# American Institute of Mathematical Sciences

ISSN:
1930-5311

eISSN:
1930-532X

All Issues

## Journal of Modern Dynamics

Open Access Articles

2019, 15(0): 425-426 doi: 10.3934/jmd.2019025 +[Abstract](1002) +[HTML](344) +[PDF](1309.08KB)
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2019, 15(0): 437-449 doi: 10.3934/jmd.2019027 +[Abstract](1322) +[HTML](418) +[PDF](178.53KB)
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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](744) +[HTML](221) +[PDF](152.12KB)
Abstract:

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

2019, 14(1): v-xxv doi: 10.3934/jmd.2019v +[Abstract](5245) +[HTML](1670) +[PDF](244.49KB)
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2019, 14(1): i-iv doi: 10.3934/jmd.2019i +[Abstract](2332) +[HTML](1138) +[PDF](77.22KB)
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2018, 13(1): i-iv doi: 10.3934/jmd.2018i +[Abstract](2458) +[HTML](722) +[PDF](72.4KB)
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2018, 13(1): v-x doi: 10.3934/jmd.2018v +[Abstract](4068) +[HTML](946) +[PDF](112.68KB)
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2016, 10(02): 175-189 doi: 10.3934/jmd.2016.10.175 +[Abstract](3246) +[PDF](409.8KB)
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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](2702) +[PDF](5152.2KB)
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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](2533) +[PDF](204.6KB)
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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](2152) +[PDF](51.8KB)
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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](3183) +[PDF](180.7KB)
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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.
2014, 8(1): i-ii doi: 10.3934/jmd.2014.8.1i +[Abstract](2643) +[PDF](9238.9KB)
Abstract:
Professor Michael Brin of the University of Maryland endowed an international prize for outstanding work in the theory of dynamical systems and related areas. The prize is given biennially for specific mathematical achievements that appear as a single publication or a series thereof in refereed journals, proceedings or monographs.

2014, 8(1): 15-24 doi: 10.3934/jmd.2014.8.15 +[Abstract](2539) +[PDF](176.5KB)
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N/A
2012, 6(2): 139-182 doi: 10.3934/jmd.2012.6.139 +[Abstract](3185) +[PDF](366.9KB)
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We review the Brin prize work of Artur Avila on Teichmüller dynamics and Interval Exchange Transformations. The paper is a nontechnical self-contained summary that intends to shed some light on Avila's early approach to the subject and on the significance of his achievements.
2012, 6(2): 183-203 doi: 10.3934/jmd.2012.6.183 +[Abstract](3194) +[PDF](307.2KB)
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The field of one-dimensional dynamics, real and complex, emerged from obscurity in the 1970s and has been intensely explored ever since. It combines the depth and complexity of chaotic phenomena with a chance to fully understand it in probabilistic terms: to describe the dynamics of typical orbits for typical maps. It also revealed fascinating universality features that had never been noticed before. The interplay between real and complex worlds illuminated by beautiful pictures of fractal structures adds special charm to the field. By now, we have reached a full probabilistic understanding of real analytic unimodal dynamics, and Artur Avila has been the key player in the final stage of the story (which roughly started with the new century). To put his work into perspective, we will begin with an overview of the main events in the field from the 1970s up to the end of the last century. Then we will describe Avila's work on unimodal dynamics that effectively closed up the field. We will finish by describing his results in the closely related direction, the geometry of Feigenbaum Julia sets, including a recent construction of a new class of Julia sets of positive area.
2012, 6(2): i-ii doi: 10.3934/jmd.2012.6.2i +[Abstract](2652) +[PDF](4673.4KB)
Abstract:
Professor Michael Brin of the University of Maryland endowed an international prize for outstanding work in the theory of dynamical systems and related areas. The prize is given biennially for specific mathematical achievements that appear as a single publication or a series thereof in refereed journals, proceedings or monographs.

2011, 5(1): 71-105 doi: 10.3934/jmd.2011.5.71 +[Abstract](4519) +[PDF](396.8KB)
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We compute the asymptotics, as $R$ tends to infinity, of the number $N(R)$ of closed geodesics of length at most $R$ in the moduli space of compact Riemann surfaces of genus $g$. In fact, $N(R)$ is the number of conjugacy classes of pseudo-Anosov elements of the mapping class group of a compact surface of genus $g$ of translation length at most $R$.
2010, 4(2): 243-255 doi: 10.3934/jmd.2010.4.243 +[Abstract](2069) +[PDF](157.2KB)
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D. Dolgopyat is the winner of the second Brin Prize in Dynamical Systems(2009). This article overviews his remarkable achievements in anontechnical manner. It complements two other surveys of Dolgopyat's workwritten by Y. Pesin and C. Liverani and published in this issue. Thissurvey covers Dolgopyat's work on various physical models, including theLorentz gas, Galton board, and some systems of hard disks.
2010, 4(2): 227-241 doi: 10.3934/jmd.2010.4.227 +[Abstract](2511) +[PDF](187.5KB)
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This paper is a nontechnical survey and aims to illustrate Dolgopyat'sprofound contributions to smooth ergodic theory. I will discuss some ofDolgopyat's work on partial hyperbolicity and nonuniform hyperbolicity withemphasis on the interaction between the two-the class of dynamical systemswith mixed hyperbolicity. On one hand, this includes uniformlypartially hyperbolic diffeomorphisms with nonzero Lyapunov exponents in thecenter direction. The study of their ergodic properties has provided analternative approach to the Pugh-Shub stable ergodicity theory for bothconservative and dissipative systems. On the other hand, ideas of mixedhyperbolicity have been used in constructing volume-preservingdiffeomorphisms with nonzero Lyapunov exponents on any manifold.

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