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Maria José Pacifico

Publications and source records attributed to Maria José Pacifico.

11 recordsLinked to original sources
arXiv2026

Komuro Expansivity and Periodic Orbit Growth for Multi-Singular Hyperbolic Flows

In this paper we prove that every multi-singular hyperbolic set of a $C^1$ flow is Komuro expansive. This is the strongest natural form of expansivity for flows with singularities accumulated by regular orbits, allowing arbitrary orientation-preserving time reparametrizations. We then use this to establish a two-sided asymptotic counting bound of the form $e^{ht}/t$ for periodic orbits, and prove that the normalized orbit measures converge to the unique measure of maximal entropy. We also establish the same results for a $C^1$ open and dense subset of star vector fields.

math.DS↗
arXiv2026

Optimal Play in Hex on Finite and Infinite Boards

The game of Hex is one of the most celebrated connection games in combinatorics. Although it is known that the first player always has a winning strategy, very little is understood about the complexity of optimal play. Following Campbell's introduction of the parameters $λ(n)$ and $δ(n)$, measuring respectively the length of the shortest guaranteed winning path and the minimum number of stones required by the first player to force a win, we establish new structural properties of these quantities. Our main result determines the exact value \[ λ(5)=7, \] thereby confirming one of Campbell's conjectures. We also show that the corresponding problem on the infinite strip $5\times\infty$ has a markedly different behavior. In this setting we prove that \[ λ(5\times\infty)=5, \] which is strictly smaller than $λ(5)$. To the best of our knowledge, this paper provides the first solution to one of Campbell's 2004 conjectures and answers a question posed by Stromquist (2006) in the affirmative. Furthermore, it uncovers a fundamental distinction between finite and infinite boards, offering new insights into the nature of optimal winning strategies in Hex.

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arXiv2026

Mean dimension of general iterated function systems

In this paper, we introduce and investigate the notions of Mean Dimension and Metric Mean Dimension for generalized iterated function systems (IFS). We establish basic properties of these invariants and prove that Mean Dimension is always bounded above by the lower Metric Mean Dimension and the upper Metric Mean Dimension in this setting. We further show that generalized iterated function systems with the Small Boundary Property have zero Mean Dimension. Finally, we introduce a Gluing Orbit Property for generalized iterated function systems and prove that, under suitable transitivity and non-rigidity assumptions, it guarantees positive topological entropy.

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arXiv2025

Partially hyperbolic diffeomorphisims with a finite number of measures of maximal entropy

We prove the finiteness of ergodic measures of maximal entropy for partially hyperbolic diffeomorphisms where the center direction has a dominated decomposition into one dimensional bundle and there is a uniform lower bound for the absolute value of the Lyapunov exponents. As applications we prove finiteness for a class derived from Anosov partially hyperbolic diffeomorphisms defined on $\mathbb{T}^4$ and that in a class of skew product over partially hyperbolic diffeomorphisms there exists a $C^1$ open and $C^r$ dense set of diffeomorphisms with a finite number of ergodic measures of maximal entropy. We also study the upper semicontinuity of the number of measures of maximal entropy with respect to the diffeomorphism.

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arXiv2022

Expansiveness, Lyapunov exponents and entropy for set valued maps

In this paper we introduce a notion of expansiveness for a set valued map defined on a topological space different from that given by Richard Williams at \cite{Wi, Wi2} and prove that the topological entropy of an expansive set valued map defined on a Peano space of positive dimension is greater than zero. We define Lyapunov exponent for set valued maps and prove that positiveness of its Lyapunov exponent implies positiveness for the topological entropy. Finally we introduce the definition of (Lyapunov) stable points for set valued maps and prove a dichotomy for the set of stable points for set valued maps defined on Peano spaces: either it is empty or the whole space.

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arXiv2021

An entropy dichotomy for singular star flows

We show that non-trivial chain recurrent classes for generic $C^1$ star flows satisfy a dichotomy: either they have zero topological entropy, or they must be isolated. Moreover, chain recurrent classes for generic star flows with zero entropy must be sectional hyperbolic, and cannot be detected by any non-trivial ergodic invariant probability. As a result, we show that $C^1$ generic star flows have only finitely many Lyapunov stable chain recurrent classes.

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arXiv2013

N-expansive homeomorphisms on surfaces

We exploit the techniques developed in [Le] to study N-expansive homeomorphisms on surfaces. We prove that when f is a 2-expansive homeomorphism defined on a compact boundaryless surface M without wandering points then f is expansive. This condition on the wandering set cannot be relaxed: we present an example of a 2-expansive homeomorphisms on the bitorus with wandering points that is not expansive.

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arXiv2011

Evolution of the population of Microtus Epiroticus: the Yoccoz-Birkeland model

We study the discretized version of a dynamical system given by a model proposed by Yoccoz and Birkeland to describe the evolution of the population of Microtus Epiroticus on Svalbard Islands, see http://zipcodezoo.com/Animals/M/Microtus_epiroticus . We prove that this discretized version has an attractor Lambda with a hyperbolic 2-periodic point p in it. For certain values of the parameters the system restricted to the attractor exhibits sensibility to initial conditions. Under certain assumptions that seems to be sustained by numerical simulations, the system is topologically mixing explaining some of the high oscillations observed in Nature. Moreover, we estimate its order-2 Kolmogorov entropy obtaining a positive value. Finally we give numerical evidence that there is a homoclinic point associated with p.

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