Scientists / Open questions

Artur Avila

Dynamical systems; spectral theory

2014 Fields Medal

21 questions

Conjugacy-class closures, analytic Cantor spectrum, and matrix-class projection for Schrödinger cocycles over generalized skew-shifts: the open problems of Avila–Bochi–Damanik (2009)
Characterizing continuity points of Lyapunov exponents for fiber-bunched smooth cocycles on surface fibers, and the shear-conjugacy refinement of the Avila–Viana invariance principle
Every-phase absolutely continuous spectrum for small analytic quasiperiodic Schrödinger operators at Liouville frequencies, and the multiplicity of dual phases
The $C^1$ endpoint of the global Livšic theorem for diffeomorphism-valued cocycles over hyperbolic systems
Minimality versus twisting for linear monoids and the geometric meaning of Rauzy-Veech pinching: the open questions from Avila-Viana's proof of the Zorich-Kontsevich conjecture
Anderson localization for general Hölder sampling over hyperbolic transformations, and the large-coupling exceptional set
Mixing with zero entropy and rigidity of pseudo-rotations: the open questions of Avila-Fayad-Le Calvez-Xu-Zhang (2020) on disc diffeomorphisms
Unboundedness of the parameter derivative of the critical orbit at non-hyperbolic parameters of the unicritical family (Avila's Problem 1)
Lebesgue measure of the set of infinitely renormalizable parameters in the Multibrot set $\mathcal{M}_d$
Minimality of projective actions of SL(2,R) cocycles non-homotopic to the identity over ergodic translations (Avila–Krikorian, Problem 3.1)
Small eigenvalues of the foliated Laplacian on moduli spaces of flat surfaces: existence below the $1/4$ barrier for Masur–Veech measures, accumulation at $1/4$ , uniform spectral gaps, and the Ruelle zeta function of the Teichmüller flow
Open fronts in the Avila–Bochi–Damanik gap-opening program for strictly ergodic Schrödinger operators: simultaneous gap opening, analytic generic Cantor spectrum, and aperiodic dynamics without finite-dimensional factors
Exact local distribution and sharp arithmetic thresholds for absolutely continuous spectral measures of one-frequency quasiperiodic Schrödinger operators at subcritical energies
Geometric realization of zero Lyapunov exponents of the Kontsevich-Zorich cocycle beyond the Forni subspace
Does the set of analytic one-frequency complex cocycles with trivial Oseledets filtration contain a nonempty open set?
Realization of Avila's acceleration for one-frequency SL(2,R) Schrödinger cocycles: degree lower bound and premonotonic characterization, and prevalence of fixed-acceleration energies
Eigenvalues and non-weak-mixing directions of non-arithmetic Veech surfaces: the Avila-Delecroix open problems
Mixing of smooth time-changes of parabolic nilflows: the four open problems of Avila–Forni–Ravotti–Ulcigrai (dichotomy beyond the dense class, effective triviality, quantitative and multiple mixing)
Positive Lyapunov exponents beyond the settled regimes: Herman's high-regularity entropy-density question, a replacement for the refuted Kotani–Last conjecture, and measure-theoretic typicality
Sharp conditions on the quantization data and sharp contraction rates for the Voros–Avila exact quantization scheme
Full hyperbolicity of the renormalization operator for unicritical maps of higher even degree: the transverse unstable direction

Back to open problems

Conjugacy-class closures, analytic Cantor spectrum, and matrix-class projection for Schrödinger cocycles over generalized skew-shifts: the open problems of Avila–Bochi–Damanik (2009)

Characterizing continuity points of Lyapunov exponents for fiber-bunched smooth cocycles on surface fibers, and the shear-conjugacy refinement of the Avila–Viana invariance principle

Every-phase absolutely continuous spectrum for small analytic quasiperiodic Schrödinger operators at Liouville frequencies, and the multiplicity of dual phases

The C1C^1C1 endpoint of the global Livšic theorem for diffeomorphism-valued cocycles over hyperbolic systems

Minimality versus twisting for linear monoids and the geometric meaning of Rauzy-Veech pinching: the open questions from Avila-Viana's proof of the Zorich-Kontsevich conjecture

Anderson localization for general Hölder sampling over hyperbolic transformations, and the large-coupling exceptional set

Mixing with zero entropy and rigidity of pseudo-rotations: the open questions of Avila-Fayad-Le Calvez-Xu-Zhang (2020) on disc diffeomorphisms

Unboundedness of the parameter derivative of the critical orbit at non-hyperbolic parameters of the unicritical family (Avila's Problem 1)

Lebesgue measure of the set of infinitely renormalizable parameters in the Multibrot set Md\mathcal{M}_dMd​

Minimality of projective actions of SL(2,R) cocycles non-homotopic to the identity over ergodic translations (Avila–Krikorian, Problem 3.1)

Small eigenvalues of the foliated Laplacian on moduli spaces of flat surfaces: existence below the 1/41/41/4 barrier for Masur–Veech measures, accumulation at 1/41/41/4, uniform spectral gaps, and the Ruelle zeta function of the Teichmüller flow

Open fronts in the Avila–Bochi–Damanik gap-opening program for strictly ergodic Schrödinger operators: simultaneous gap opening, analytic generic Cantor spectrum, and aperiodic dynamics without finite-dimensional factors

Exact local distribution and sharp arithmetic thresholds for absolutely continuous spectral measures of one-frequency quasiperiodic Schrödinger operators at subcritical energies

Geometric realization of zero Lyapunov exponents of the Kontsevich-Zorich cocycle beyond the Forni subspace

Does the set of analytic one-frequency complex cocycles with trivial Oseledets filtration contain a nonempty open set?

Realization of Avila's acceleration for one-frequency SL(2,R) Schrödinger cocycles: degree lower bound and premonotonic characterization, and prevalence of fixed-acceleration energies

Eigenvalues and non-weak-mixing directions of non-arithmetic Veech surfaces: the Avila-Delecroix open problems

Mixing of smooth time-changes of parabolic nilflows: the four open problems of Avila–Forni–Ravotti–Ulcigrai (dichotomy beyond the dense class, effective triviality, quantitative and multiple mixing)

Positive Lyapunov exponents beyond the settled regimes: Herman's high-regularity entropy-density question, a replacement for the refuted Kotani–Last conjecture, and measure-theoretic typicality

Sharp conditions on the quantization data and sharp contraction rates for the Voros–Avila exact quantization scheme

Full hyperbolicity of the renormalization operator for unicritical maps of higher even degree: the transverse unstable direction

The $C^1$ endpoint of the global Livšic theorem for diffeomorphism-valued cocycles over hyperbolic systems

Lebesgue measure of the set of infinitely renormalizable parameters in the Multibrot set $\mathcal{M}_d$

Small eigenvalues of the foliated Laplacian on moduli spaces of flat surfaces: existence below the $1/4$ barrier for Masur–Veech measures, accumulation at $1/4$ , uniform spectral gaps, and the Ruelle zeta function of the Teichmüller flow