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Saprykina, Mariaorcid.org/0000-0003-1810-4900

Open this publication in new window or tab >>Noncommutative coboundary equations over integrable systems### de la Llave, Rafael

### Saprykina, Maria

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##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Mathematical Sciences (AIMS), 2023
##### Keywords

Coboundaries, cohomology equations, Livshits theorems, Livšic theorems, rigidity
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-338353 (URN)10.3934/jmd.2023020 (DOI)2-s2.0-85173000234 (Scopus ID)
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##### Note

School of Mathematics, Georgia Institute of Technology, 686 Cherry St., Atlanta, GA, 30332-1160, USA, 686 Cherry St.

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

We prove an analog of the Livshits theorem for real-analytic families of cocycles over an integrable system with values in a Banach algebra (Formula Presented) or a Lie group. Namely, we consider an integrable dynamical system (Formula Presented), and a real-analytic family of cocycles (Formula Presented) indexed by a complex parameter ε in an open ball (Formula Presented). We show that if ηε is close to identity and has trivial periodic data, i.e., (Formula Presented) for each periodic point p = fn p and each (Formula Presented), then there exists a real-analytic family of maps (Formula Presented) satisfying the coboundary equation (Formula Presented) for all (Formula Presented) and (Formula Presented). We also show that if the coboundary equation above with an analytic left-hand side ηε has a solution in the sense of formal power series in ε, then it has an analytic solution.

QC 20231020

Available from: 2023-10-20 Created: 2023-10-20 Last updated: 2023-10-20Bibliographically approvedOpen this publication in new window or tab >>Convergence of the Birkhoff normal form sometimes implies convergence of a normalizing transformation### De la Llave, Rafael

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_1_j_idt240_some",{id:"formSmash:j_idt233:1:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_1_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_1_j_idt240_otherAuthors",{id:"formSmash:j_idt233:1:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_1_j_idt240_otherAuthors",multiple:true}); 2022 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, no 3, p. 1166-1187Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cambridge University Press (CUP), 2022
##### Keywords

nearly integrable Hamiltonian systems, Birkhoff normal form, convergence of the normalizing transformations
##### National Category

Mathematical Analysis
##### Identifiers

urn:nbn:se:kth:diva-309059 (URN)10.1017/etds.2021.71 (DOI)000750488900012 ()2-s2.0-85112391799 (Scopus ID)
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##### Note

Georgia Inst Technol, Sch Math, Atlanta, GA 30332 USA..

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Consider an analytic Hamiltonian system near its analytic invariant torus T-0 carrying zero frequency. We assume that the Birkhoff normal form of the Hamiltonian at T-0 is convergent and has a particular form: it is an analytic function of its non-degenerate quadratic part. We prove that in this case there is an analytic canonical transformation-not just a formal power series-bringing the Hamiltonian into its Birkhoff normal form.

QC 20220221

Available from: 2022-02-21 Created: 2022-02-21 Last updated: 2022-06-25Bibliographically approvedOpen this publication in new window or tab >>Realizing Arbitrary D-Dimensional Dynamics By Renormalization Of Cd-Perturbations Of Identity### Fayad, B.

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_2_j_idt240_some",{id:"formSmash:j_idt233:2:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_2_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_2_j_idt240_otherAuthors",{id:"formSmash:j_idt233:2:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_2_j_idt240_otherAuthors",multiple:true}); 2022 (English)In: Discrete and Continuous Dynamical Systems, ISSN 1078-0947, E-ISSN 1553-5231, Vol. 42, no 2, p. 597-604Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Institute of Mathematical Sciences (AIMS), 2022
##### Keywords

Anosov-Katok method, Realization by iterations, Renormalization
##### National Category

Computer Engineering Other Physics Topics Subatomic Physics
##### Identifiers

urn:nbn:se:kth:diva-319967 (URN)10.3934/dcds.2021129 (DOI)000706622700001 ()2-s2.0-85123509731 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

Any Cd conservative map f of the d-dimensional unit ball Bd, d ≥ 2, can be realized by renormalized iteration of a Cd perturbation of identity: there exists a conservative diffeomorphism of Bd, arbitrarily close to identity in the Cd topology, that has a periodic disc on which the return dynamics after a Cd change of coordinates is exactly f.

QC 20221017

Available from: 2022-10-17 Created: 2022-10-17 Last updated: 2022-10-17Bibliographically approvedOpen this publication in new window or tab >>Topological weak mixing and diffusion at all times for a class of Hamiltonian systems### Fayad, Bassam

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_3_j_idt240_some",{id:"formSmash:j_idt233:3:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_3_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_3_j_idt240_otherAuthors",{id:"formSmash:j_idt233:3:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_3_j_idt240_otherAuthors",multiple:true}); 2022 (English)In: Ergodic Theory and Dynamical Systems, ISSN 0143-3857, E-ISSN 1469-4417, Vol. 42, no 2, p. 777-791, article id PII S0143385721000122Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Cambridge University Press (CUP), 2022
##### Keywords

Hamiltonian systems, diffusion, topological weak mixing, Anosov-Katok method, AbC method
##### National Category

Computational Mathematics Condensed Matter Physics Subatomic Physics
##### Identifiers

urn:nbn:se:kth:diva-307765 (URN)10.1017/etds.2021.12 (DOI)000741433500016 ()2-s2.0-85106995298 (Scopus ID)
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##### Note

IMJ PRG CNRS, UP7D, 58-56 Ave France,Boite Courrier 7012, F-75205 Paris 13, France..

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

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.

QC 20220208

Available from: 2022-02-08 Created: 2022-02-08 Last updated: 2022-06-25Bibliographically approvedOpen this publication in new window or tab >>Erratic behavior for 1-dimensional random walks in a Liouville quasi-periodic environment### Dolgopyat, Dmitry

### Fayad, Bassam

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_4_j_idt240_some",{id:"formSmash:j_idt233:4:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_4_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_4_j_idt240_otherAuthors",{id:"formSmash:j_idt233:4:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_4_j_idt240_otherAuthors",multiple:true}); 2021 (English)In: Electronic Journal of Probability, E-ISSN 1083-6489, Vol. 26, article id 66Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

INST MATHEMATICAL STATISTICS-IMS, 2021
##### Keywords

random walks in random environment, random walks in random potential, Liouville phenomena, localization
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:kth:diva-296870 (URN)10.1214/21-EJP622 (DOI)000654413100001 ()2-s2.0-85109088007 (Scopus ID)
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##### Note

Univ Maryland, College Pk, MD 20742 USA..

CNRS UMR7586 IMJ PRG Paris, Paris, France..

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).

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.

QC 20210611

Available from: 2021-06-11 Created: 2021-06-11 Last updated: 2024-07-04Bibliographically approvedOpen this publication in new window or tab >>Isolated elliptic fixed points for smooth Hamiltonians### Fayad, B.

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_5_j_idt240_some",{id:"formSmash:j_idt233:5:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_5_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_5_j_idt240_otherAuthors",{id:"formSmash:j_idt233:5:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_5_j_idt240_otherAuthors",multiple:true}); 2017 (English)In: Contemporary Mathematics, ISSN 0271-4132, E-ISSN 1098-3627, Vol. 692, p. 67-82Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

American Mathematical Society (AMS), 2017
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-216742 (URN)10.1090/conm/692/13924 (DOI)2-s2.0-85029553747 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

We construct on ℝ2d, for any d ≥ 3, smooth Hamiltonians having an elliptic equilibrium with an arbitrary frequency, that is not accumulated by a positive measure set of invariant tori. For d ≥ 4, the Hamiltonians we construct have not any invariant torus of dimension d. Our examples are obtained by a version of the successive conjugation scheme à la Anosov-Katok.

QC 20171024

Available from: 2017-10-24 Created: 2017-10-24 Last updated: 2024-03-18Bibliographically approvedOpen this publication in new window or tab >>Arnol ' d Diffusion in a Pendulum Lattice### Kaloshin, Vadim

### Levi, Mark

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_6_j_idt240_some",{id:"formSmash:j_idt233:6:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_6_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_6_j_idt240_otherAuthors",{id:"formSmash:j_idt233:6:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_6_j_idt240_otherAuthors",multiple:true}); 2014 (English)In: Communications on Pure and Applied Mathematics, ISSN 0010-3640, E-ISSN 1097-0312, Vol. 67, no 5, p. 748-775Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Hamiltonian-Systems, Unbounded Energy, Localization, Growth, Kink
##### National Category

Mathematics
##### Identifiers

urn:nbn:se:kth:diva-145808 (URN)10.1002/cpa.21509 (DOI)000332144200002 ()2-s2.0-84895096830 (Scopus ID)
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##### Funder

Swedish Research Council, VR 2006-3264
##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

The main model studied in this paper is a lattice of pendula with a nearest-neighbor coupling. If the coupling is weak, then the system is near-integrable and KAM tori fill most of the phase space. For all KAM trajectories the energy of each pendulum stays within a narrow band for all time. Still, we show that for an arbitrarily weak coupling of a certain localized type, the neighboring pendula can exchange energy. In fact, the energy can be transferred between the pendula in any prescribed way.

QC 20140604

Available from: 2014-06-04 Created: 2014-06-02 Last updated: 2024-03-18Bibliographically approvedOpen this publication in new window or tab >>An Example of a Nearly Integrable Hamiltonian System with a Trajectory Dense in a Set of Maximal Hausdorff Dimension### Kaloshin, Vadim

### Saprykina, Maria

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_7_j_idt240_some",{id:"formSmash:j_idt233:7:j_idt240:some",widgetVar:"widget_formSmash_j_idt233_7_j_idt240_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt233_7_j_idt240_otherAuthors",{id:"formSmash:j_idt233:7:j_idt240:otherAuthors",widgetVar:"widget_formSmash_j_idt233_7_j_idt240_otherAuthors",multiple:true}); 2012 (English)In: Communications in Mathematical Physics, ISSN 0010-3616, E-ISSN 1432-0916, Vol. 315, no 3, p. 643-697Article in journal (Refereed) Published
##### Abstract [en]

##### Keywords

Arnold Diffusion, Lagrangian Systems, Instability, Stability, Points
##### National Category

Mathematics Physical Sciences
##### Identifiers

urn:nbn:se:kth:diva-104994 (URN)10.1007/s00220-012-1532-x (DOI)000309718600003 ()2-s2.0-84867441458 (Scopus ID)
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##### Note

KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).

The famous ergodic hypothesis suggests that for a typical Hamiltonian on a typical energy surface nearly all trajectories are dense. KAM theory disproves it. Ehrenfest (The Conceptual Foundations of the Statistical Approach in Mechanics. Ithaca, NY: Cornell University Press, 1959) and Birkhoff (Collected Math Papers. Vol 2, New York: Dover, pp 462-465, 1968) stated the quasi-ergodic hypothesis claiming that a typical Hamiltonian on a typical energy surface has a dense orbit. This question is wide open. Herman (Proceedings of the International Congress of Mathematicians, Vol II (Berlin, 1998). Doc Math 1998, Extra Vol II, Berlin: Int Math Union, pp 797-808, 1998) proposed to look for an example of a Hamiltonian near with a dense orbit on the unit energy surface. In this paper we construct a Hamiltonian which has an orbit dense in a set of maximal Hausdorff dimension equal to 5 on the unit energy surface.

QC 20121116

Available from: 2012-11-16 Created: 2012-11-15 Last updated: 2024-03-18Bibliographically approved