On the regularity of integrable conformal structures invariant under Anosov systems
Rafael De La Llave Victoria Sadovskaya
Discrete & Continuous Dynamical Systems - A 2005, 12(3): 377-385 doi: 10.3934/dcds.2005.12.377
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
Regularity of the composition operator in spaces of Hölder functions
Rafael De La Llave R. Obaya
Discrete & Continuous Dynamical Systems - A 1999, 5(1): 157-184 doi: 10.3934/dcds.1999.5.157
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.
A parameterization method for the computation of invariant tori and their whiskers in quasi-periodic maps: Numerical algorithms
Àlex Haro Rafael de la Llave
Discrete & Continuous Dynamical Systems - B 2006, 6(6): 1261-1300 doi: 10.3934/dcdsb.2006.6.1261
In this paper we develop several numerical algorithms for the computation of invariant manifolds in quasi-periodically forced systems. The invariant manifolds we consider are invariant tori and the asymptotic invariant manifolds (whiskers) to these tori.
    The algorithms are based on the parameterization method described in [36], where some rigorous results are proved. In this paper, we concentrate on numerical issues of algorithms. Examples of implementations appear in the companion paper [34].
    The algorithms for invariant tori are based essentially on Newton method, but taking advantage of dynamical properties of the torus, such as hyperbolicity or reducibility as well as geometric properties.
    The algorithms for whiskers are based on power-matching expansions of the parameterizations. Whiskers include as particular cases the usual (strong) stable and (strong) unstable manifolds, and also, in some cases, the slow manifolds which dominate the asymptotic behavior of solutions converging to the torus.
keywords: Quasi-periodic systems numerical methods. invariant manifolds invariant tori
The parameterization method for one- dimensional invariant manifolds of higher dimensional parabolic fixed points
Inmaculada Baldomá Ernest Fontich Rafael de la Llave Pau Martín
Discrete & Continuous Dynamical Systems - A 2007, 17(4): 835-865 doi: 10.3934/dcds.2007.17.835
We use the parameterization method to prove the existence and properties of one-dimensional submanifolds of the center manifold associated to the fixed point of $C^r$ maps with linear part equal to the identity. We also provide some numerical experiments to test the method in these cases.
keywords: Parabolic point parameterization method. invariant manifold
Convergence of differentiable functions on closed sets and remarks on the proofs of the "Converse Approximation Lemmas''
Xuemei Li Rafael de la Llave
Discrete & Continuous Dynamical Systems - S 2010, 3(4): 623-641 doi: 10.3934/dcdss.2010.3.623
In KAM theory and other areas of analysis, one is often led to consider sums of functions defined in decreasing domains. A question of interest is whether the limit function is differentiable or not.
   We present examples showing that the answer cannot be based just on the size of the derivatives but that it also has to include considerations of the geometry of the domains.
   We also present some sufficient conditions on the geometry of the domains that ensure that indeed the sum of the derivatives is a Whitney derivative of the sum of the functions.
keywords: converse approximation. KAM theory Whitney differentiability
Renormalization and central limit theorem for critical dynamical systems with weak external noise
Oliver Díaz-Espinosa Rafael de la Llave
Journal of Modern Dynamics 2007, 1(3): 477-543 doi: 10.3934/jmd.2007.1.477
We study the effect of weak noise on critical one-dimensional maps; that is, maps with a renormalization theory.
    We establish a one-dimensional central limit theorem for weak noise and obtain Berry--Esseen estimates for the rate of this convergence.
    We analyze in detail maps at the accumulation of period doubling and critical circle maps with golden mean rotation number. Using renormalization group methods, we derive scaling relations for several features of the effective noise after long periods. We use these scaling relations to show that the central limit theorem for weak noise holds in both examples.
    We note that, for the results presented here, it is essential that the maps have parabolic behavior. They are false for hyperbolic orbits.
keywords: transfer operators period doubling central limit theorem effective noise. renormalization critical circle maps
An application of topological multiple recurrence to tiling
Rafael De La Llave A. Windsor
Discrete & Continuous Dynamical Systems - S 2009, 2(2): 315-324 doi: 10.3934/dcdss.2009.2.315
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
Discrete & Continuous Dynamical Systems - A 2006, 14(2): 295-328 doi: 10.3934/dcds.2006.14.295
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
Smooth dependence on parameters of solutions to cohomology equations over Anosov systems with applications to cohomology equations on diffeomorphism groups
Rafael de la Llave A. Windsor
Discrete & Continuous Dynamical Systems - A 2011, 29(3): 1141-1154 doi: 10.3934/dcds.2011.29.1141
We consider the dependence on parameters of the solutions of cohomology equations over Anosov diffeomorphisms. We show that the solutions depend on parameters as smoothly as the data. As a consequence we prove optimal regularity results for the solutions of cohomology equations taking value in diffeomorphism groups. These results are motivated by applications to rigidity theory, dynamical systems, and geometry.
    In particular, in the context of diffeomorphism groups we show: Let $f$ be a transitive Anosov diffeomorphism of a compact manifold $M$. Suppose that $\eta \in C$k+α$(M,$Diff$^r(N))$ for a compact manifold $N$, $k,r \in \N$, $r \geq 1$, and $0 < \alpha \leq \Lip$. We show that if there exists a $\varphi\in C$k+α$(M,$Diff$^1(N))$ solving

$ \varphi_{f(x)} = \eta_x \circ \varphi_x$

then in fact $\varphi \in C$k+α$(M,$Diff$^r(N))$. The existence of this solutions for some range of regularities is studied in the literature.

keywords: Anosov diffeomorphisms Cohomology equations rigidity. diffeomorphism groups Livšic theory
Perturbation and numerical methods for computing the minimal average energy
Timothy Blass Rafael de la Llave
Networks & Heterogeneous Media 2011, 6(2): 241-255 doi: 10.3934/nhm.2011.6.241
We investigate the differentiability of minimal average energy associated to the functionals $S_\epsilon (u) = \int_{\mathbb{R}^d} \frac{1}{2}|\nabla u|^2 + \epsilon V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\epsilon$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.
keywords: Lindstedt series Minimal average energy Sobolev gradient descent Plane-like minimizers Cell problem quasiperiodic solutions of PDE.

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