arXiv subjects

Bernardo San Martín

Publications and source records attributed to Bernardo San Martín.

4 recordsLinked to original sources

arXiv2026

On the Lorenz-Fibonacci sequences and substitutions

In the present article, we consider two families of integer sequences, the $(k,r)$-Lorenz-Fibonacci sequences of the first and second kind, whose characteristic polynomial is $$p_{k,r}(x)=x^k-x^{k-1}-\cdots- x^r+x^{r-1}+\cdots +x+1,$$ where $k\geq 2r+1$, and $k\geq 3$. These families include the well-known $k$-bonacci sequence, when $r=0$. These sequences arise naturally in the study of the dynamics of the Lorenz attractor. We introduce two families of substitutions (in an alphabet of $k$ symbols) so that they are associated with each of the families of integer sequences and share the same polynomial. We study the main combinatorial properties of these substitutions.

math.NT↗

arXiv2025

On the Use of the Schwarzian derivative in Real One-Dimensional Dynamics

In the study of properties within one dimensional dynamics, the assumption of a negative Schwarzian derivative has been shown to be very useful. However, this condition may seem somewhat arbitrary, as it is not inherently a dynamical condition, except for the fact that it is preserved under iteration. In this brief work, we show that the negative Schwarzian derivative condition is not arbitrary in any sense but is instead strictly related to the fulfillment of the Minimum Principle for the derivative of the map and its iterates, which plays a key role in the proof of Singer's Theorem.

math.DS↗

arXiv2019

Topological entropy and metric entropy for regular impulsive semiflows

In arXiv:1801.01238 a variation of Bowen's topological entropy that can be applied to the study of discontinuous semiflows on compact metric spaces was introduced. The main novetly is the use of certain family of pseudosemimetrics associated to the semiflow, in a such a way that in the continuous case they coincide with the classical Bowen's entropies. It was proved that regular impulsive semiflow can be semiconjugated by uniformly continuous bijection to certain continuous semiflow defined in a compact metric space. In the cited paper, the entropy is defined by the growth rate of separated sets. The main results of the present work is show that, for regular impulsive semiflows, the entropy defined by using spanning sets agrees with the entropy using separated sets. Moreover, this notion of entropy satisfy a variational principle. This results are obtained introducing a little change on the family of pseudosemimetrics associated to.

math.DS↗

arXiv2018

Topological Entropy for Discontinuous Semiflows

We study two variations of Bowen's definitions of topological entropy based on separated and spanning sets which can be applied to the study of discontinuous semiflows on compact metric spaces. We prove that these definitions reduce to Bowen's ones in the case of continuous semiflows. As a second result, we prove that our entropies give a lower bound for the $τ$-entropy defined by Alves, Carvalho and Vásquez (2015). Finally, we prove that for impulsive semiflows satisfying certain regularity condition, there exists a continuous semiflow defined on another compact metric space which is related to the first one by a semiconjugation, and whose topological entropy equals our extended notion of topological entropy by using separated sets for the original semiflow.

math.DS↗