| 1. |
- Damjanović, Danijela, et al.
(author)
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On local rigidity of partially hyperbolic affine Z(k) actions
- 2019
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In: Journal für die Reine und Angewandte Mathematik. - : WALTER DE GRUYTER GMBH. - 0075-4102 .- 1435-5345. ; 751, s. 1-26
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Journal article (peer-reviewed)abstract
- The following dichotomy for affine Z(k) actions on the torus T-d, k, d is an element of N, is shown to hold: (i) The linear part of the action has no rank-one factors, and then the affine action is locally rigid. (ii) The linear part of the action has a rank-one factor, and then the affine action is locally rigid in a probabilistic sense if and only if the rank-one factors are trivial. Local rigidity in a probabilistic sense means that rigidity holds for a set of full measure of translation vectors in the rank-one factors.
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| 2. |
- Dolgopyat, Dmitry, et al.
(author)
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Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment
- 2021
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In: Electronic Journal of Probability. - : INST MATHEMATICAL STATISTICS-IMS. - 1083-6489. ; 26
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Journal article (peer-reviewed)abstract
- We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.
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| 3. |
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| 4. |
- Farré Puiggalí, Gerard, et al.
(author)
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Instabilities of invariant quasi-periodic tori
- 2022
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In: Journal of the European Mathematical Society (Print). - : European Mathematical Society - EMS - Publishing House GmbH. - 1435-9855 .- 1435-9863. ; 24:12, s. 4363-4383
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Journal article (peer-reviewed)abstract
- We prove the existence of real analytic Hamiltonians with topologically unstable quasi -periodic invariant tori. Using various versions of our examples, we solve the following problems in the stability theory of analytic quasi-periodic motion:(1) Show the existence of topologically unstable tori of arbitrary frequency. Moreover, the Birkhoff Normal Form at the invariant torus can be chosen to be convergent, equal to a planar or non -planar polynomial.(2) Show the optimality of the exponential stability for Diophantine tori.(3) Show the existence of real analytic Hamiltonians that are integrable on half of the phase space, and such that all orbits on the other half accumulate at infinity.(4) For sufficiently Liouville vectors, obtain invariant tori that are not accumulated by a positive measure set of quasi-periodic invariant tori.
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| 5. |
- Fayad, Bassam, et al.
(author)
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Topological weak mixing and diffusion at all times for a class of Hamiltonian systems
- 2022
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In: Ergodic Theory and Dynamical Systems. - : Cambridge University Press (CUP). - 0143-3857 .- 1469-4417. ; 42:2, s. 777-791
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Journal article (peer-reviewed)abstract
- We present examples of nearly integrable analytic Hamiltonian systems with several strong diffusion properties: topological weak mixing and diffusion at all times. These examples are obtained by AbC constructions with several frequencies.
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