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Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation

Harry Furstenberg
1967 Mathematical Systems Theory  
The objects of ergodic t h e o r y -m e a s u r e spaces with measure-preserving transformation groups-will be called processes, those of topological dynamics-compact metric spaces with groups of homeomor  ...  T h e same class Y of flows, as well as the notion of disjointness, arises in connection with a problem in Diophantine approximation (Part IV).  ...  Is the condition in T h e o r e m III.6 sufficient as well as necessary for an R0-flow? Part IV. A Problem in Diophantine Approximation. 1. Minimal Sets on Tori.  ... 
doi:10.1007/bf01692494 fatcat:yqtlop5t6nekrkke2rngg5n7fq

Page 797 of Mathematical Reviews Vol. 35, Issue 4 [page]

1968 Mathematical Reviews  
Furstenberg, Harry , 4369 Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation. Math.  ...  The paper is divided into four parts: (I) Disjoint processes ; (II) Disjoint flows; (III) Properties of minimal sets; (IV) A problem in diophantine approximation.  ... 

ETS volume 32 issue 4 Cover and Back matter

2012 Ergodic Theory and Dynamical Systems  
On the self-similarity problem for smooth flows on orientable surfaces Kwietniak, D. and Oprocha, P. On weak mixing, minimality and weak disjointness of all iterates Li, Z. and Góra, P.  ...  Postcritical sets and saddle basic sets for Axiom A polynomial skew products on C 2 Navas, A.  ...  Ergodic theory and dynamical systems VOLUME 32 PART 4 AUGUST 2012 Cambridge Journals Online For further information about this journal please go to the journal website at: journals.cambridge.org/ets  ... 
doi:10.1017/etds.2012.104 fatcat:qb4tkc7or5a6bk3dizpnkervnq

Quantitative ergodic theorems and their number-theoretic applications [article]

Alex Gorodnik, Amos Nevo
2013 arXiv   pre-print
This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory.  ...  of intrinsic Diophantine approximation, and results on fast distribution of dense orbits on homogeneous spaces.  ...  Both authors would like to express their gratitude to the Ergodic Theory Group at the Fédération Denis Poisson for the opportunity to explain the present work in the lecture series "Théorie ergodique des  ... 
arXiv:1304.6847v1 fatcat:ba7thy5qfzbp3pj5xkuf3xfkzi

Quantitative ergodic theorems and their number-theoretic applications

Alexander Gorodnik, Amos Nevo
2014 Bulletin of the American Mathematical Society  
We present an account of some recent applications of ergodic theorems for actions of algebraic and arithmetic groups to the solution of natural problems in Diophantine approximation and number theory.  ...  This allows the derivation of ergodic theorems with a rate of convergence, an important phenomenon which does not arise in classical ergodic theory.  ...  Both authors would like to express their gratitude to the Ergodic Theory Group at the Fédération Denis Poisson for the opportunity to explain the present work in the lecture series "Théorie ergodique des  ... 
doi:10.1090/s0273-0979-2014-01462-4 fatcat:pcicudistbgljm4lriwzsbp32q

Furstenberg and Margulis Awarded 2020 Abel Prize

Elaine Kehoe
2020 Notices of the American Mathematical Society  
In his 1967 paper, "Disjointness in Ergodic Theory, Minimal Sets, and a Problem in Diophantine Approximation," Furstenberg introduced the notion of "disjointness," a notion in ergodic systems that is analogous  ...  Motivated by Diophantine approximation, in 1967, Furstenberg introduced the notion of disjointness of ergodic systems, a notion akin to that of being coprime for integers.  ... 
doi:10.1090/noti2110 fatcat:sgwngh4djrcgfax7ia2uzjyrhy

ETS volume 32 issue 2 Cover and Back matter

2012 Ergodic Theory and Dynamical Systems  
On weak mixing, minimality and weak disjointness of all iterates Li, Y., Chen, E. and Cheng, W.-C.  ...  BOOK REVIEW 'Ergodic Theory: with a view towards Number Theory' by Manfred Einsiedler and Thomas Ward Danilenko, A. I.  ... 
doi:10.1017/s0143385712000120 fatcat:4nprlakesrhwtb6kqlppkjw4pi

ETS volume 32 issue 3 Cover and Back matter

2012 Ergodic Theory and Dynamical Systems  
Infinite-step nilsystems, independence and complexity Farina, A. and Valdinoci, E. Some results on minimizers and stable solutions of a variational problem Fiebig, D.  ...  On weak mixing, minimality and weak disjointness of all iterates Li, Y., Chen, E. and Cheng, W.-C. Tail pressure and the tail entropy function Li, Z. and Góra, P.  ... 
doi:10.1017/s0143385712000314 fatcat:2cuo7w5fyff4tmc4pi2anb7yje

Slow chaos in surface flows

Corinna Ulcigrai
2020 Bolletino dell Unione Matematica Italiana  
In the last decade, there have been a lot of advances in our understanding of the chaotic properties of smooth area-preserving flows (a class which include locally Hamiltonian flows), thanks to the connection  ...  to Teichmueller dynamics and, very recently, to the influence of the work of Marina Ratner in homogeneous dynamics.  ...  Acknowledgements The author is part of SwissMAP (The Mathematics of Physics National Centre for Compentence in Research) and is currently supported by a SNSF (Swiss National Science Foundation) Grant No  ... 
doi:10.1007/s40574-020-00267-0 fatcat:bggpqjrdwzeatpgqgkwmnwu2se

Constructions in elliptic dynamics

BASSAM FAYAD, ANATOLE KATOK
2004 Ergodic Theory and Dynamical Systems  
Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi in [7] and  ...  continuing with exotic invariant sets of rational maps of the Riemann sphere [21] , and the construction of invariant tori with nonstandard and unexpected behavior in the context of KAM theory [22, 23  ...  and ergodic properties for exotic minimal sets.  ... 
doi:10.1017/s0143385703000798 fatcat:pqsdttgpvvfqrksfs3jcwscz54

Constructions in elliptic dynamics [article]

Bassam Fayad, Anatole Katok
2005 arXiv   pre-print
Michel Herman made important contributions to the development and applications of this method beginning from the construction of minimal and uniquely ergodic diffeomorphisms jointly with Fathi in FH and  ...  continuing with exotic invariant sets of rational maps of the Riemann sphere H3, and the construction of invariant tori with nonstandard and unexpected behavior in the context of KAM theory H1, H2.  ...  and ergodic properties for exotic minimal sets.  ... 
arXiv:math/0501362v1 fatcat:lnpuuw4jkjbe3piwzx723f5y5m

On the disjointness property of groups and a conjecture of Furstenberg [article]

Eli Glasner, Benjamin Weiss
2019 arXiv   pre-print
In his seminal 1967 paper "Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation" Furstenberg introduced the notion of disjointness of dynamical systems, both topological  ...  He conjectured that a similar result holds in general and in our 1983 work "Interpolation sets for subalgebras of l^∞(Z)" we confirmed this by showing that the closed subalgebra A of l^∞(Z), generated  ...  Acknowledgement: We would like to thank Dana Bǎrtosová whose talk -"On a problem of Ellis and Pestov's conjecture" -at the "Twelfth Symposium on General Topology and  ... 
arXiv:1807.08493v5 fatcat:7dhhvonhqzhhjohxh6c5o65qei

Slow chaos in surface flows [article]

Corinna Ulcigrai
2020 arXiv   pre-print
This is a survey of recent advances in the study of chaotic and spectral properties of smooth area-preserving flows on surfaces, written in occasion of a plenary talk given at the XXI Congress of the Italian  ...  Mathematical Union, in Pavia in September 2020.  ...  The author is part of SwissMAP (The Mathematics of Physics National Centre for Compentence in Research) and is currently supported by a SNSF (Swiss National Science Foundation) Grant No. 200021_ 188617  ... 
arXiv:2010.06231v1 fatcat:iujzooe7bzfanlnf63acriy63m

A note on zero-one laws in metrical Diophantine approximation

Victor Beresnevich, Sanju Velani
2008 Acta Arithmetica  
In this paper we discuss a general problem on metrical Diophantine approximation associated with a system of linear forms.  ...  The main result is a zero-one law that extends one-dimensional results of Cassels and Gallagher.  ...  Finally and most importantly-happy number seventy five Wolfgang!  ... 
doi:10.4064/aa133-4-5 fatcat:rthykbmjkfesne4xfodluf244q

Convergence to the Mahler measure and the distribution of periodic points for algebraic Noetherian Z^d-actions [article]

Vesselin Dimitrov
2017 arXiv   pre-print
It is best possible in such a generality, where an exceptional set is an inevitable feature.  ...  Our main arithmetic result extends to Diophantine approximation by points of sufficiently small canonical height.  ...  In Habegger's case, it is due to working with the theory of the exponential function in a general context of rational approximation to sets definable in a polynomially bounded o-minimal expansion of the  ... 
arXiv:1611.04664v2 fatcat:x3fnr23bknfovmy7ll7kproln4
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