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Discrete & Continuous Dynamical Systems - S

2009 , Volume 2 , Issue 2

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Preface
Ayşe Şahin, Ilie Ugarcovici and Marian Gidea
2009, 2(2): i-ii doi: 10.3934/dcdss.2009.2.2i +[Abstract](411) +[PDF](49.0KB)
Abstract:
In October, 2007 the AMS Central Sectional Meeting was held at DePaul University in Chicago, IL. At that time, the guest editors of this issue of DCDS-S organized two special sessions dedicated to dynamical systems: “Smooth dynamical systems” and “Ergodic theory and symbolic dynamical systems”. The meeting was preceded by a workshop titled “Applications of measurable and smooth dynamical systems to number theory”, hosted jointly by DePaul and Northeastern Illinois universities, where M. Einsiedler, A. Katok, and A. Venkatesh gave expository lectures. The goal of the workshop was to disseminate the tools and ideas of their work on the Littlewood conjecture and related topics to the larger dynamical systems community. This confluence of events encouraged us to put together this volume which gives a small snapshot of research conducted in dynamical systems around the time of the workshop and conference.
   The volume does not represent the entire scope of what is a very large and active field but does span a variety of distinct areas in dynamical systems. It is worth noting, however, that there are natural groupings around some central ideas which cut across sub-disciplines.

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Measured topological orbit and Kakutani equivalence
Andres del Junco, Daniel J. Rudolph and Benjamin Weiss
2009, 2(2): 221-238 doi: 10.3934/dcdss.2009.2.221 +[Abstract](419) +[PDF](232.2KB)
Abstract:
Suppose $X$ and $Y$ are Polish spaces each endowed with Borel probability measures $\mu$ and $\nu$. We call these Polish probability spaces. We say a map $\phi$ is a nearly continuous if there are measurable subsets $X_0\subseteq X$ and $Y_0\subseteq Y$, each of full measure, and $\phi:X_0\to Y_0$ is measure-preserving and continuous in the relative topologies on these subsets. We show that this is a natural context to study morphisms between ergodic homeomorphisms of Polish probability spaces. In previous work such maps have been called almost continuous or finitary. We propose the name measured topological dynamics for this area of study. Suppose one has measure-preserving and ergodic maps $T$ and $S$ acting on $X$ and $Y$ respectively. Suppose $\phi$ is a measure-preserving bijection defined between subsets of full measure on these two spaces. Our main result is that such a $\phi$ can always be regularized in the following sense. Both $T$ and $S$ have full groups ($FG(T)$ and $FG(S)$) consisting of those measurable bijections that carry a point to a point on the same orbit. We will show that there exists $f\in FG(T)$ and $h\in FG(S)$ so that $h\phi f$ is nearly continuous. This comes close to giving an alternate proof of the result of del Junco and Şahin, that any two measure-preserving ergodic homeomorphisms of nonatomic Polish probability spaces are continuously orbit equivalent on invariant $G_\delta$ subsets of full measure. One says $T$ and $S$ are evenly Kakutani equivalent if one has an orbit equivalence $\phi$ which restricted to some subset is a conjugacy of the induced maps. Our main result implies that any such measurable Kakutani equivalence can be regularized to a Kakutani equivalence that is nearly continuous. We describe a natural nearly continuous analogue of Kakutani equivalence and prove it strictly stronger than Kakutani equivalence. To do this we introduce a concept of nearly unique ergodicity.
An example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate
Brett M. Werner
2009, 2(2): 239-249 doi: 10.3934/dcdss.2009.2.239 +[Abstract](378) +[PDF](187.3KB)
Abstract:
We present an example of Kakutani equivalent and strong orbit equivalent substitution systems that are not conjugate.
An Ambrose-Kakutani representation theorem for countable-to-1 semiflows
David M. McClendon
2009, 2(2): 251-268 doi: 10.3934/dcdss.2009.2.251 +[Abstract](281) +[PDF](241.3KB)
Abstract:
Let $X$ be a Polish space and $T_t$ a jointly Borel measurable action of $\mathbb{R}^+ = [0, \infty)$ on $X$ by surjective maps preserving some Borel probability measure $\mu$ on $X$. We show that if each $T_t$ is countable-to-1 and if $T_t$ has the "discrete orbit branching property'' (described in the introduction), then $(X, T_t)$ is isomorphic to a "semiflow under a function''.
Limited scope adic transformations
Sarah Bailey Frick
2009, 2(2): 269-285 doi: 10.3934/dcdss.2009.2.269 +[Abstract](346) +[PDF](237.7KB)
Abstract:
We introduce a family of adic transformations on diagrams that are nonstationary and nonsimple. This family includes some previously studied adic transformations such as the Pascal and Euler adic transformations. We give examples of particular adic transformations with roots of unity and we show as well that the Euler adic is totally ergodic. We show that the Euler adic and a disjoint subfamily of adic transformations are loosely Bernoulli.
Heaviness in symbolic dynamics: Substitution and Sturmian systems
David Ralston
2009, 2(2): 287-300 doi: 10.3934/dcdss.2009.2.287 +[Abstract](340) +[PDF](201.9KB)
Abstract:
Heaviness refers to a sequence of partial sums maintaining a certain lower bound and was recently introduced and studied in [11]. After a review of basic properties to familiarize the reader with the ideas of heaviness, general principles of heaviness in symbolic dynamics are introduced. The classical Morse sequence is used to study a specific example of heaviness in a system with nontrivial rational eigenvalues. To contrast, Sturmian sequences are examined, including a new condition for a sequence to be Sturmian.
A note on universality in multidimensional symbolic dynamics
Michael Hochman
2009, 2(2): 301-314 doi: 10.3934/dcdss.2009.2.301 +[Abstract](474) +[PDF](191.9KB)
Abstract:
We show that in the category of effective $\mathbb{Z}$-dynamical systems there is a universal system, i.e. one that factors onto every other effective system. In particular, for $d\geq3$ there exist $d$-dimensional shifts of finite type which are universal for $1$-dimensional subactions of SFTs. On the other hand, we show that there is no universal effective $\mathbb{Z}^{d}$-system for $d\geq2$, and in particular SFTs cannot be universal for subactions of rank $\geq2$. As a consequence, a decrease in entropy and Medvedev degree and periodic data are not sufficient for a factor map to exists between SFTs.
   We also discuss dynamics of cellular automata on their limit sets and show that (except for the unavoidable presence of a periodic point) they can model a large class of physical systems.
An application of topological multiple recurrence to tiling
Rafael De La Llave and A. Windsor
2009, 2(2): 315-324 doi: 10.3934/dcdss.2009.2.315 +[Abstract](307) +[PDF](225.3KB)
Abstract:
We show that given any tiling of Euclidean space, any geometric pattern of points, we can find a patch of tiles (of arbitrarily large size) so that copies of this patch appear in the tiling nearly centered on a scaled and translated version of the pattern. The rather simple proof uses Furstenberg's topological multiple recurrence theorem.
The modulated ergodic Hilbert transform
Doǧan Çömez
2009, 2(2): 325-336 doi: 10.3934/dcdss.2009.2.325 +[Abstract](334) +[PDF](241.9KB)
Abstract:
In this article we study the ergodic Hilbert transform modulated by bounded sequences. We prove that sequences satisfying some variation conditions and are universally good for ordinary ergodic averages, such as the sequences defined by the Fourier coefficients of $L_p$ functions, are universally good modulating sequences for the ergodic Hilbert transform. We also prove that sequences belonging to the subfamily $B_1^{\alpha} $ of the two-sided bounded Besicovitch class $B_1$ are good modulating sequences for the ergodic Hilbert transform.
Ultrametric logarithm laws I
J. S. Athreya, Anish Ghosh and Amritanshu Prasad
2009, 2(2): 337-348 doi: 10.3934/dcdss.2009.2.337 +[Abstract](663) +[PDF](206.0KB)
Abstract:
We announce ultrametric analogues of the results of Kleinbock-Margulis for shrinking target properties of semisimple group actions on symmetric spaces. The main applications are $S$-arithmetic Diophantine approximation results and logarithm laws for buildings, generalizing the work of Hersonsky-Paulin on trees.
Note on homology of expanding foliations
Radu Saghin
2009, 2(2): 349-360 doi: 10.3934/dcdss.2009.2.349 +[Abstract](304) +[PDF](164.5KB)
Abstract:
This note contains some remarks about the homologies that can be associated to a foliation which is invariant and uniformly expanded by a diffeomorphism. We construct a family of 'dynamical' closed currents supported on the foliation which help us relate the geometric volume growth of the leaves under the diffeomorphism with the map induced on homology in the case when these currents have nonzero homology.
Generating forms for exact volume-preserving maps
H. E. Lomelí and J. D. Meiss
2009, 2(2): 361-377 doi: 10.3934/dcdss.2009.2.361 +[Abstract](365) +[PDF](241.8KB)
Abstract:
We develop a general theory of implicit generating forms for volume-preserving diffeomorphisms on a manifold. Our results generalize the classical formulas for generating functions of symplectic twist maps and examples of Carroll for volume-preserving maps on $\R^n$.
Hyperbolicity for symmetric periodic orbits in the isosceles three body problem
Daniel Offin and Hildeberto Cabral
2009, 2(2): 379-392 doi: 10.3934/dcdss.2009.2.379 +[Abstract](501) +[PDF](205.8KB)
Abstract:
We study the isosceles three body problem with fixed symmetry line for arbitrary masses, as a subsystem of the N-body problem. Our goal is to construct minimizing noncollision periodic orbits using a symmetric variational method with a finite order symmetry group. The solution of this variational problem gives existence of noncollision periodic orbits which realize certain symbolic sequences of rotations and oscillations in the isosceles three body problem for any choice of the mass ratio. The Maslov index for these periodic orbits is used to prove the main result, Theorem 4.1, which states that the minimizing curves in the three dimensional reduced energy momentum surface can be extended to periodic curves which are generically hyperbolic. This reminds one of a theorem of Poincaré [8], concerning minimizing periodic geodesics on orientable 2D surfaces. The results in this paper are novel in two directions: in addition to the higher dimensional setting, the minimization in the current problem is over a symmetry class, rather than a loop space.
Obstruction argument for transition chains of tori interspersed with gaps
Marian Gidea and Clark Robinson
2009, 2(2): 393-416 doi: 10.3934/dcdss.2009.2.393 +[Abstract](333) +[PDF](335.3KB)
Abstract:
We consider a dynamical system whose phase space contains a two-dimensional normally hyperbolic invariant manifold diffeomorphic to an annulus. We assume that the dynamics restricted to the annulus is given by an area preserving monotone twist map. We assume that in the annulus there exist finite sequences of primary invariant Lipschitz tori of dimension $1$, with the property that the unstable manifold of each torus has a topologically crossing intersection with the stable manifold of the next torus in the sequence. We assume that the dynamics along these tori is topologically transitive. We assume that the tori in these sequences, possibly with the exception of the tori at the ends of the sequences, can be $C^0$-approximated from both sides by other primary invariant tori in the annulus. We assume that the region in the annulus between two successive sequences of tori is a Birkhoff zone of instability. We prove the existence of orbits that follow the sequences of invariant tori and cross the Birkhoff zones of instability.

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