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Journal of Modern Dynamics

Open Access Articles

The work of Sébastien Gouëzel on limit theorems and on weighted Banach spaces
Dmitry Dolgopyat
2020, 16(0): 351-371 doi: 10.3934/jmd.2020014 +[Abstract](1694) +[HTML](575) +[PDF](249.0KB)

We review recent advances in the spectral approach to studying statistical properties of dynamical systems highlighting, in particular, the role played by Sébastien Gouëzel.

Three problems solved by Sébastien Gouëzel
François Ledrappier
2020, 16(0): 373-387 doi: 10.3934/jmd.2020015 +[Abstract](921) +[HTML](339) +[PDF](192.11KB)

We present three results of Sébastien Gouëzel's: the local limit theorem for random walks on hyperbolic groups, a multiplicative ergodic theorem for non-expansive mappings (joint work with Anders Karlsson), and the description of the essential spectrum of the Laplacian on \begin{document}$ SL(2,{\mathbb R}) $\end{document} orbits in the moduli space (joint work with Artur Avila).

The 2019 Michael Brin Prize in Dynamical Systems
The Editors
2020, 16(0): 349-350 doi: 10.3934/jmd.2020013 +[Abstract](1137) +[HTML](366) +[PDF](2517.79KB)
The 2018 Michael Brin Prize in Dynamical Systems
The Editors
2019, 15(0): 425-426 doi: 10.3934/jmd.2019025 +[Abstract](1641) +[HTML](617) +[PDF](1309.08KB)
From invariance to self-similarity: The work of Michael Hochman on fractal dimension and its aftermath
Hillel Furstenberg
2019, 15(0): 437-449 doi: 10.3934/jmd.2019027 +[Abstract](2572) +[HTML](866) +[PDF](178.53KB)

M. Hochman's work on the dimension of self-similar sets has given impetus to resolving other questions regarding fractal dimension. We describe Hochman's approach and its influence on the subsequent resolution by P. Shmerkin of the conjecture on the dimension of the intersection of \begin{document}$ \times p $\end{document}- and \begin{document}$ \times q $\end{document}-Cantor sets.

The work of Mike Hochman on multidimensional symbolic dynamics and Borel dynamics
Mike Boyle
2019, 15(0): 427-435 doi: 10.3934/jmd.2019026 +[Abstract](1248) +[HTML](389) +[PDF](152.12KB)

We review the impact of Mike Hochman's work on mutlidimensional symbolic dynamics and Borel dynamics.

The 2017 Michael Brin Prize in Dynamical Systems
The Editors
2019, 15(0): 131-132 doi: 10.3934/jmd.2019015 +[Abstract](2859) +[HTML](1070) +[PDF](1902.58KB)
The work of Lewis Bowen on the entropy theory of non-amenable group actions
Jean-Paul Thouvenot
2019, 15(0): 133-141 doi: 10.3934/jmd.2019016 +[Abstract](2926) +[HTML](842) +[PDF](151.84KB)

We present the achievements of Lewis Bowen, or, more precisely, his breakthrough works after which a theory started to develop. The focus will therefore be made here on the isomorphism problem for Bernoulli actions of countable non-amenable groups which he solved brilliantly in two remarkable papers. Here two invariants were introduced, which led to many developments.

Bill Veech's contributions to dynamical systems
Giovanni Forni, Howard Masur and John Smillie
2019, 14(1): v-xxv doi: 10.3934/jmd.2019v +[Abstract](7372) +[HTML](2186) +[PDF](244.49KB)
William A. Veech's publications
The Editors
2019, 14(1): i-iv doi: 10.3934/jmd.2019i +[Abstract](3148) +[HTML](1559) +[PDF](77.22KB)
Roy Adler's publications and patents
The Editors
2018, 13(1): i-iv doi: 10.3934/jmd.2018i +[Abstract](2996) +[HTML](1120) +[PDF](72.4KB)
Symmetry of entropy in higher rank diagonalizable actions and measure classification
Manfred Einsiedler and Elon Lindenstrauss
2018, 13(1): 163-185 doi: 10.3934/jmd.2018016 +[Abstract](2783) +[HTML](755) +[PDF](270.75KB)

An important consequence of the theory of entropy of \begin{document}$ \mathbb{Z}$\end{document}-actions is that the events measurable with respect to the far future coincide (modulo null sets) with those measurable with respect to the distant past, and that measuring the entropy using the past will give the same value as measuring it using the future. In this paper we show that for measures invariant under multiparameter algebraic actions if the entropy attached to coarse Lyapunov foliations fail to display a stronger symmetry property of a similar type this forces the measure to be invariant under non-trivial unipotent groups. Some consequences of this phenomenon are noted.

Roy Adler and the lasting impact of his work
Bruce Kitchens, Brian Marcus and Benjamin Weiss
2018, 13(1): v-x doi: 10.3934/jmd.2018v +[Abstract](5302) +[HTML](1292) +[PDF](112.68KB)
Values of random polynomials at integer points
Jayadev S. Athreya and Gregory A. Margulis
2018, 12(1): 9-16 doi: 10.3934/jmd.2018002 +[Abstract](6104) +[HTML](1467) +[PDF](142.38KB)

Using classical results of Rogers [12, Theorem 1] bounding the L2-norm of Siegel transforms, we give bounds on the heights of approximate integral solutions of quadratic equations and error terms in the quantitative Oppenheim theorem of Eskin-Margulis-Mozes [6] for almost every quadratic form. Further applications yield quantitative information on the distribution of values of random polynomials at integral points.

Logarithm laws for unipotent flows, Ⅱ
Jayadev S. Athreya and Gregory A. Margulis
2017, 11(1): 1-16 doi: 10.3934/jmd.2017001 +[Abstract](6584) +[HTML](274) +[PDF](186.6KB)

We prove analogs of the logarithm laws of Sullivan and KleinbockMargulis in the context of unipotent flows. In particular, we prove results for horospherical actions on homogeneous spaces G/Γ.

The work of Federico Rodriguez Hertz on ergodicity of dynamical systems
Dmitry Dolgopyat
2016, 10(02): 175-189 doi: 10.3934/jmd.2016.10.175 +[Abstract](4070) +[PDF](409.8KB)
We review recent advances on ergodicity of partially and nonuniformly hyperbolic systemsdescribing, in particular, important contributions of Federico Rodriguez Hertz and his collaborators.
The 2015 Michael Brin Prize in Dynamical Systems
The Editors
2016, 10(02): 173-174 doi: 10.3934/jmd.2016.10.173 +[Abstract](4054) +[PDF](5152.2KB)
Professor Michael Brin of the University of Maryland endowed an internationalprize for outstanding work in the theory of dynamical systems and relatedareas. The prize is given biennially for specific mathematical achievements thatappear as a single publication or a series thereof in refereed journals, proceedingsor monographs.

For more information please click the “Full Text” above.
On the work of Rodriguez Hertz on rigidity in dynamics
Ralf Spatzier
2016, 10(02): 191-207 doi: 10.3934/jmd.2016.10.191 +[Abstract](3221) +[PDF](204.6KB)
This paper is a survey about recent progress in measure rigidity and global rigidity of Anosov actions, and celebrates the profound contributions by Federico Rodriguez Hertz to rigidity in dynamical systems.
Federico Rodriguez Hertz
2014, 8(3&4): i-i doi: 10.3934/jmd.2014.8.3i +[Abstract](2867) +[PDF](51.8KB)
This special issue presents some of the lecture notes of the courses held in the 2008 and 2011 Summer Institutes at the Mathematics Research and Conference Center of Polish Academy of Sciences at Będlewo, Poland. The school was structured as daily courses with a double lecture each, in two parts of 45-50 minutes with a break in between.

For more information please click the “Full Text” above.
On the work of Sarig on countable Markov chains and thermodynamic formalism
Yakov Pesin
2014, 8(1): 1-14 doi: 10.3934/jmd.2014.8.1 +[Abstract](4315) +[PDF](180.7KB)
The paper is a nontechnical survey and is aimed to illustrate Sarig'sprofound contributions to statistical physics and in particular,thermodynamic formalism for countable Markov shifts. I will discusssome of Sarig's work on characterization of existence of Gibbsmeasures, existence and uniqueness of equilibrium states as well asphase transitions for Markov shifts on a countable set of states.

2021 Impact Factor: 0.641
5 Year Impact Factor: 0.894
2021 CiteScore: 1.1


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