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CMP 1 809 305
(2001:07)
37Jxx
Iturriaga, Renato(MEX-CIM); Sánchez-Morgado, Héctor(MEX-NAM-IM)
A minimax selector for a class of Hamiltonians on cotangent bundles.
(English. English summary)
Internat. J. Math. 11 (2000), no. 9, 1147--1162.
{A review for this item is in process.}
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CMP 1 785 184
(2001:02)
37J50 (37J45 58E10)
Contreras, G.(MEX-CIM); Iturriaga, R.(MEX-CIM); Paternain, G. P.(MEX-CIM); Paternain, M.(UR-UREPS-CM)
The Palais-Smale condition and Mañé's critical values.
(English. English summary)
Ann. Henri Poincaré 1 (2000), no. 4, 655--684.
Let $N$ be a closed connected smooth manifold, and $\scr L\colon TN
\to {\Bbb R}$ a smooth convex superlinear autonomous Lagrangian. Let
$M$ be a covering for $N$ and $L$ a lift of $\scr L$ to $TM$.
Denote by $S\sb {L}(\gamma) = \int\sb {a}\sp {b} L(\gamma(t), \dot{\gamma}(t))
\,dt$ the action of the curve $\gamma \colon [a,b] \to M$, by
$C(q\sb {1}, q\sb {2}; T)$ the set $\{\gamma \colon [0, T] \to M\vert\
\gamma(0) = q\sb {1}, \gamma(T) = q\sb {2}\}$ and by $\Phi\sb {k}(q\sb {1},
q\sb {2})
= \inf\{S\sb {L+k}(\gamma)\vert \gamma \in \bigcap\sb {T > 0} C(q\sb {1},
q\sb {2}; T)\}$ the action potential.
The critical value $c(L)$ was introduced by Mane as the infimum of
$k \in {\Bbb R}$ such that, for some $q \in M$, $\Phi\sb {k}(q,q) >
-\infty$.
The authors show that such a critical value can also be characterized
(whenever the Peierls barrier $h\sb {c}(q\sb {1}, q\sb {2})$ is finite) as the
infimum of the levels $k$ such that the functional ${\scr A}\sb {L}
\colon {\Bbb R} \times \{x \in H\sp {1}(0,T; M)\vert x(0) = q\sb {1}, x(T) =
q\sb {2}\}\to {\Bbb R}$, ${\scr A}\sb {L}(b, x) = b\int\sb {0}\sp {1} L(x(t),
\dot{x}(t)/b) \,dt$, satisfies the (PS) condition at level $k$. The
condition on the Peierls barrier is shown to hold whenever the static
set is not empty, in particular whenever $M$ is compact.
If $M$ is the universal covering of $N$ and $q\sb {0} \in N$ has no
conjugate point, then it is also shown in the paper that for every $q
\in M$ there is a unique solution with energy $k$ joining $q\sb {0}$ and
$q$ provided $k > c(L)$, and, conversely, that the existence of such a
unique solution for every $q \in M$ implies that $k \geq c(L)$.
Reviewed by Vittorio Coti Zelati
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CMP 1 720 372
(2000:03)
37J50 (37J45 53D25)
Contreras, Gonzalo(MEX-CIM); Iturriaga, Renato(MEX-CIM)
Global minimizers of autonomous Lagrangians.
22$\sp {\rm o}$ Colóquio Brasileiro de Matemática. [22nd Brazilian Mathematics Colloquium]
Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 1999. 148 pp. ISBN 85-244-0151-6
This book gives an up-to-date presentation of J. Mather's theory of
minimizing measures for positive definite
Lagrangian systems
[see J. N. Mather, Math. Z. 207 (1991), no. 2, 169--207; MR 92m:58048;
Ann. Inst. Fourier (Grenoble) 43 (1993), no. 5, 1349--1386; MR 95c:58075]
and surrounding areas. It is based on R. Mane's approach to this
topic
[see R. Mane, in Proceedings of the International Congress of
Mathematicians, Vol. 1, 2 (Zurich, 1994), 1216--1220, Birkhauser, Basel,
1995; MR 97e:58090; Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2,
141--153; MR 98i:58092].
The introduction presents classes of interesting examples to which the
theory developed in the book applies:
mechanical Lagrangians, magnetic Lagrangians and twisted geodesic flows.
In Section 4-2 relations to Finsler geometry are treated via the Maupertuis
principle.
Chapters 2 and 3 form the core of the book.
Here the authors present complete proofs for existence and properties of
minimizing measures, minimizing orbits,
Mather's minimal average action and Mane's critical value, Peierls's
energy barrier, etc.
Chapter 4 treats the relation to weak solutions of the Hamilton-Jacobi
equation as developed by
A. Fathi [C. R. Acad. Sci. Paris Ser. I Math. 324 (1997), no. 9,
1043--1046; MR 98g:58151] and to
G. P. and M. Paternain's work
[Math. Z. 217 (1994), no. 3, 367--376; MR 95k:58117]
on Anosov energy levels.
The final chapter is based on work by Mane and by the authors and
investigates how the theory is
simplified and sharpened if one assumes the Lagrangian to be generic.
A non-expert will often have to refer to the original literature in order
to find more motivation for concepts and
results.
Apart from this the book is a very useful and almost everywhere reliable
source of information.
Reviewed by Victor Bangert
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2001f:37073
37J05 (37J45 37J50 70H03)
Iturriaga, Renato(MEX-CIM)
A face of Lagrangian systems.
(Spanish)
Fourth Summer School on Geometry and Dynamical Systems (Spanish) (Guanajuato, 1997),
143--164,
Aportaciones Mat. Comun., 21,
Soc. Mat. Mexicana, México, 1998.
From the text (translated from the Spanish): "These notes are the result of an
introductory minicourse of five sessions on Lagrangian systems. When
such a broad area of study is restricted in this way, we can only hope to see
one of its facets. The idea behind this course was to interest students in this
branch of mathematics. To that end, we presented three talks covering
classical material: Lagrangian systems, Hamiltonian systems and the Tonelli
theorem, and two on recent contributions to the topic by J. Mather and R.
Ma ne. The more ambitious goal of these notes is to have the idea
of the course reach more people than those who were in attendance."
\{For the entire collection see MR 2001e:00023.\}
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2001e:00023
00B25
IV Escuela de Verano de Geometría y Sistemas Dinámicos.
(Spanish) [Fourth Summer School on Geometry and Dynamical Systems]
Proceedings of the school held in Guanajuato, 1997.
Edited by Omegar Calvo and Renato Iturriaga.
Aportaciones Matemáticas: Comunicaciones [Mathematical Contributions: Communications], 21.
Sociedad Matemática Mexicana, México, 1998. ii+313 pp. ISBN 968-36-6853-4
Contents:
Ricardo Berlanga Zubiaga, Luis Hernandez Lamoneda and Adolfo Sanchez
Valenzuela, Introduction to the geometry of Lie groups (1--93);
Omegar Calvo Andrade, Complex linear systems (95--120);
Xavier Gomez-Mont, Geodesic flow: the Frida Kahlo of mathematics (121--142);
Renato Iturriaga, A face of Lagrangian systems (143--164);
Jesus Mucino-Raymundo, Hyperbolic geometry: an introduction using calculus
and complex variables (165--196);
Victor Nunez, Knots (197--250);
Fausto Ongay [Fausto Ongay-Larios], Electromagnetism and differential forms
(251--288);
Adolfo Sanchez Valenzuela, Electromagnetism, harmonic analysis and the
conformal group (289--313).
\{The papers, all in Spanish, are being reviewed individually.\}
Cited in: 1 787 550 1 787 545 1 787 543 2001g:37067 2001f:57008 2001f:37073
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2001d:37099
37J50 (53C60 70H03)
Iturriaga, Renato(MEX-CIM); Sánchez-Morgado, Héctor(MEX-NAM-IM)
Finsler metrics and action potentials.
(English. English summary)
Proc. Amer. Math. Soc. 128 (2000), no. 11, 3311--3316 (electronic).
Summary: "We study the behavior of Mane's action potential
$\Phi\sb k$ associated to a convex superlinear Lagrangian, for $k$ bigger
than the critical value $c(L)$. We obtain growth estimates for the
action potential as a function of $k$. We also prove that the action
potential can be written as
$\Phi\sb k(x,y)=D\sb {\rm F}(x,y)+f(y)-f(x)$ where $f$
is a smooth function and $D\sb {\rm F}$ is the distance function associated to
a Finsler metric."
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2001c:37022
37C27 (55M25 57M25)
Contreras, Gonzalo(MEX-CIM); Iturriaga, Renato(MEX-CIM)
Average linking numbers.
(English. English summary)
Ergodic Theory Dynam. Systems 19 (1999), no. 6, 1425--1435.
This paper investigates the average linking number (which takes the
periods of the orbits into account) of orbits in a flow generated by a
vector field on $S\sp 3$. In case orbits are not closed, a "good set of
short curves" is used to close the orbits, and limits are taken as
time goes to infinity. A definition that closely resembles the Gauss
integral, but uses the given vector field for tangent vectors, is
shown to be independent of the good set of short curves, under quite
general conditions on invariant measures on $S\sp 3$.
Reviewed by Mark E. Kidwell
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2000h:37102
37J99 (37D25 37D40 37J50)
Contreras, Gonzalo(MEX-CIM); Iturriaga, Renato(MEX-CIM)
Convex Hamiltonians without conjugate points.
(English. English summary)
Ergodic Theory Dynam. Systems 19 (1999), no. 4, 901--952.
The authors study Hamiltonian flows associated
to superlinear convex Hamiltonians, restricted to regular energy
levels. They define a notion
of conjugate points for such a flow, generalizing the usual notion
of conjugate points in the setting of Riemannian geometry.
They construct canonical invariant bundles along orbits
without conjugate points, analogous to the Green bundles
in the particular case of geodesic flows
[L. W. Green, Michigan Math. J. 5 (1958), 31--34; MR 20 #4300].
They give necessary and sufficient conditions for these
bundles to define an Anosov splitting for the flow,
extending the previously known results in the case of geodesic flows.
By comparing the Green bundles with the splitting provided
by Oseledets and Pesin theory, they obtain a formula for the metric entropy
of the Liouville measure, generalizing the results of
A. Freire and R. Mane [Invent. Math. 69 (1982), no. 3,
375--392; MR 84d:58063] for the case of
geodesic flows.
In the dual language of Lagrangian flows, Mane [Nonlinearity 9
(1996), no. 2, 273--310; MR 97d:58118]
proved that
for any Lagrangian $L\colon TM \toR$, there is a generic
set of potentials $\Phi$ for which the Lagrangian flow of $L+\Phi$
has a unique minimizing measure.
In the paper under review, the authors prove the
following conjecture of Mane: potentials for which
the support of the unique minimizing measure is a periodic orbit
can be arbitrarly approximated by
potentials for which the minimizing measure is still supported
by a periodic orbit, and with the following additionnal
properties: the periodic orbit is hyperbolic, and its
stable and unstable manifolds
have only transverse intersections.
Reviewed by Thierry Barbot
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99f:58075
58F05 (58F17 58F27)
Contreras, G.(MEX-CIM); Iturriaga, R.(MEX-CIM); Paternain, G. P.(UR-UREPS-CM); Paternain, M.(UR-UREPS-CM)
Lagrangian graphs, minimizing measures and Mañé's critical values.
(English. English summary)
Geom. Funct. Anal. 8 (1998), no. 5, 788--809.
For a convex superlinear Lagrangian $L$ on a closed connected manifold
$N$, R. Mane defined the critical value $c(L)$ and the action
potential $\Phi\sb {k}$. His results can be extended to any lift
$\tilde{L}$ on a covering $\tilde{N}$. The first theorem in this
paper states that for any lift, $c(\tilde{L})$ is the infimum of the
values $k$ such that $\tilde{H}\sp {-1}(-\infty,k)$ contains an exact
Lagrangian graph, where $\tilde{H}$ is the Hamiltonian associated with
$\tilde{L}$ . As a consequence, for $k>c(\tilde{L})$ the
Euler-Lagrange flow of $\tilde{L}$ on the energy level $k$ can be
reparametrized as the geodesic flow on the unit tangent bundle of an
appropriately chosen Finsler metric on $\tilde{N}$. The second
theorem states that if the Euler-Lagrange flow of $L$ on the energy
level $k$ is Anosov, then $k>c\sb {u}(L)=$ critical value of the lift of
$L$ to the universal covering of $N$. It follows that given
$k<c\sb {u}(L)$, there exists a potential $\psi$ with arbitrarily small
$C\sp {2}$-norm such that the energy level $k$ of $L+\psi$ possesses
conjugate points. In the last theorem, a weak KAM solution for any
covering is explicitly given in terms of the action potential.
Reviewed by Hector Sanchez-Morgado
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99a:58063
58F05 (58F15 58F17)
Contreras, G.(BR-PCRJ); Iturriaga, R.(MEX-CIM); Sánchez-Morgado, H.(MEX-NAM-IM)
On the creation of conjugate points for Hamiltonian systems.
(English. English summary)
Nonlinearity 11 (1998), no. 2, 355--361.
Let $M$ be a closed connected smooth manifold and let $H\colon
T\sp {*}M\to {R}$ be a convex superlinear Hamiltonian. For $e\in
{R}$, let ${\scr A}\sb {e}$ be the set of $\phi\in C\sp {\infty}(M)$
such that the flow of the Hamiltonian $H+\phi$ is Anosov on the energy
level $(H+\phi)\sp {-1}(e)$ and let ${\scr B}\sb {e}$ be the set of $\phi\in
C\sp {\infty}(M)$ such that the energy level $(H+\phi)\sp {-1}(e)$ contains
no conjugate points. The authors show that ${\scr A}\sb {e}$ is
precisely the interior of ${\scr B}\sb {e}$ in the $C\sp {2}$ topology.
This theorem extends to the Hamiltonian setting a result of
R. O. Ruggiero [Math. Z. 208 (1991), no. 1, 41--55; MR 92i:58143] for
the geodesic flow.
The proof goes as follows. It is well known that
in the $C\sp {k}$ topology the set ${\scr A}\sb {e}$ is open and the set
${\scr B}\sb {e}$ is closed. G. P. Paternain and M. Paternain [Math. Z.
217 (1994), no. 3, 367--376; MR 95k:58117] showed
that ${\scr A}\sb {e}$ is always
contained in ${\scr B}\sb {e}$, hence to prove their theorem the authors
need to show that an interior point of ${\scr B}\sb {e}$ lies in ${\scr
A}\sb {e}$. For this they take a system without conjugate points which
is not Anosov, then they show that they can make a $C\sp {2}$ small
perturbation in a neighborhood of an orbit segment to end up with a
system that has conjugate points. This is achieved by showing that the
index form of the perturbed system is not positive definite on that
segment.
Reviewed by Gabriel P. Paternain
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98i:58093
58F05 (49Q20 58E05)
Contreras, Gonzalo(BR-PCRJ); Delgado, Jorge(BR-PCRJ); Iturriaga, Renato(MEX-CIM)
Lagrangian flows: the dynamics of globally minimizing orbits. II.
(English. English summary)
Bol. Soc. Brasil. Mat. (N.S.) 28 (1997), no. 2, 155--196.
In this work the authors provide proofs of most of the theorems of
R. Mane's last and unfinished work [Bol. Soc. Brasil. Mat. (N.S.)
28 (1997), no. 2, 141--153; MR 98i:58092; see
the preceding review].
Cited in: 98i:58092
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97d:58077
58F05 (58F11 70H30)
Iturriaga, Renato(MEX-CIM)
Minimizing measures for time-dependent Lagrangians.
Proc. London Math. Soc. (3) 73 (1996), no. 1, 216--240.
This is a very nice paper in which the author develops R. Mane's
ideas towards generalizing the Aubry-Mather theory to the context of
time-dependent, but not periodic, Lagrangians.
In a seminal paper [Math. Z. 207 (1991), no. 2, 169--207; MR 92m:58048], J. N. Mather studied
periodic Lagrangians on closed Riemannian manifolds, introducing
concepts such as the homology of an invariant measure, minimizing
measures, etc. Here the author considers a Lagrangian system
$(d/dt)(\partial L/\partial v)(x,x',y)=(\partial L/\partial
x)(x,x',y)$, where the parameter $y$ belongs to a compact Riemannian
manifold $N$ and varies according to an autonomous system
$y'=X(y)$. This setting covers for instance the case where $N$ is a
torus and $X$ an irrational linear flow (the quasiperiodic case) or
when the flow on $N$ is hyperbolic. In the latter, the results come
closer to those of Mather.
Reviewed by Jair Koiller
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(c) 2001, American Mathematical Society