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Frank Trujillo

Publications and source records attributed to Frank Trujillo.

17 recordsLinked to original sources
arXiv2026

Rigidity of Mather's $\beta$-function on a KAM set for analytic billiards-like maps and unique quasi-analytic continuation

In his seminal paper, Kac famously asked whether "one can hear the shape of a drum" - that is whether the isometry class of a bounded domain in a Euclidean space is uniquely determined by the spectrum of its Laplace spectrum. The Laplace spectrum is closely related to the length spectrum of the associated billiard. For a convex bounded planar domain, to each billiard periodic orbit one can associate not only its length but also its rotation number. The set of pairs of length and rotation number of each periodic orbit is called the marked length spectrum. Using the marked length spectrum of the domain one can associate with it its minimal action function also known as Mathers $\beta$-function denoted by $\beta_\Omega$. Via unique quasianalytic continuation, we prove the following rigidity problem: knowing that Mather's $\beta$-functions of two billiards in two analytic planar domains coincide on a set of positive measure of KAM diophantine numbers implies that they coincide on a set of all KAM diophantine numbers. In particular, knowing that Mather's $\beta$-functions of two domains coincide on a positive measure set of KAM diophantine numbers implies that the Marvizi-Melrose invariants of these domains coincide.

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arXiv2026

Non-autonomous KAM theory for lower dimensional invariant tori (I): Normally elliptic and hyperbolic cases

Many physical phenomena are naturally modeled by dynamical systems subject to non-autonomous perturbations that decay in time, including the interaction of a laser pulse with a molecule, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In the present paper, we consider time-dependent perturbations decaying in time of Hamiltonian systems having a lower-dimensional isotropic normally elliptic (resp. hyperbolic) invariant torus with quasiperiodic solutions. Under suitable rates of decay in time of the perturbation, we prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for H\"older, smooth, and analytic Hamiltonian systems.

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arXiv2026

Non-autonomous KAM theory for lower dimensional invariant tori (II): Normally parabolic case

Dynamical systems subject to non-autonomous perturbations decaying in time arise naturally in many physical contexts, including laser-molecule interactions, epidemiological models, nonlinear oscillatory systems, and celestial mechanics. In this paper, we consider time-dependent perturbations decaying polynomially fast in time of Hamiltonian systems having a lower-dimensional isotropic normally parabolic invariant torus supporting quasiperiodic solutions. We prove the existence of invariant manifolds in the extended phase space, and determine the asymptotic behavior of the associated transverse dynamics. The above results are obtained for H\"older, smooth, and analytic Hamiltonian systems.

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arXiv2024

On the ergodicity of anti-symmetric skew products with singularities and its applications

We introduce a novel method for proving ergodicity for skew products of interval exchange transformations (IETs) with piecewise smooth cocycles having singularities at the ends of exchanged intervals. This approach is inspired by Borel-Cantelli-type arguments from Fayad and Lema\'nczyk (2006). The key innovation of our method lies in its applicability to singularities beyond the logarithmic type, whereas previous techniques were restricted to logarithmic singularities. Our approach is particularly effective for proving the ergodicity of skew products for symmetric IETs and antisymmetric cocycles. Moreover, its most significant advantage is its ability to study the equidistribution of error terms in the spectral decomposition of Birkhoff integrals for locally Hamiltonian flows on compact surfaces, applicable not only when all saddles are perfect (harmonic) but also in the case of some non-perfect saddles.

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arXiv2024

On the uniqueness of affine IETs semi-conjugated to IETs

We prove that for almost every irreducible interval exchange transformation $T$ and for any vector $ω$ in its associated central-stable space (with respect to the Kontsevich-Zorich cocycle) there exists a unique AIET, up to normalization of its domain, semi-conjugated to $T$ and whose log-slope vector equals $ω$. This provides a partial answer to a question raised by S. Marmi, P. Moussa, and J.-C. Yoccoz.

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arXiv2024

Ergodic properties of infinite extension of symmetric interval exchange transformations

We prove that skew products with the cocycle given by the function $f(x)=a(x-1/2)$ with $a\neq 0$ are ergodic for every ergodic symmetric IET in the base, thus giving the full characterization of ergodic extensions in this family. Moreover, we prove that under an additional natural assumption of unique ergodicity on the IET, we can replace $f$ with any differentiable function with a non-zero sum of jumps. Finally, by considering weakly mixing IETs instead of just ergodic, we show that the skew products with cocycle given by $f$ have infinite ergodic index.

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arXiv2024

Ergodicity of skew-products over typical IETs

We prove ergodicity of a class of infinite measure preserving systems, called skew-products. More precisely, we consider systems of the form \[ {T_f}:{[0, 1) \times \mathbb{R}}\to{[0, 1) \times \mathbb{R}},\quad {T_f(x, t)}:={(T(x), t+f(x))}, \] where $T$ is an interval exchange transformation and $f$ is a piece-wise constant function with a finite number of discontinuities. We show that such system is ergodic with respect to ${Leb}_{[0,1)\times \mathbb{R}}$ for a typical choice of parameters of $T$ and $f$.

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arXiv2024

On the ergodicity of infinite antisymmetric extensions of symmetric IETs

In this article, we consider skew product extensions over symmetric interval exchange transformations with respect to the cocycle $f(x)=χ_{(0,1/2)}-χ_{(1/2,1)}$. More precisely, we prove that for almost every interval exchange transformation $T$ with symmetric combinatorial data, the skew product $T_f: [0, 1) \times \mathbb Z \to [0, 1) \times \mathbb Z$ given by $T_f(x,r)=(T(x),r+f(x))$ is ergodic with respect to the product of the Lebesgue and counting measure.

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arXiv2024

Rigidity for piece-wise smooth circle maps and certain GIETs

The goal of this article is to show a rigidity property of conjugacies of generalized interval exchange transformations (GIETs). More precisely, we show that if two piecewise $C^3$ GIETs $f$ and $g$ of generic rotation number with mean-non-linearity 0 are homeomorphic, boundary-equivalent and their renormalizations approach in an appropriate way the set of affine interval exchange transformations, then their respective renormalizations converge to each other and the conjugating map is $C^1$. Moreover, if $f$ and $g$ are GIETs with rotation type combinatorial data, generic rotation number and they are break-equivalent as piecewise circle diffeomorphisms, they are actually $C^1$-conjugated as circle diffeomorphisms. These results generalize the work of K. Cunha and D. Smania \cite{cunha_rigidity_2014} in the case of piecewise $C^3$ circle maps, where the authors prove an analogous result for GIETs with rotation type combinatorial data and bounded rotation number.

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arXiv2023

Ergodicity of explicit logarithmic cocycles over IETs

We prove ergodicity in a class of skew-product extensions of interval exchange transformations given by cocycles with logarithmic singularities. This, in particular, gives explicit examples of ergodic $\mathbb{R}$-extensions of minimal locally Hamiltonian flows with non-degenerate saddles in genus two. More generally, given any symmetric irreducible permutation, we show that for almost every choice of lengths vector, the skew-product built over the IET with the given permutation and lengths vector given by a cocycle, with symmetric, logarithmic singularities, which is \emph{odd} when restricted to each continuity subinterval is ergodic.

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arXiv2023

Affine IETs with a singular conjugacy to an IET

We produce affine interval exchange transformations (AIETs) which are topologically conjugated to (standard) interval exchange maps (IETs) via a singular conjugacy, i.e. a diffeomorphism $h$ of $[0,1]$ which is $C^0$ but not $C^1$ and such that the pull-back of the Lebesgue measure is a singular invariant measure for the AIET. In particular, we show that for almost every IET $T_0$ of at least two intervals and any vector $w$ belonging to the central-stable space $E_{cs}(T_0)$ for the Rauzy-Veech renormalization, any AIET T with log-slopes given by $w$ and semi-conjugated to $T_0$ is topologically conjugated to $T$. If in addition, if $w$ does not belong to $E_s(T_0)$, the conjugacy between $T$ and $T_0$ is singular.

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arXiv2022

On the Hausdorff dimension of invariant measures of piecewise smooth circle homeomorphisms

We show that, generically, the unique invariant measure of a sufficiently regular piecewise smooth circle homeomorphism with irrational rotation number and zero mean nonlinearity (e.g., piecewise linear) has zero Hausdorff dimension. To encode this generic condition, we consider piecewise smooth homeomorphisms as generalized interval exchange transformations (GIETs) of the interval and rely on the notion of combinatorial rotation number for GIETs, which can be seen as an extension of the classical notion of rotation number for circle homeomorphisms to the GIET setting.

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arXiv2020

Lyapunov instability in KAM stable Hamiltonians with two degrees of freedom

For a fixed frequency vector $ω\in \mathbb{R}^2 \, \setminus \, \lbrace 0 \rbrace$ obeying $ω_1 ω_2 < 0$ we show the existence of Gevrey-smooth Hamiltonians, arbitrarily close to an integrable Kolmogorov non-degenerate analytic Hamiltonian, having a Lyapunov unstable elliptic equilibrium with frequency $ω$. In particular, the elliptic fixed points thus constructed will be KAM stable, i.e. accumulated by invariant tori whose Lebesgue density tend to one in the neighbourhood of the point and whose frequencies cover a set of positive measure. Similar examples for near-integrable Hamiltonians in action-angle coordinates in the neighbourhood of a Lagragian invariant torus with arbitrary rotation vector are also given in this work.

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arXiv2020

Quasi-analytic properties of the KAM curve

Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong quasi-analyticity properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.

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arXiv2019

Smooth mixing transformations with loosely Bernoulli cartesian product

A zero-entropy system is said to be loosely Bernoulli if it can be induced from an irrational rotation of the circle. We provide a criterion for zero-entropy systems to be loosely Bernoulli that is compatible with mixing. Using these criteria, we show the existence of smooth mixing zero-entropy loosely Bernoulli transformations whose cartesian product with themselves is loosely Bernoulli.

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