Instituto Nacional de Matemática Pura e Aplicada Thesis Markov and Lagrange Dynamical Spectra ...

108 Pages·2013·4.78 MB·English

Instituto Nacional de Matema´tica Pura e Aplicada
Thesis
Markov and Lagrange Dynamical Spectra for Geodesic Flows in Surfaces
with Negative Curvature.
Sergio Augusto Roman˜a Ibarra
Rio de Janeiro
April, 2013
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Instituto Nacional Matem´atica Pura e Aplicada
Sergio Augusto Roman˜a Ibarra
Markov and Lagrange Dynamical Spectra for Geodesic Flows in Surfaces
with Negative Curvature.
Thesis presented to the Post-graduate Program in Mathematics at
Instituto de Matem´atica Pura e Aplicada as partial fulfillment of
the requirements for the degree of Doctor in Mathematics.
Advisor: Carlos Gustavo T. de A. Moreira
Rio de Janeiro
March, 2013
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Agradecimentos
Em primeiro lugar come¸co agradecendo ao grande matema´tico, Carlos Gustavo T. de A.
Moreira (Gugu) por sua orienta¸ca˜o (meu orientador ´e f*** cara) e pelo envolvimento com
os problemas de Tese. Foi um grande prazer ser seu aluno de doutorado.
Agrade¸co de todo o cora¸c˜ao a minha linda fam´ılia, meus pais, Epifanio Roman˜a e Nancy
Ibarra, meus irm˜aos, Eugenio Camacho Ibarra, Zoila Roman˜a Ibarra e Sindy Roman˜a
Ibarra, minha cunhada, Yennis Gomez e meus sobrinhos, Juleicy Camacho, Johan Julio
Roman˜a e Juliane Camacho, pelo apoio durante toda a vida.
Agrade¸co tamb´em a minha linda e amada namorada (futura esposa), Joseilda Batista de
Brito (mais conhecida como Josy) por aguentar todo o estresse ocasionado pelo processo
de doutorado, agrade¸co a esta mulher por me apoiar em cada momento e sempre estar
junto dando for¸cas e na˜o deixando a vontade de seguir em frente ir embora, obrigado
amor.
Agrade¸co a os professores Jacob Palis, Maria Jos´e Pac´ıfico, Alexander Arbieto, Fernando
Cod´a, membros da Banca examinadora, pelos valiosos coment´arios. Queria tamb´em
agradecer aos professores do IMPA, Enrique Pujals, Luis Florit, por sempre estarem
dispon´ıveis para discutir assuntos de matema´tica, discusso˜es que contribu´ıram para meu
trabalho.
Queria agradecer de forma especial, a minha querida cunhis, Mary Brito por estar sempre
atenta e dando for¸cas, alem das diversas festas que alegraram alguns momentos e a meus
amigos Bruno Gois, Renan Lima, Marlon Lopez e Lucas Backes, por sua colaborac¸˜ao e
paciˆencia para escutar minhas conclus˜oes (loucuras).
Agrade¸co aos funcion´arios do IMPA, pois eles fazem um trabalho excelente para o sucesso
do instituto.
Tamb´em agrade¸co aos membros da repu´blica, Carlos Bocker, Mitchael Plaza, Eduardo,
Bernadete, Isabelle, Rafael, Ricardo e Susana.
Finalmente agrade¸co ao CNPq pelo apoio financeiro durante o doutorado.
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Abstract
The current work consists in two parts, both of them related to the study of the fractal
geometry.
The first part focuses on showing that the Lagrange and the Markov dynamical spec-
trum has a non-empty stable interior. First, we study the Lagrange and the Markov
dynamical spectrum for diffeomorphisms in surface that have a horseshoe with Hausdorff
dimension greater than 1 and the property V. We show that for a “large” set of real
functions on the surface and for a diffeomorphism with a horseshoe associated and Haus-
dorff dimension greater than 1, with the property V, both, the Lagrange and the Markov
dynamical spectrum have persistently non-empty interior. Then, we find hyperbolic sets
for the geodesic flow of surfaces of pinched negative curvature and finite volume, with
Hausdorff dimension close to 3. Associated with this hyperbolic set, we find a horseshoe
of Hausdorff dimension close to 2 for Poincar´e map. Finally, we prove that the Lagrange
and the Markov dynamical spectrum (associated to geodesic flow) have persistently non-
empty interior.
The second part focuses on showing that the Marstrand’s thoerem is true in surfaces
simply connected and non-positive curvature.
Keywords: Lagrange and Markov dynamical spectrum, Horseshoe, Hausdorff dimen-
sion, pinched curvature negative, Poincar´e map, persistently non-empty interior, the
Marstrand’s theorem.
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Contents
List of Figures 3
1 Introduction 6
I Lagrange and Markov Dynamical Spectra: For Geodesic
Flows in Surfaces with Negative Curvature. 12
2 The Lagrange and Markov Dynamical Spectrum 14
2.1 The “Large” Subset of C1(M,R) . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Regular Cantor Set and Expanding Maps Associated to a Horseshoe . . . . 20
2.2.1 Regular Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.2.2 Expanding Maps Associated to a Horseshoe . . . . . . . . . . . . . 20
2.3 The Interior of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.3.1 Intersections of Regular Cantor Sets . . . . . . . . . . . . . . . . . . 29
2.4 The Image of the Product of Two Regular Cantor Sets by a Real Function
and Behavior of the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 Hausdorff Dimension Greater than 1 for Hyperbolic Set in Cross-section
for Geodesic Flow 35
3.1 Cross-sections and Poincar´e Maps . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.1 Hyperbolicity of Poincar´e Maps . . . . . . . . . . . . . . . . . . . . 38
3.2 Good Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3 Separation of GCS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.4 Global Poincar´e Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Hausdorff Dimension of (cid:84) R−n((cid:83)l Σ ) . . . . . . . . . . . . . . . . . . 56
n∈Z i=1 i
4 Markov and Lagrange Spectrum For Geodesic Flow 58
4.1 The Interior of Spectrum for Perturbations of φ . . . . . . . . . . . . . . . 59
4.1.1 The Family of Perturbation . . . . . . . . . . . . . . . . . . . . . . 59
4.1.2 Realization of the Perturbation . . . . . . . . . . . . . . . . . . . . 60
4.1.3 Regaining the Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 66
4.2 The Interior of Spectrum for Geodesic Flow . . . . . . . . . . . . . . . . . 70
4.2.1 The Set of Geodesics With Transversal Self-Intersection . . . . . . . 71
4.2.2 Perturbation of the Metric . . . . . . . . . . . . . . . . . . . . . . . 72
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II Geometric Marstrand 80
5 The Marstrand Theorem in Nonposisive Curvature 82
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
5.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.2.1 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.3 Behavior of the Projection π . . . . . . . . . . . . . . . . . . . . . . . . . . 85
5.3.1 Differentiability of π in θ and w. . . . . . . . . . . . . . . . . . . . . 85
5.3.2 The Bessel Function Associated to π (w) . . . . . . . . . . . . . . . 91
θ
5.4 Proof of the Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 95
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