DCDS

Under the condition that unstable manifolds are one dimensional, the derivative formula of the potential function of the generalized SRB measure with respect to the underlying dynamical system is extended from the hyperbolic attractor case to the general case when the hyperbolic set intersecting with unstable manifolds is a Cantor set. It leads to derivative formulas of objects and quantities that characterize a uniformly hyperbolic system, including the generalized SBR measure and its entropy, the root of the Bowen's equation, and the Hausdorff dimension of the hyperbolic set on a dimension two Riemannian manifold.

DCDS-B

We study the coupled map lattice model of tree dispersion. Under quite general conditions on the nonlinearity of the local growth function and the dispersion (coupling) function, we show that when the maximal dispersal distance is finite and the spatial redistribution pattern remains unchanged in time, the moving front will always converge in the strongest sense to an asymptotic state: a traveling wave with finite length of the wavefront. We also show that when the climate becomes more favorable to growth or germination, the front at any nonzero density level will have a positive acceleration. An estimation of the magnitude of the acceleration is given.

DCDS

Let $M^{n}$ be a compact $C^{\infty}$ Riemannian manifold of dimension
$n\geq 2$. Let $\text{Diff}^{\r }(M^{n})$ be the space of all
$C^{\r }$ diffeomorphisms of $M^{n}$, where $ 1 < r \le \infty$.
For a $C^{\r }$ diffeomorphism $f$ in
$\text{Diff}^{\r }(M^{n})$ with a hyperbolic attractor
$\Lambda_{f}$ on which $f$ is topologically transitive, let $U(f)$
be the $C^{1}$ open set of $\text{Diff}^{\r }(M^{n})$ such that
each element in $U(f)$ can be connected to $f$ by finitely many
$C^{1}$ structural stability balls in
$\text{Diff}^{\r }(M^{n})$. Then by the structural stability,
any element $g$ in $U(f)$ has a hyperbolic attractor $\Lambda_{g}$
and $g|\Lambda_{g}$ is topologically conjugate to $f|\Lambda_{f}$.
Therefore, the topological entropy $h(g|\Lambda_{g})$
is a constant function when it is restricted to $U(f)$.
However, the metric entropy $h_{\mu}(g)$ with respect to the SRB measure
$\mu=\mu_{g}$ can vary.
We prove that the infimum of the metric entropy $h_{\mu}(g)$ on
$U(f)$ is zero.

DCDS

In this paper we continue our study [9] of the infimum of the metric entropy of the SRB measure in the space of hyperbolic dynamical systems on a smooth Riemannian manifold of higher dimension. We restrict our study to the space of volume preserving Anosov diffeomorphisms and the space of volume preserving expanding endomorphisms. In our previous paper, we use the perturbation method at a hyperbolic periodic point. It raises the question whether the volume can be preserved. In this paper, we answer this question affirmatively. We first construct a smooth path starting from any point in the space of volume preserving Anosov diffeomorphisms such that the metric entropy tends to zero as the path approaches the boundary of the space. Similarly, we construct a smooth path starting from any point in the space of volume preserving expanding endomorphisms with a fixed degree greater than one such that the metric entropy tends to zero as the path approaches the boundary of the space. Therefore, the infimum of the metric entropy as a functional is zero in both spaces.