Limit theorems for toral translations

Dolgopyat, Dmitry; Fayad, Bassam

doi:10.1090/pspum/089/01492

Cited by 26 publications

(17 citation statements)

References 110 publications

(197 reference statements)

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“…There is a large literature of results on limiting distributions for entropy zero dynamical systems, see for example [13,14,12,18,20,35]. Let us mention two recent results in the context of substitution systems which are related to our work.…”

Section: 3mentioning

confidence: 90%

A temporal central limit theorem for real-valued cocycles over rotations

Bromberg¹,

Ulcigrai²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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We consider deterministic random walks on the real line driven by irrational rotations, or equivalently, skew product extensions of a rotation by α where the skewing cocycle is a piecewise constant mean zero function with a jump by one at a point β. When α is badly approximable and β is badly approximable with respect to α, we prove a Temporal Central Limit theorem (in the terminology recently introduced by D.Dolgopyat and O.Sarig), namely we show that for any fixed initial point, the occupancy random variables, suitably rescaled, converge to a Gaussian random variable. This result generalizes and extends a theorem by J. Beck for the special case when α is quadratic irrational, β is rational and the initial point is the origin, recently reproved and then generalized to cover any initial point using geometric renormalization arguments by Avila-Dolgopyat-Duryev-Sarig (Israel J., 2015) and Dolgopyat-Sarig (J. Stat. Physics, 2016). We also use renormalization, but in order to treat irrational values of β, instead of geometric arguments, we use the renormalization associated to the continued fraction algorithm and dynamical Ostrowski expansions. This yields a suitable symbolic coding framework which allows us to reduce the main result to a CLT for non homogeneous Markov chains.

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Section: 3mentioning

confidence: 90%

A temporal central limit theorem for real-valued cocycles over rotations

Bromberg¹,

Ulcigrai²

2018

Ann. Inst. H. Poincaré Probab. Statist.

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“…Many papers explore just how far one can push a quantitative Borel-Cantelli lemma, including the recent work of Dolgopyat, Fayad, and Liu [5] and Dolgopyat and Fayad [6]. Additionally, much work has been done classifying shrinking target properties.…”

Section: Brief History Of Shrinking Targetsmentioning

confidence: 99%

Shrinking targets on square-tiled surfaces

Southerland¹

2021

Preprint

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We prove a shrinking target property for the action of subgroups of Veech groups on square-tiled surfaces. Our main tool, of independent interest, is the construction of a Fourier-like transform that we use to show that the L 2 -norm of the Koopman operator for the action is equivalent to the L 2 -norm of a Markov operator related to a random walk on a subgroup. This generalizes work of V. Finkelshtein on the flat torus.

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“…We mention here a list of questions related to the statistical properties of Kronecker sequences that can be approached using this same principle. More details and questions can be found in [28]. Uniform distribution of the sequence x+kα on T d is equivalent to the fact that, for regular sets C, D(α, x, C, N )/N → 0 as N → ∞.…”

Section: Ergodic Theory Of Diagonal Actions On the Space Of Lattices mentioning

confidence: 99%

Some Questions Around Quasi-Periodic Dynamics

Fayad¹,

Krikorian²

2019

Proceedings of the International Congress of Mathematicians (ICM 2018)

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We propose in these notes a list of some old and new questions related to quasi-periodic dynamics. A main aspect of quasi-periodic dynamics is the crucial influence of arithmetics on the dynamical features, with a strong duality in general between Diophantine and Liouville behavior. We will discuss rigidity and stability in Diophantine dynamics as well as their absence in Liouville ones. Beyond this classical dichotomy between the Diophantine and the Liouville worlds, we discuss some unified approaches and some phenomena that are valid in both worlds. Our focus is mainly on low dimensional dynamics such as circle diffeomorphisms, disc dynamics, quasi-periodic cocycles, or surface flows, as well as finite dimensional Hamiltonian systems.In an opposite direction, the study of the dynamical properties of some diagonal and unipotent actions on the space of lattices can be applied to arithmetics, namely to the theory of Diophantine approximations. We will mention in the last section some problems related to that topic.The field of quasi-periodic dynamics is very extensive and has a wide range of interactions with other mathematical domains. The list of questions we propose is naturally far from exhaustive and our choice was often motivated by our research involvements.

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Limit theorems for toral translations

Cited by 26 publications

References 110 publications

A temporal central limit theorem for real-valued cocycles over rotations

A temporal central limit theorem for real-valued cocycles over rotations

Shrinking targets on square-tiled surfaces

Some Questions Around Quasi-Periodic Dynamics

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