# American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

All Issues

## Journal of Modern Dynamics

Open Access Articles

2020, 16(0): 351-371 doi: 10.3934/jmd.2020014 +[Abstract](852) +[HTML](252) +[PDF](249.0KB)
Abstract:

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

2020, 16(0): 373-387 doi: 10.3934/jmd.2020015 +[Abstract](553) +[HTML](203) +[PDF](192.11KB)
Abstract:

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).

2020, 16(0): 349-350 doi: 10.3934/jmd.2020013 +[Abstract](750) +[HTML](199) +[PDF](2517.79KB)
Abstract:
2019, 15(0): 425-426 doi: 10.3934/jmd.2019025 +[Abstract](1136) +[HTML](378) +[PDF](1309.08KB)
Abstract:
2019, 15(0): 437-449 doi: 10.3934/jmd.2019027 +[Abstract](1638) +[HTML](550) +[PDF](178.53KB)
Abstract:

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.

2019, 15(0): 427-435 doi: 10.3934/jmd.2019026 +[Abstract](880) +[HTML](256) +[PDF](152.12KB)
Abstract:

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

2019, 15(0): 131-132 doi: 10.3934/jmd.2019015 +[Abstract](2163) +[HTML](786) +[PDF](1902.58KB)
Abstract:
2019, 15(0): 133-141 doi: 10.3934/jmd.2019016 +[Abstract](2107) +[HTML](666) +[PDF](151.84KB)
Abstract:

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.

2019, 14(1): v-xxv doi: 10.3934/jmd.2019v +[Abstract](5663) +[HTML](1795) +[PDF](244.49KB)
Abstract:
2019, 14(1): i-iv doi: 10.3934/jmd.2019i +[Abstract](2488) +[HTML](1208) +[PDF](77.22KB)
Abstract:
2018, 13(1): i-iv doi: 10.3934/jmd.2018i +[Abstract](2553) +[HTML](787) +[PDF](72.4KB)
Abstract:
2018, 13(1): 163-185 doi: 10.3934/jmd.2018016 +[Abstract](2370) +[HTML](501) +[PDF](270.75KB)
Abstract:

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.

2018, 13(1): v-x doi: 10.3934/jmd.2018v +[Abstract](4369) +[HTML](1045) +[PDF](112.68KB)
Abstract:
2018, 12(1): 9-16 doi: 10.3934/jmd.2018002 +[Abstract](5356) +[HTML](1298) +[PDF](142.38KB)
Abstract:

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.

2017, 11(1): 1-16 doi: 10.3934/jmd.2017001 +[Abstract](5667) +[HTML](134) +[PDF](186.6KB)
Abstract:

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.

2016, 10(02): 175-189 doi: 10.3934/jmd.2016.10.175 +[Abstract](3424) +[PDF](409.8KB)
Abstract:
We review recent advances on ergodicity of partially and nonuniformly hyperbolic systemsdescribing, in particular, important contributions of Federico Rodriguez Hertz and his collaborators.
2016, 10(02): 173-174 doi: 10.3934/jmd.2016.10.173 +[Abstract](2960) +[PDF](5152.2KB)
Abstract:
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.

2016, 10(02): 191-207 doi: 10.3934/jmd.2016.10.191 +[Abstract](2663) +[PDF](204.6KB)
Abstract:
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.
2014, 8(3&4): i-i doi: 10.3934/jmd.2014.8.3i +[Abstract](2304) +[PDF](51.8KB)
Abstract:
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.

2014, 8(1): 1-14 doi: 10.3934/jmd.2014.8.1 +[Abstract](3404) +[PDF](180.7KB)
Abstract:
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.

2020 Impact Factor: 0.848
5 Year Impact Factor: 0.815
2020 CiteScore: 0.9