arXiv subjects

Eduardo Santana

Publications and source records attributed to Eduardo Santana.

11 recordsLinked to original sources

arXiv2026

Thermodynamic Dictionary for Syracuse-like Maps

In this paper we estabilsh a dictionary between the behavior of a general map $f: X \to X$ which is defined by using a decomposition into disjoint subsets $X = A \cup B$ and the thermodynamic formalism. The approach of this paper is set theoretical, topological and ergodic. We define a key topology and consider its Borel $σ$-algebra in order to study the thermodynamic formalism for the general setting, proving that recurrence implies periodicity, the topological entropy is zero, we can decompose the invariant measures into a general sum of ergodic ones and we establish a dictionary between the cycles and the equilibrium states of continuous potentials. All this is done for the general setting, with applications for the Syracuse maps and the Collatz map in particular, as a significant application of the dictionary, proving the finiteness of cycles. Finally, we prove that any Syracuse map has absence of divergent orbits, proving the weak Collatz Conjecture.

math.DS↗

arXiv2025

Equilibrium States for Random Zooming Systems

In this work, based on Pinheiro for deterministic systems, we extend the notion of zooming systems to the random context and based on the technique of Arbieto-Matheus-Oliveira we prove the existence of equilibrium states for which we call random zooming potentials, that include the hyperbolic ones, possibly with the presence of a critical set. With a mild condition, we obtain uniqueness. As an example of existence, we have the so-called random Viana maps with critical points. We also prove that the classes of random zooming potentials and random hyperbolic potentials are equivalent and also contain the null potential, giving measures of maximal entropy.

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arXiv2025

Evolution on Sets

We introduce the notion of evolution on sets and study several sets endowed with this structure and obtain some results about this new notion.

math.GM↗

arXiv2025

Open Zooming Systems: Uniqueness of Equilibrium States

We study open zooming systems and potentials with uniqueness of equilibrium states. The uniqueness is established for a certain class of zooming potentials when the map is topologically exact, including the null one. Also, with equilibrium stability, we prove that there exists a countable and an open sets of continuous potentials with uniqueness which are both dense in the set of continuous potentials with finiteness. The results here are related to the works [14] and [49] where finiteness and stability are studied.

math.DS↗

arXiv2025

Subcohomology and a Livsic Theorem for Zooming Systems

In the context of continuous zooming systems $f:M \to M$ on a compact metric space $M$, which include the non-uniformly expanding ones, possibly with the presence of a critical set, with the zooming set dense in $M$, we prove that any Hölder potential $ϕ: M \to \mathbb{R}$ for which the integrals $\int ϕdμ\geq 0$ with respect to any $f$-invariant probability $μ$, admits a continuous function $λ_{0} : M \to \mathbb{R}$ (which can be Hölder if some integral is positive) such that \[ ϕ\geq λ_{0}- λ_{0} \circ f. \] This extends a result in [9] for $C^{1}$-expanding maps on the circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ to important classes of maps as uniformly expanding, local diffeomorphisms with non-uniform expansion, Viana maps, Benedicks-Carleson maps and Rovella maps. We also give an example beyond the exponential contractions context. Moreover, in the case of the integrals $\int ϕdμ= 0$ with respect to any $f$-invariant probability $μ$ and the set of periodic points to be dense in $M$, we obtain a version of the Livsic Theorem, that is, the functions $λ_{0}$ can be taken such that \[ ϕ= λ_{0}- λ_{0} \circ f. \] Additionally, we also prove that the measure which maximizes the integrals is unique for a residual set of potentials.

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arXiv2025

Uniqueness of Equilibrium Measure for a Family of Partially Hyperbolic Horseshoes

In this paper, we show the uniqueness of equilibrium state for a family of partially hyperbolic horseshoes, introduced in [12] for some classes of continuous potentials. For the first class, the method used here is making use of the Sarig's theory for countable shifts. For this purpose, we study the dynamics of an induced map associated to the horseshoe map, we build a symbolic system with infinitely many symbols that is topologically conjugated to this induced map and we show that the induced potential is locally Hölder and recurrent. For the second class, by following ideas of [29] and [31], we prove that uniqueness for the horseshoe is equivalent to uniqueness for the restriction to a non-uniformly expanding map, which is a hyperbolic potential and then has uniqueness. Both classes include potentials with high variation, differently from previous results for potentials with low variation. We also prove uniqueness when the potential presents its supremum at a special fixed point and less than the pressure.

math.DS↗

arXiv2025

Equilibrium States for Open Zooming Systems

In this work, we construct Markov structures for zooming systems adapted to holes of a special type. Our construction is based on backward contractions provided by zooming times. These Markov structures may be used to code the open zooming systems. In the context of open zooming systems, possibly with the presence of a critical/singular set, we prove the existence of finitely many ergodic zooming equilibrium states for zooming potentials whose induced potential is locally Hölder. For example, the zooming Hölder continuous. Among the zooming ones are the so-called \textit{hyperbolic potentials} and also what we call \textit{bounded distortion potentials}, having as a particular case the \textit{pseudo-geometric potentials} $ϕ_{t} = -t \log J_μf $, where $J_μf$ is a Jacobian of the reference zooming measure. Moreover, for this last class of potentials, we show the existence of what we call pseudo-conformal measures. Afterwards, with a mild condition, we prove uniqueness of equilibrium state with no requirement of transitivity. The technique consists in using an inducing scheme in a finite Markov structure with infinitely many symbols to code the dynamics to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. To obtain a pseudo-conformal measure, we "spread" the conformal one which exists for the symbolic dynamics. The uniqueness is obtained by showing that the equilibrium states are liftable to the same inducing scheme. Finally, we show that the class of hyperbolic potentials is equivalent to the class of continuous zooming potentials. Moreover, we give a class of examples of hyperbolic potentials (including the null one). It implies the existence and uniqueness of equilibrium state. Among the maps considered is the important class known as Viana maps.

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arXiv2025

Equilibrium Stability for Open Zooming Systems

We prove that for a wide family of open zooming systems and zooming potentials we have equilibrium stability, i.e., the equilibrium states depend continuously on the dynamics and the potential. We consider the open zooming systems with special holes and quite general contractions and zooming potentials with locally Hölder induced potential, which include the Hölder ones. We also prove stability for skew-products with the base being a zooming system like above. As a consequence of finiteness and stability, we obtain uniqueness of equilibrium state.

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arXiv2024

Random Expansive Measures

The notion of expansivity and its generalizations (measure expansive, measure positively expansive, continuum-wise expansive, countably-expansive) are well known for deterministic systems and can be a useful property for studying significant type of behavior, such as chaotic one. This study aims to extend these notions into a random context and prove a relationship between relative positive entropy and random expansive measures and apply it to show that if a random dynamical system has positive relative topological entropy then the $w$-stable classes have zero measure for the conditional measures. We also prove that there exists a probability measure that is both invariant and expansive. Moreover, we obtain a relation between the notions of random expansive measures and random countably-expansive systems.

math.DS↗

arXiv2024

Equilibrium states for hyperbolic potentials via inducing schemes

In a context of non-uniformly expanding maps, possibly with the presence of a critical set, we prove the existence of finitely many ergodic equilibrium states for hyperbolic potentials. Moreover, the equilibrium states are expanding measures. This generalizes a result due to Ramos and Viana, where analytical methods are used for maps with no critical sets. The strategy here consists in using a finite number of inducing schemes with a Markov structure in infinitely many symbols to code the dynamics, to obtain an equilibrium state for the associated symbolic dynamics and then projecting it to obtain an equilibrium state for the original map. We apply our results to the important class of multidimensional Viana maps.

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arXiv2023

A Livšic-type theorem and some regularity properties for nonadditive sequences of potentials

We study some notions of cohomology for asymptotically additive sequences and prove a Livšic-type result for almost additive sequences of potentials. As a consequence, we are able to characterize almost additive sequences based on their equilibrium measures and also show the existence of almost (and asymptotically) additive sequences of Hölder continuous functions satisfying the bounded variation condition (with a unique equilibrium measure) and which are not physically equivalent to any additive sequence generated by a Hölder continuous function. None of these examples were previously known, even in the case of full shifts of finite type. Moreover, we also use our main result to suggest a classification of almost additive sequences based on physical equivalence relations with respect to the classical additive setup.

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