## Journals

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### Open Access Journals

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 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-B

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.

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.

DCDS

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.

DCDS-S

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.

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.

JMD

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.

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.

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 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 $\eta \in C$

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

NHM

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.

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