Die Entkommensrate des n-Zentrenproblems [Elektronische Ressource] = The escape rate of the n-centre problem / vorgelegt von Markus Krapf

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• Publié par: friedrich-alexander-universitat_erlangen-nurnberg
• Publié le: 01 janvier 2007
• Nombre de lectures: 8
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Die Entkommensrate des n-Zentrenproblems
The Escape Rate of the n-centre Problem
Den Naturwissenschaftlichen Fakult¨aten
der Friedrich-Alexander-Universit¨at Erlangen-Nu¨rnberg
zur
Erlangung des Doktorgrades
vorgelegt von
Markus Krapf
aus Weiden i.d. Opf.Als Dissertation genehmigt von den Naturwissenschaftlichen
Fakult¨aten der Universit¨at Erlangen Nu¨rnberg
Tag der mu¨ndlichen Pru¨fung: 17.07.07
Vorsitzender der
Promotionskommission: Prof. Dr. Eberhard B¨ansch
Erstberichterstatter: Prof. Dr. A. Knauf
Zweitberichterstatter: Prof. Dr. G. KellerContents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Zusammenfassung . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 Introduction 3
1.1 General Definitions and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Definition of the n-Centre Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Classification of States in Phase Space . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 The Interaction Zone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Scattering Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.5.1 Asymptotics and Asymptotic Maps . . . . . . . . . . . . . . . . . . . . . . 8
1.5.2 Møller Transformation and Scattering Map . . . . . . . . . . . . . . . . . 9
1.6 Time Delay and Duration within the Interaction Zone . . . . . . . . . . . . . . . 12
1.7 First Definition of the Escape Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Escape Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 General Notations and Theoretical Background 22
2.1 Topological Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.1.1 Measure-theoretical Entropy and Entropy Function . . . . . . . . . . . . . 22
2.1.2 Topological Pressure and Topological Entropy . . . . . . . . . . . . . . . . 24
2.2 Shift Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.2.1 Block Shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2.2 Classes of Functions on Shift Spaces . . . . . . . . . . . . . . . . . . . . . 30
2.2.3 Topological Dynamics of Shift Spaces . . . . . . . . . . . . . . . . . . . . 31
2.3 Locally Constant Functions on Shift Spaces . . . . . . . . . . . . . . . . . . . . . 34
2.3.1 Topological Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.2 Markov Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.3.3 Gibbs Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 Thermodynamical Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.1 Roof Functions and Suspension Flows . . . . . . . . . . . . . . . . . . . . 38
2.4.2 Thermodynamical Formalism on One-Sided Shifts . . . . . . . . . . . . . 39
2.5 Renewal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.5.1 Renewal Theory for the Stick Model . . . . . . . . . . . . . . . . . . . . . 44
2.5.2 An Abstract Renewal Theorem . . . . . . . . . . . . . . . . . . . . . . . . 45
2.5.3 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
i3 Dynamics Over the Interaction Zone 51
3.1 Poincar´e Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.1 General Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Poincar´e Surfaces of the n-Centre Problem . . . . . . . . . . . . . . . . . 53
3.2 Outer Poincar´e Surfaces and Space of Asymptotics . . . . . . . . . . . . . . . . . 54
3.3 Symbolic Dynamics of the n-Centre Problem . . . . . . . . . . . . . . . . . . . . 57
3.3.1 Poincar´e Times and -Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.3.2 Shift Space of Bounded States on the Inner Poincar´e Surfaces . . . . . . . 58
3.4 Symbolic Dynamics and the Escape Rate . . . . . . . . . . . . . . . . . . . . . . 60
3.4.1 Total Inner Poincar´e Time τ vs. Duration τ . . . . . . . . . . . . . . 60I d,EE±3.4.2 Approximation of V (t) by Tubes . . . . . . . . . . . . . . . . . . . . . . 61IE
3.5 The Inner P-Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.1 Adapted Coordinates and Metric . . . . . . . . . . . . . . . . . . . . . . . 64
3.5.2 Further Structures onI . . . . . . . . . . . . . . . . . . . . . . . . . . . 65E
3.6 Volume Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6.1 Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.6.2 Volume Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.6.3 Volume Estimates by the Unstable Jacobian . . . . . . . . . . . . . . . . . 70
4 Results About the Escape Rate 73
4.1 The Input Data of the n-Centre Problem . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Thermodynamical Formalism: First Approach . . . . . . . . . . . . . . . . . . . . 74
4.3 Second Approach: Renewal Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Computational Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
˜4.4.1 A Priori Estimates for β . . . . . . . . . . . . . . . . . . . . . . . . . . . 78E
4.4.2 Quality of the Approximation . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.5 Additional Result About The Free Energy Function . . . . . . . . . . . . . . . . 83
4.6 Further Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.6.1 Statistics of Words in the Set of Best Fitting Sequences . . . . . . . . . . 84
4.6.2 Spatial Distribution of Long Interacting States . . . . . . . . . . . . . . . 84
4.6.3 Ergodic Properties and Suspended Flow . . . . . . . . . . . . . . . . . . . 85
A Statistics of Sticks 86
A.1 Unweighted Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
A.1.1 Refinement of the Bounds for the Asymptotics in the Lattice Case . . . . 95
A.2 Weighted Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
Bibliography 101
Index of Symbols 104
Index 106
Danksagung 107
iiAbstract
We show existence and give a implict formula for the escape-rate of the n-centre problem in
celestial mechanics for high energies.
In the first chapter we introduce and define the n-centre problem and cite some properties
needed for further studying. Then we define the time-delay for a scattered state from which
we deduce the definition of the escape-rate as some integral over an appropriate ’asymptotic
space’. Asafirststepoftheapproachforestimatingtheescape-rateweshowastrongcorrelation
betweenthedurationspentwithinthesocalledinteractionzoneandthetime-delayandtherefore
also with the escape-rate.
Thesecond chapter is to give the theoretical background and introducesthe notation needed
for this work. We will provide two different approaches showing the existence and giving an
implicit formula of the escape rate. The first is by using Thermodynamical Formalism with its
mainingredient, thefreeenergyfunction. Thesecond isbyusingRenewal Theoryinan adapted
version of Lalley [Lal]. The main ingredient here is a special form of a Renewal Equation. Both
approachesleadtoanimplicitformulafortheescaperate. Asacomputationaltoolforcomputing
the escape rate with this formula, in Chapter 4 we determine the unique equilibrium measure
for a locally constant function on the shift space as a Markov measure.
The third chapter serves as preparation for the application approaches of Chapter 2. We
consider the dynamics within the interaction zone more precisely by introducing Poincar´e sur-
faces, times and maps needed for applying symbolic dynamics. Of particular interest is the set
of bounded states on the so called “inner” Poincar´e surfaces which is a hyperbolic set for the
Poincar´e map. By introducing so called “best fitting” sequences and the unstable Jacobian we
are able to estimate an appropriate volume whose asymptotic rate equals the escape rate.
The two approaches developed in Chapter 2 and the preparing considerations in Chapter 3
are merged in the fourth chapter. It turns out that data of the n-centre problem, i.e. the mass
distribution of then centres, are coded in two functions: the Poincar´e time between appropriate
Poincar´e surfaces in the interaction zone and (the logarithm of) the unstable Jacobian of the
associated Poincar´e map,bothdefinedon thehyperbolicset ofboundedstates onthesesurfaces.
After an appropriate adaption of these data we use the Thermodynamical formalism of Chapter
2. This gives an implicit formula for the escape-rate. Furthermore using Perturbation Theory
allows for computable approximations of this rate. We end this chapter by raising a subsequent
question concerning so called “spatial distribution of long interaction states”, which is meant as
a further prospect.
Appendix A is devoted to a ”toy-model”, for which we provide a slight refinement of the
statement of renewal theory, based on explicit estimates.
1Zusammenfassung
IndieserArbeitzeigen wirdieExistenzderEntkommensratefu¨rdasn-ZentrenproblemderHim-
melsmechanik. Des weiteren geben