# American Institute of Mathematical Sciences

ISSN:
1078-0947

eISSN:
1553-5231

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## Discrete & Continuous Dynamical Systems - A

2008 , Volume 20 , Issue 3

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2008, 20(3): 425-457 doi: 10.3934/dcds.2008.20.425 +[Abstract](117) +[PDF](405.8KB)
Abstract:
We describe several topics within the theory of linear and nonlinear second order elliptic Partial Differential Equations. Through elementary approaches, we first explain how elliptic and parabolic PDEs are related to central issues in Probability and Geometry. This leads to several concrete equations. We classify them and describe their regularity theories. After this, most of the paper focuses on the ABP technique and its applications to the classical isoperimetric problem for which we present a new original proof, the symmetry result of Gidas-Ni-Nirenberg, and the regularity theory for fully nonlinear elliptic equations.
2008, 20(3): 459-509 doi: 10.3934/dcds.2008.20.459 +[Abstract](132) +[PDF](732.7KB)
Abstract:
This article addresses long-term behavior of solutions to a semilinear damped wave equation with a critical source term. A distinctive feature of the model is the geometrically constrained dissipation: it only affects a small subset of the domain adjacent to a connected portion of the boundary. The main result of the paper provides an affirmative answer to the open question whether global attractors for a wave equation with critical source and geometrically constrained damping are smooth and finite-dimensional. A positive answer to the same question in the case of subcritical sources was given in [9]. However, critical exponent of the source term combined with weak geometrically restricted dissipation constitutes the major new difficulty of the problem. To overcome this issue we develop a new version of Carleman's estimates and apply them in the context of recent results [12] on fractal dimension of global attractors.
2008, 20(3): 511-541 doi: 10.3934/dcds.2008.20.511 +[Abstract](73) +[PDF](397.0KB)
Abstract:
We prove that, given a tree pattern $\mathcal{P}$, the set of periods of a minimal representative $f: T\rightarrow T$ of $\mathcal{P}$ is contained in the set of periods of any other representative. This statement is an immediate corollary of the following stronger result: there is a period-preserving injection from the set of periodic points of $f$ into that of any other representative of $\mathcal{P}$. We prove this result by extending the main theorem of [6] to negative cycles.
2008, 20(3): 543-576 doi: 10.3934/dcds.2008.20.543 +[Abstract](80) +[PDF](399.5KB)
Abstract:
In this paper we discuss initial-boundary problems for second order parabolic equations with rapidly oscillating coefficients in a bounded convex domain. The asymptotic expansions of the solutions for problems with multiple spatial and temporal scales are presented in four different cases. Higher order corrector methods are constructed and associated explicit convergence rates obtained.
2008, 20(3): 577-587 doi: 10.3934/dcds.2008.20.577 +[Abstract](73) +[PDF](195.4KB)
Abstract:
In [10] the author provided a generalization of the classical Desch-Schappacher-Webb sufficient criterion which ensures hypercyclicity of linear semigroups. In this paper we simplify assumptions of [10], obtaining new criteria for hypercyclicity of a $C_0$ semigroup in a subspace (sub-hypercyclicity), and also for its sub-chaoticity. Moreover, we provide full characterization of chaoticity and hypercyclicity spaces of semigroups satisfying the assumptions of these new criteria. We also present examples showing that, in general, these assumptions cannot be weakened.
2008, 20(3): 589-604 doi: 10.3934/dcds.2008.20.589 +[Abstract](125) +[PDF](265.7KB)
Abstract:
We show that there is a residual subset $\S (M)$ of Diff$^1$ (M) such that, for every $f\in \S(M)$, any homoclinic class of $f$ containing periodic saddles $p$ and $q$ of indices $\alpha$ and $\beta$ respectively, where $\alpha< \beta$, has superexponential growth of the number of periodic points inside the homoclinic class. Furthermore, it is shown that the super-exponential growth occurs for hyperbolic periodic points of index $\gamma$ inside the homoclinic class for every $\gamma\in[\alpha,\beta]$.
2008, 20(3): 605-616 doi: 10.3934/dcds.2008.20.605 +[Abstract](95) +[PDF](190.9KB)
Abstract:
We study a nonlinear Dirac system in one space dimension with a quadratic nonlinearity which exhibits null structure in the sense of Klainerman. Using an $L^{p}$ variant of the $L^2$ restriction method of Bourgain and Klainerman-Machedon, we prove local well-posedness for initial data in a Sobolev-like space $\hat{H^{s,p}}(\R)$ whose scaling dimension is arbitrarily close to the critical scaling dimension.
2008, 20(3): 617-637 doi: 10.3934/dcds.2008.20.617 +[Abstract](83) +[PDF](277.0KB)
Abstract:
We obtain an optimal $W^{1,p}$, $2 \leq p < \infty$, regularity theory on the conormal derivative problem for a nonlinear parabolic equation in divergence form with small BMO nonlinearity in a $\delta$-Reifenberg flat domain.
2008, 20(3): 639-657 doi: 10.3934/dcds.2008.20.639 +[Abstract](82) +[PDF](247.0KB)
Abstract:
The topological pressure is defined for sub-additive potentials via separated sets and open covers in general compact dynamical systems. A variational principle for the topological pressure is set up without any additional assumptions. The relations between different approaches in defining the topological pressure are discussed. The result will have some potential applications in the multifractal analysis of iterated function systems with overlaps, the distribution of Lyapunov exponents and the dimension theory in dynamical systems.
2008, 20(3): 659-672 doi: 10.3934/dcds.2008.20.659 +[Abstract](60) +[PDF](181.6KB)
Abstract:
Hetzer and Shen [3] considered a system of a two-species Lotka-Volterra competition model with an inhibitor, investigated its long-term behavior and proposed two open questions: one is whether the system has a nontrivial periodic solution; the other is whether one of two positive equilibria is non-hyperbolic in the case that the system has exactly two positive equilibria. The goal of this paper is first to give these questions clear answers, then to present a complete classification for its dynamics in terms of coefficients. As a result, all solutions are convergent as $t$ goes to infinity.
2008, 20(3): 673-711 doi: 10.3934/dcds.2008.20.673 +[Abstract](88) +[PDF](574.2KB)
Abstract:
We consider a one-parameter family of expanding interval maps $\{T_{\alpha}\}_{\alpha \in [0,1]}$ (Japanese continued fractions) which include the Gauss map ($\alpha=1$) and the nearest integer and by-excess continued fraction maps ($\alpha=\frac{1}{2},\,\alpha=0$). We prove that the Kolmogorov-Sinai entropy $h(\alpha)$ of these maps depends continuously on the parameter and that $h(\alpha) \to 0$ as $\alpha \to 0$. Numerical results suggest that this convergence is not monotone and that the entropy function has infinitely many phase transitions and a self-similar structure. Finally, we find the natural extension and the invariant densities of the maps $T_{\alpha}$ for $\alpha=\frac{1}{n}$.
2008, 20(3): 713-724 doi: 10.3934/dcds.2008.20.713 +[Abstract](57) +[PDF](246.0KB)
Abstract:
In this work we exhibit a one-parameter family of $C^1$-diffeomorphisms $F_\alpha$ of the 2-sphere, where $\alpha>1$, such that the equator $\S^1$ is an attracting set for every $F_\alpha$ and $F_\alpha|_{\S^1}$ is the identity. For $\alpha>2$ the Lebesgue measure on the equator is a non ergodic physical measure having uncountably many ergodic components. On the other hand, for $1<\alpha\leq 2$ there is no physical measure for $F_\alpha$. If $\alpha<2$ this follows directly from the fact that the $\omega$-limit of almost every point is a single point on the equator (and the basin of each of these points has zero Lebesgue measure). This is no longer true for $\alpha=2$, and the non existence of physical measure in this critical case is a more subtle issue.
2008, 20(3): 725-738 doi: 10.3934/dcds.2008.20.725 +[Abstract](72) +[PDF](191.5KB)
Abstract:
We study nearest-neighbor shifts of finite type (NNSOFT) on a free group $\G$. We determine when a NNSOFT on $\G$ admits a periodic coloring and give an example of a NNSOFT that does not allow a periodic coloring. Then, we find an expression for the entropy of the golden mean shift on $\G$. In doing so, we study a new generalization of Fibonacci numbers and analyze their asymptotics with a one-dimensional iterated map that is related to generalized continued fractions.

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