## Journals

- Advances in Mathematics of Communications
- Big Data & Information Analytics
- Communications on Pure & Applied Analysis
- Discrete & Continuous Dynamical Systems - A
- Discrete & Continuous Dynamical Systems - B
- Discrete & Continuous Dynamical Systems - S
- Evolution Equations & Control Theory
- Inverse Problems & Imaging
- Foundations of Data Science
- Journal of Computational Dynamics
- Journal of Dynamics & Games
- Journal of Geometric Mechanics
- Journal of Industrial & Management Optimization
- Journal of Modern Dynamics
- Kinetic & Related Models
- Mathematical Biosciences & Engineering
- Mathematical Control & Related Fields
- Mathematical Foundations of Computing
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- AIMS Mathematics
- Conference Publications
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- Mathematics in Engineering

### Open Access Journals

ERA-MS

We study a two-parameter family of one-dimensional maps and the related $(a,b)$-continued fractions suggested for consideration by Don Zagier and announce the following results and outline their proofs: (i) the associated natural extension maps
have attractors with finite rectangular structure for the entire parameter set except for a Cantor-like set of one-dimensional zero measure that we completely describe; (ii) for a dense open set of parameters the Reduction theory conjecture holds, i.e. every point is mapped to the attractor after finitely many iterations.
We also give an application of this theory to coding geodesics on the modular surface and outline the computation of the smooth invariant measures associated with these transformations.

DCDS

We prove that if a diffeomorphism on a compact manifold preserves
a nonatomic ergodic hyperbolic Borel probability measure, then
there exists a hyperbolic periodic point such that the closure of
its unstable manifold has positive measure. Moreover, the support
of the measure is contained in the closure of all such hyperbolic
periodic points. We also show that if an ergodic hyperbolic
probability measure does not locally maximize entropy in the space
of invariant ergodic hyperbolic measures, then there exist
hyperbolic periodic points that satisfy a multiplicative
asymptotic growth and are uniformly distributed with respect to
this measure.

JMD

We study a two-parameter family of one-dimensional maps and related
$(a,b)$-continued fractions suggested for consideration by Don Zagier. We prove
that the associated natural extension maps have attractors with finite
rectangular structure for the entire parameter set except for a Cantor-like set
of one-dimensional Lebesgue zero measure that we completely describe. We show
that the structure of these attractors can be "computed'' from the data
$(a,b)$, and that for a dense open set of parameters the Reduction theory
conjecture holds,

*i.e.*, every point is mapped to the attractor after finitely many iterations. We also show how this theory can be applied to the study of invariant measures and ergodic properties of the associated Gauss-like maps.
DCDS-S

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.

For more information please click the “Full Text” above.

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.

For more information please click the “Full Text” above.

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