arXiv subjects

Godofredo Iommi

Publications and source records attributed to Godofredo Iommi.

At least 19 recordsLinked to original sources

arXiv2026

Equidistribution and thermodynamics at infinity

We prove level-2 large deviation upper bounds for potentials on countable Markov shifts and for suspension semi-flows over countable Markov shifts. For strongly positive recurrent potentials, we establish equidistribution of weighted empirical measures toward the corresponding equilibrium state. We then apply these results to interval maps, obtaining, in particular, equidistribution and large-deviation estimates for measures supported on boundary points. For the Gauss map, this yields equidistribution results on rational numbers, including a theorem of David and Shapira.

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arXiv2026

Continuous pointwise ergodicity for semigroup actions on locally compact spaces

We investigate proper actions of arbitrary semigroups on separable locally compact metric spaces, where point orbits are allowed to escape to infinity. An action is pointwise uniquely ergodic when every compact orbit closure supports exactly one invariant probability measure and non-compact orbit closures support none. The associated ergodic map therefore assigns the selected probability measure to non-escaping points and the zero subprobability to escaping ones. Under the hypothesis that compact orbit closures admit at least one invariant measure, we establish that the weak* continuity of this ergodic map together with a vanishing at infinity condition is equivalent to the mean ergodicity of the Koopman representation on the space of continuous functions vanishing at infinity. In consequence, every such function and every finite signed measure split uniquely into invariant components and limits of coboundaries. The corresponding projections are obtained by integration against the ergodic map. Because this operator-theoretic characterization avoids explicit averaging schemes, it remains applicable even to semigroups without F{\o}lner sequences. When restricted to countable, discrete, bicancellative, and left amenable semigroups, these properties are shown to be equivalent to the uniform convergence of F{\o}lner averages and the weak-star continuity of their dual limits, extending classical results for group actions on compact spaces. Furthermore, we identify the space of ergodic measures with a compactified ergodic quotient, prove that the invariant measure simplex is Bauer, and show that these structural properties descend through proper factor maps. The theoretical framework is complemented by dynamical examples, including a continuously pointwise ergodic subshift that exhibits discontinuous entropy along the ergodic map.

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arXiv2025

Convergence of the Birkhoff spectrum for nonintegrable observables

We consider interval maps with countably many full branches and observables with polynomial tails. We show that the Birkhoff spectrum is real analytic and that its convergence to the Hausdorff dimension of the repeller is governed by the polynomial tail exponent. This result extends previous work by Arima on more regular observables and demonstrates how the tail behaviour influences the structure of the Birkhoff spectrum. Our proof relies on techniques from thermodynamic formalism and tail estimates for the observable and our applications are to natural observations on Gauss maps, Lüroth transformations as well as to a the first return time for a class of induced Manneville-Pomeau maps.

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arXiv2025

Non-compact spaces of invariant measures

We study a compactification of the space of invariant probability measures for a transitive countable Markov shift. We prove that it is affine homeomorphic to the Poulsen simplex. Furthermore, we establish that, depending on a combinatorial property of the shift space, the compactification contains either a single new ergodic measure or a dense set of them. As an application of our results, we prove that the space of ergodic probability measures of a transitive countable Markov shift is homeomorphic to $\ell_2$, extending to the non-compact setting a known result for subshifts of finite type. Additionally, we explore implications for thermodynamic formalism, including a version of the dual variational principle for transitive countable Markov shifts with uniformly continuous potentials.

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arXiv2025

Ergodic theorems for set maps under weak forms of additivity

We investigate various relaxations of additivity for set maps into Banach spaces in the context of representations of amenable groups. Specifically, we establish conditions under which asymptotically additive and almost additive set maps are equivalent. For Banach lattices, we further show that these notions are related to a third weak form of additivity adapted to the order structure of the space. By utilizing these equivalences and reducing non-additive settings to the additive one by finding suitable additive realizations, we derive new non-additive ergodic theorems for amenable group representations into Banach spaces and streamline proofs of existing results in certain cases.

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arXiv2025

Additive realizations of asymptotically additive set maps

Given a countable discrete amenable group, we study conditions under which a set map into a Banach space (or more generally, a complete semi-normed space) can be realized as the ergodic sum of a vector under a group representation, such that the realization is asymptotically indistinguishable from the original map. We show that for uniformly bounded group representations, this property is characterized by the class of bounded asymptotically additive set maps, extending previous work for sequences in Banach spaces and on the case of a single non-expansive linear map. Additionally, we develop a relative version of this characterization, identifying when the additive realization can be chosen within a prescribed target set. As an application, our results generalize central aspects of thermodynamic formalism, bridging the additive and asymptotically additive frameworks.

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arXiv2024

Hölder continuity of measures for heavy tail potentials

For a class of potentials $ψ$ satisfying a condition depending on the roof function of a suspension (semi)flow, we show an EKP inequality, which can be interpreted as a Hölder continuity property in the weak${^*}$ norm of measures, with respect to the pressure of those measures, where the Hölder exponent depends on the $L^q$-space that $ψ$ belongs to. This also captures a new type of phase transition for intermittent (semi)flows (and maps).

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arXiv2024

Odometers, Backward continued fractions and counting rationals

It has been more than twenty years since Moshe Newman, based on work by Neil Calkin and Herbert Wilf, introduced an explicit bijection between the rational and natural numbers. Interestingly, this bijection is dynamic in nature. Indeed, Newman's map has the property that the orbit of zero provides the required bijection. Claudio Bonanno and Stefano Isola, using continued fractions expansions, described the dynamics of its first return time map T. They proved that it is topologically conjugated to the dyadic odometer. In this article, we prove that the correct numerical system needed to analyze this map is the backward continued fractions. Indeed, this approach has the advantage that it provides explicitly the action of T on the expansion. As a by-product, we naturally obtain an explicit formula for Minkowski's question mark function in terms of backward continued fractions. The whole point of Newman was to provide an explicit bijection, our approach shares the same taste for the explicit.

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arXiv2024

Odometers in non-compact spaces

We define an odometer in the Baire space. That is the non-compact space of one sided sequences of natural numbers. We go on to prove that it is topologically conjugated to the dyadic odometer restricted to an appropriate non-compact subset of the space of one sided sequences on two symbols. We extend the action of the odometer to finite words. It turns out that, arranging the finite words in appropriate binary tree, this extension runs through the tree from left-right and top-down. Several well known binary trees made out of rational numbers, such as the Kepler tree or the Calkin and Wilf tree, are recovered in this way. Indeed, it suffices to identify finite words with rational numbers by means of certain continued fractions. The odometric action allow us to recover, in a unified way, several counting results for the rational numbers. Moreover, associated to certain interval maps with countably many branches we construct their corresponding odometers. Explicit formulas are provided in the cases of the Gauss map (continued fractions) and the Renyi map (backward continued fractions).

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arXiv2022

Escape of entropy for countable Markov shifts

In this paper we study ergodic theory of countable Markov shifts. These are dynamical systems defined over non-compact spaces. Our main result relates the escape of mass, the measure theoretic entropy, and the entropy at infinity of the system. This relation has several consequences. For example we obtain that the entropy map is upper semi-continuous and that the ergodic measures form an entropy dense subset. Our results also provide new proofs of results describing the existence and stability of the measure of maximal entropy. We relate the entropy at infinity with the Hausdorff dimension of the set of recurrent points that escape on average. Of independent interest, we prove a version of Katok's entropy formula in this non-compact setting.

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arXiv2022

Differentiability of the pressure in non-compact spaces

Regularity properties of the pressure are related to phase transitions. In this article we study thermodynamic formalism for systems defined in non-compact phase spaces, our main focus being countable Markov shifts. We produce metric compactifications of the space which allow us to prove that the pressure is differentiable on a residual set and outside an Aronszajn null set in the space of uniformly continuous functions. We establish a criterion, the so called sectorially arranged property, which implies that the pressure in the original system and in the compactification coincide. Examples showing that the compactifications can have rich boundaries, for example a Cantor set, are provided.

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arXiv2021

Time change for flows and thermodynamic formalism

This paper is devoted to study how do thermodynamic formalism quantities varies for time changes of suspension flows defined over countable Markov shifts. We prove that in general no quantity is preserved. We also make a topological description of the space of suspension flows according to certain thermodynamic quantities. For example, we show that the set of suspension flows defined over the full shift on a countable alphabet having finite entropy is open. Of independent interest might be a set of analytic tools we use to construct examples with prescribed thermodynamic behaviour.

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arXiv2021

The Halász-Székely Barycenter

We introduce a notion of barycenter of a probability measure related to the symmetric mean of a collection of nonnegative real numbers. Our definition is inspired by the work of Halász and Székely, who in 1976 proved a law of large numbers for symmetric means. We study analytic properties of this Halász-Székely barycenter. We establish fundamental inequalities that relate the symmetric mean of a list of nonnegative real numbers with the barycenter of the measure uniformly supported on these points. As consequence, we go on to establish an ergodic theorem stating that the symmetric means of a sequence of dynamical observations converges to the Halász-Székely barycenter of the corresponding distribution.

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arXiv2019

Measures of maximal entropy for suspension flows

We study suspension flows defined over sub-shifts of finite type with continuous roof functions. We prove the existence of suspension flows with uncountably many ergodic measures of maximal entropy. More generally, we prove that any suspension flow defined over a sub-shift of finite type can be perturbed (by an arbitrarily small perturbation) so that the resulting flow has uncountably many ergodic measures of maximal entropy, and that the same can be arranged so that the new flow has a unique measure of maximal entropy.

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arXiv2019

The space of invariant measures for countable Markov shifts

It is well known that the space of invariant probability measures for transitive sub-shifts of finite type is a Poulsen simplex. In this article we prove that in the non-compact setting, for a large family of transitive countable Markov shifts, the space of invariant sub-probability measures is a Poulsen simplex and that its extreme points are the ergodic invariant probability measures together with the zero measure. In particular we obtain that the space of invariant probability measures is a Poulsen simplex minus a vertex and the corresponding convex combinations. Our results apply to finite entropy non-locally compact transitive countable Markov shifts and to every locally compact transitive countable Markov shift. In order to prove these results we introduce a topology on the space of measures that generalizes the vague topology to a class of non-locally compact spaces, the topology of convergence on cylinders. We also prove analogous results for suspension flows defined over countable Markov shifts.

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arXiv2019

Pressure, Poincaré series and box dimension of the boundary

In this note we prove two related results. First, we show that for certain Markov interval maps with infinitely many branches the upper box dimension of the boundary can be read from the pressure of the geometric potential. Secondly, we prove that the box dimension of the set of iterates of a point in H^n with respect to a parabolic subgroup of isometries equals the critical exponent of the Poincare series of the associated group. This establishes a relationship between the entropy at infinity and dimension theory.

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arXiv2018

Upper semi-continuity of entropy in non-compact settings

We prove that the entropy map for countable Markov shifts of finite entropy is upper semi-continuous at ergodic measures. Note that the phase space is non-compact. Applications to systems that can be coded by these shifts, such as positive entropy diffeomorphisms on compact manifolds, are given. We also discuss the related problem of existence of measures of maximal entropy.

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arXiv2018

Hidden Gibbs measures on shift spaces over countable alphabets

We study the thermodynamic formalism for particular types of sub-additive sequences on a class of subshifts over countable alphabets. The subshifts we consider include factors of irreducible countable Markov shifts under certain conditions. We show the variational principle for topological pressure. We also study conditions for the existence and uniqueness of invariant ergodic Gibbs measures and the uniqueness of equilibrium states. As an application, we extend the theory of factors of (generalized) Gibbs measures on subshifts on finite alphabets to that on certain subshifts over countable alphabets.

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