arXiv subjects

Federico Rodriguez-Hertz

Publications and source records attributed to Federico Rodriguez-Hertz.

12 recordsLinked to original sources

arXiv2026

Thermodynamic Formalism for Quasimorphisms: Lattices in Higher Rank Semisimple Lie Groups

We give a new proof, based on thermodynamic formalism, of a foundational result of Burger and Monod in bounded cohomology. Let $G$ be a noncompact connected semisimple real Lie group with finite center and no factors of real rank one, and let $\Gamma<G$ be a uniform lattice. We prove that, for every orthogonal representation $\pi:\Gamma\to\operatorname{O}_N$, every $\pi$-quasimorphism $L:\Gamma\to\mathbb{R}^N$ is bounded.

math.DS↗

arXiv2025

Thermodynamic Formalism Out of Equilibrium, and Gibbs Processes

We study the thermodynamic formalism of systems where the potential depends randomly on an exterior system. We define the {\em pressure out of equilibrium} for such a family of potentials, and prove a corresponding variational principle. We present an application to random dynamical systems. In particular, we study an open condition for random dynamical systems where the randomness is driven by a Gibbs process, and prove hyperbolicity estimates that were previously only known in the i.i.d setting.

math.DS↗

arXiv2025

Thermodynamic formalism for Quasimorphisms: Bounded Cohomology and Statistics

For a compact negatively curved space, we develop a thermodynamic formalism framework to study the space of quasimorphisms of its fundamental group modulo bounded functions. We prove that this space is Banach isomorphic to the space of Bowen functions on the associated Gromov geodesic flow, modulo a weak form of Liv\v{s}ic cohomology. We also show that each unbounded quasimorphism is associated with a unique invariant measure for the flow, which uniquely determines the cohomology class. As a consequence, we establish the Central Limit Theorem and the invariance principle for any unbounded quasimorphism with respect to Markov measures, and we prove that the associated equilibrium state has the Bernoulli property.

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arXiv2025

Mixed quantization and partial hyperbolicity

We establish stable quantum ergodicity for spin Hamiltonians, also known as Pauli-Schrödinger operators. Our approach combines new analytic techniques of mixed quantization, inspired by local index theory, with stable ergodicity results for partially hyperbolic systems.

math.DS↗

arXiv2025

Rigidity of equilibrium states and unique quasi-ergodicity for horocyclic foliations

In this paper we prove that for topologically mixing metric Anosov flows their equilibrium states corresponding to Hölder potentials satisfy a strong rigidity property: they are determined only by their disintegrations on (strong) stable or unstable leaves. As a consequence we deduce: the corresponding horocyclic foliations of such systems are uniquely quasi-ergodic, provided that the corresponding Jacobian is Hölder, without any restriction on the dimension of the invariant distributions. This gives another proof of a result of Babillott-Ledrappier.

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arXiv2023

Neutralized local entropy, and dimension bounds for invariant measurs

We introduce a notion of a point-wise entropy of measures (i.e local entropy) called neutralized local entropy, and compare it with the Brin-Katok local entropy. We show that the neutralized local entropy coincides with Brin-Katok local entropy almost everywhere. Neutralized local entropy is computed by measuring open sets with a relatively simple geometric description. Our proof uses a measure density lemma for Bowen balls, and a version of a Besicovitch covering lemma for Bowen balls. As an application, we prove a lower point-wise dimension bound for invariant measures, complementing the previously established bounds for upper point-wise dimension.

math.DS↗

arXiv2023

Contributions to the ergodic theory of hyperbolic flows: unique ergodicity for quasi-invariant measures and equilibrium states for the time-one map

We consider the horocyclic flow corresponding to a (topologically mixing) Anosov flow or diffeomorphism, and establish the uniqueness of transverse quasi-invariant measures with Hölder Jacobians. In the same setting, we give a precise characterization of the equilibrium states of the hyperbolic system, showing that existence of a family of Radon measures on the horocyclic foliation such that any probability (invariant or not) having conditionals given by this family, necessarily is the unique equilibrium state of the system.

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arXiv2023

Equilibrium States for Center Isometries

We develop a geometric method to establish existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular the entropy maximizing measure and the SRB measure. It is also given a characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

math.DS↗

arXiv2021

Exponential mixing implies Bernoulli

Let $f$ be a $C^{1+α}$ diffeomorphism of a compact manifold $M$ preserving a smooth measure $μ$. We show that if $f:(M,μ)\to (M,μ)$ is exponentially mixing then it is Bernoulli.

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arXiv2021

Geometrical constructions of equilibrium states

In this note we report some advances in the study of thermodynamic formalism for a class of partially hyperbolic system -- center isometries, that includes regular elements in Anosov actions. The techniques are of geometric flavor (in particular, not relying in symbolic dynamics) and even provide new information in the classical case. For such systems, we give in particular a constructive proof of the existence of the SRB measure and of the entropy maximizing measure. It is also established very fine statistical properties (Bernoulliness), and it is given a characterization of equilibrium states in terms of their conditional measures in the stable/unstable lamination, similar to the SRB case. The construction is applied to obtain the uniqueness of quasi-invariant measures associated to Hölder Jacobian for the horocyclic flow.

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arXiv2020

Classification of partially hyperbolic diffeomorphisms under some rigid conditions

Consider a three dimensional partially hyperbolic diffeomorphism. It is proven that under some rigid hypothesis on the tangent bundle dynamics, the map is (modulo finite covers and iterates) either an Anosov diffeomorphism, a skew-product or the time-one map of an Anosov flow, thus recovering a well known classification conjecture of the second author to this restricted setting.

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arXiv2016

On the non-equivalence of the Bernoulli and K properties in dimension four

We study skew products where the base is a hyperbolic automorphism of $\mathbb{T}^2$, the fiber is a smooth area preserving flow on $\mathbb{T}^2$ with one fixed point (of high degeneracy) and the skewing function is a smooth non coboundary with non-zero integral. The fiber dynamics can be represented as a special flow over an irrational rotation and a roof function with one power singularity. We show that for a full measure set of rotations the corresponding skew product is $K$ and not Bernoulli. As a consequence we get a natural class of volume-preserving diffeomorphisms of $\mathbb{T}^4$ which are $K$ and not Bernoulli.

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