Regularity of the composition operator in spaces of Hölder functions
Rafael De La Llave R. Obaya
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.
keywords: Composition operator Hölder spaces differentiability properties.
An application of topological multiple recurrence to tiling
Rafael De La Llave A. Windsor
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.
keywords: multiple topological recurrence. Pattern recurrence Tiling
Topological methods in the instability problem of Hamiltonian systems
Marian Gidea Rafael De La Llave
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.
keywords: Instability correctly aligned windows. Arnold diffusion
On the regularity of integrable conformal structures invariant under Anosov systems
Rafael De La Llave Victoria Sadovskaya
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.
keywords: Sobolev spaces. Anosov systems Conformal structures
Aubry-Mather theory for functions on lattices
Hans Koch Rafael De La Llave Charles Radin
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.
keywords: functions on lattices. Aubry-Mather theory
Entropy estimates for a family of expanding maps of the circle
Rafael De La Llave Michael Shub Carles Simó
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.
keywords: entropy estimates expanding circle maps.
A comparison principle for a Sobolev gradient semi-flow
Timothy Blass Rafael De La Llave Enrico Valdinoci
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.
keywords: Comparison principle fractional powers of elliptic operators. Sobolev gradient semigroups of linear operators

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