. These upper bounds are obtained expressing the corresponding Abelian integrals in terms of polynomials and the complete elliptic functions of first and second kind. Some further results give sharp bounds on the number of zeros of these integrals by identifying subfamilies which are shown to be Chebyshev systems.

math.DS↗

arXiv2014

Forcing and entropy of strip patterns of quasiperiodic skew products in the cylinder

We extend the results and techniques from \cite{FJJK} to study the combinatorial dynamics (\emph{forcing}) and entropy of quasiperiodically forced skew-products on the cylinder. For these maps we prove that a cyclic permutation $τ$ forces a cyclic permutation $ν$ as interval patterns if and only if $τ$ forces $ν$ as cylinder patterns. This result gives as a corollary the Sharkovski\uı Theorem for quasiperiodically forced skew-products on the cylinder proved in \cite{FJJK}. Next, the notion of $s$-horseshoe is defined for quasiperiodically forced skew-products on the cylinder and it is proved, as in the interval case, that if a quasiperiodically forced skew-product on the cylinder has an $s$-horseshoe then its topological entropy is larger than or equals to $\log(s).$ Finally, if a quasiperiodically forced skew-product on the cylinder has a periodic orbit with pattern $τ,$ then $h(F) \ge h(f_τ),$ where $f_τ$ denotes the \emph{connect-the-dots} interval map over a periodic orbit with pattern $τ.$ This implies that if the period of $τ$ is $2^n q$ with $n \ge 0$ and $q \ge 1$ odd, then $h(F) \ge \tfrac{\log(λ_q)}{2^n}$, where $λ_1 = 1$ and, for each $q \ge 3,$ $λ_q$ is the largest root of the polynomial $x^{q} - 2x^{q-2} - 1.$ Moreover, for every $m=2^n q$ with $n \ge 0$ and $q \ge 1$ odd, there exists a quasiperiodically forced skew-product on the cylinder $F_m$ with a periodic orbit of period $m$ such that $h(F_m) = \tfrac{\log(λ_q)}{2^n}.$ This extends the analogous result for interval maps to quasiperiodically forced skew-products on the cylinder.

math.DS↗

arXiv2014

Volume entropy for minimal presentations of surface groups in all ranks

We study the volume entropy of a class of presentations (including the classical ones) for all surface groups, called \emph{minimal geometric presentations}. We rediscover a formula first obtained by Cannon and Wagreich with the computation in a non published manuscript by Cannon. The result is surprising: an explicit polynomial of degree $n$, the rank of the group, encodes the volume entropy of all classical presentations of surface groups. The approach we use is completely different. It is based on a dynamical system construction following an idea due to Bowen and Series and extended to all geometric presentations in. The result is an explicit formula for the volume entropy of minimal presentations for all surface groups, showing a polynomial dependence in the rank $n>2$. We prove that for a surface group $G_n$ of rank $n$ with a classical presentation $P_n$ the volume entropy is $\log(λ_n)$, where $λ_n$ is the unique real root larger than one of the polynomial \[ x^{n} - 2(n - 1) \sum_{j=1}^{n-1} x^{j} + 1. \]

math.DS↗

arXiv2008

A Chebyshev criterion for Abelian integrals

We present a criterion that provides an easy sufficient condition in order that a collection of Abelian integrals has the Chebyshev property. This condition involves the functions in the integrand of the Abelian integrals and can be checked, in many cases, in a purely algebraic way. By using this criterion, several known results are obtained in a shorter way and some new results, which could not be tackled by the known standard methods, can also be deduced.

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