ISSN:
1930-5311
eISSN:
1930-532X
All Issues
Journal of Modern Dynamics
Open Access Articles
2019, 14(1): ⅴ-xxv
doi: 10.3934/jmd.2019v
+[Abstract](4609)
+[HTML](1418)
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Abstract:
2014, 8(3&4): i-i
doi: 10.3934/jmd.2014.8.3i
+[Abstract](1901)
+[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.
For more information please click the “Full Text” above.
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.
2014, 8(1): i-ii
doi: 10.3934/jmd.2014.8.1i
+[Abstract](2192)
+[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.
For more information please click the “Full Text” above.
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.
For more information please click the “Full Text” above.
2012, 6(2): 139-182
doi: 10.3934/jmd.2012.6.139
+[Abstract](2909)
+[PDF](366.9KB)
Abstract:
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.
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](2919)
+[PDF](307.2KB)
Abstract:
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.
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](2327)
+[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.
For more information please click the "Full Text" above.
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.
For more information please click the "Full Text" above.
2011, 5(1): 71-105
doi: 10.3934/jmd.2011.5.71
+[Abstract](4098)
+[PDF](396.8KB)
Abstract:
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 $R$.
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): i-ii
doi: 10.3934/jmd.2010.4.2i
+[Abstract](2311)
+[PDF](740.0KB)
Abstract:
Professor Michael Brin of the University of Maryland endowed an international prize for outstandingwork 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.
The prize recognizes mathematicians who have made substantial impact in the field at an early stage of their careers.
The prize is awarded by an international committee of experts chaired by Anatole Katok. Its members are Jean Bourgain, John N. Mather, Yakov Pesin, Marina Ratner, Marcelo Viana and BenjaminWeiss.
For more information please click the “Full Text” above.
Professor Michael Brin of the University of Maryland endowed an international prize for outstandingwork 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.
The prize recognizes mathematicians who have made substantial impact in the field at an early stage of their careers.
The prize is awarded by an international committee of experts chaired by Anatole Katok. Its members are Jean Bourgain, John N. Mather, Yakov Pesin, Marina Ratner, Marcelo Viana and BenjaminWeiss.
For more information please click the “Full Text” above.
2009, 3(4): 631-636
doi: 10.3934/jmd.2009.3.631
+[Abstract](2434)
+[PDF](87.4KB)
Abstract:
We show that given a fixed irrational rotation of the $d$-dimensional torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [6] and Fayad-Krikorian [5]. The key technique is the analyticity of $m$-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten-Martini Problem [2].
We show that given a fixed irrational rotation of the $d$-dimensional torus, any analytic SL(2, R)-cocycle can be perturbed in such a way that the Lyapunov exponent becomes positive. This result strengthens and generalizes previous results of Krikorian [6] and Fayad-Krikorian [5]. The key technique is the analyticity of $m$-functions (under the hypothesis of stability of zero Lyapunov exponents), first observed and used in the solution of the Ten-Martini Problem [2].
2009, 3(4): 549-554
doi: 10.3934/jmd.2009.3.549
+[Abstract](2333)
+[PDF](89.0KB)
Abstract:
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs less than $o(\frac{n}{\log n\log\log n})$ times.
Given a hyperbolic matrix $H\in SL(2,\R)$, we prove that for almost every $R\in SL(2,\R)$, any product of length $n$ of $H$ and $R$ grows exponentially fast with $n$ provided the matrix $R$ occurs less than $o(\frac{n}{\log n\log\log n})$ times.
2008, 2(3): i-ii
doi: 10.3934/jmd.2008.2.3i
+[Abstract](2162)
+[PDF](1045.5KB)
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.
For more information please click the “Full Text” above.
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.
For more information please click the “Full Text” above.
2008, 2(1): 83-128
doi: 10.3934/jmd.2008.2.83
+[Abstract](2381)
+[PDF](523.7KB)
Abstract:
We consider measures on locally homogeneous spaces $\Gamma \backslash G$ which are invariant and have positive entropy with respect to the action of a single diagonalizable element $a \in G$ by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action.
This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [14,3] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for $G=G_1 \times G_2$ with $G_1$ a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.
We consider measures on locally homogeneous spaces $\Gamma \backslash G$ which are invariant and have positive entropy with respect to the action of a single diagonalizable element $a \in G$ by translations, and prove a rigidity statement regarding a certain type of measurable factors of this action.
This rigidity theorem, which is a generalized and more conceptual form of the low entropy method of [14,3] is used to classify positive entropy measures invariant under a one parameter group with an additional recurrence condition for $G=G_1 \times G_2$ with $G_1$ a rank one algebraic group. Further applications of this rigidity statement will appear in forthcoming papers.
2008, 2(1): i-v
doi: 10.3934/jmd.2008.2.1i
+[Abstract](2273)
+[PDF](723.1KB)
Abstract:
The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics.
For more information please click the “Full Text” above.
Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
The editors of the Journal of Modern Dynamics are happy to dedicate this issue to Gregory Margulis, who, over the last four decades, has influenced dynamical systems as deeply as few others have, and who has blazed broad trails in the application of dynamical systems to other fields of core mathematics.
For more information please click the “Full Text” above.
Additional editors: Leonid Polterovich, Ralf Spatzier, Amie Wilkinson and Anton Zorich.
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