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DCDS

We study the regularity of the composition operator
$((f, g)\to g \circ f)$
in spaces of Hölder differentiable functions. Depending on the smooth norms used
to topologize $f, g$ and their composition, the operator has different differentiability
properties. We give complete and sharp results for the classical Hölder spaces of
functions defined on geometrically well behaved open sets in Banach spaces. We
also provide examples that show that the regularity conclusions are sharp and also
that if the geometric conditions fail, even in finite dimensions, many elements of
the theory of functions (smoothing, interpolation, extensions) can have somewhat
unexpected properties.

DCDS-S

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.

DCDS

We use topological methods to investigate some recently proposed
mechanisms of instability (Arnol'd diffusion) in Hamiltonian
systems.

In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit non-degeneracy assumptions, which hold generically.

In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount.

As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability.

Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.

In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold $\Lambda$, so that: (a) the manifold $\Lambda$ is covered rather densely by transitive tori (possibly of different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies some explicit non-degeneracy assumptions, which hold generically.

In this paper we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount.

As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on differentiability.

Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.

DCDS

We consider conformal structures invariant under
a volume-preserving Anosov system. We show that if such a
structure is in $L^p$ for sufficiently large $p$, then it is
continuous.

DCDS

We generalize the Aubry-Mather theorem on the existence of quasi-periodic solutions of one dimensional difference equations to situations in which the independent variable ranges over more complicated lattices. This is a natural generalization of Frenkel-Kontorova models to physical situations in a higher dimensional space. We also consider generalizations in which the interactions among the particles are not just nearest neighbor, and indeed do not have finite range.

DCDS-B

In this paper we consider the family of circle maps
$f_{k,\alpha,\epsilon}:\mathbb{S}^1\rightarrow \mathbb{S}^1$ which when
written mod 1 are of the form
$f_{k,\alpha,\epsilon}: x \mapsto k x + \alpha + \epsilon \sin(2\pi
x)$, where the parameter $\alpha$ ranges in $\mathbb{S}^1$
and $k\geq 2.$ We prove that for small $\epsilon$ the average over $\alpha$ of the
entropy of $f_{k,\alpha,\epsilon}$ with respect to the natural absolutely
continuous measure is smaller than $\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx,$ while
the maximum with respect to $\alpha$ is larger. In the case of the average the
difference is of order of $\epsilon^{2k+2}.$ This result is in contrast to families
of expanding Blaschke products depending on rotations where the averages are
equal and for which the inequality for averages goes in the other direction when
the expanding property does not hold, see [4]. A striking fact for both
results is that the maximum of the entropies is greater than or equal to
$\int_0^1 \log|Df_{k,0,\epsilon}(x)|dx$. These results should also be compared with
[3], where similar questions are considered for a family of
diffeomorphisms of the two sphere.

CPAA

We consider gradient descent equations for energy functionals of
the type $S(u) = \frac{1}{2} < u(x), A(x)u(x)>_{L^2} +
\int_{\Omega} V(x,u) dx$, where $A$ is a
uniformly elliptic operator of order 2, with smooth coefficients.
The gradient descent equation for such a functional depends on the
metric under consideration.

We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.

We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

We consider the steepest descent equation for $S$ where the gradient is an element of the Sobolev space $H^{\beta}$, $\beta \in (0,1)$, with a metric that depends on $A$ and a positive number $\gamma >$sup$|V_{2 2}|$. We prove a weak comparison principle for such a gradient flow.

We extend our methods to the case where $A$ is a fractional power of an elliptic operator, and provide an application to the Aubry-Mather theory for partial differential equations and pseudo-differential equations by finding plane-like minimizers of the energy functional.

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