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• W1882052079 abstract "For a linear flow <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi> <mml:semantics> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:annotation encoding=application/x-tex>Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on a vector bundle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=pi colon upper E right-arrow upper S> <mml:semantics> <mml:mrow> <mml:mi>π<!-- π --></mml:mi> <mml:mo>:</mml:mo> <mml:mi>E</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi>S</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>pi : E rightarrow S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> a spectrum can be defined in the following way: For a chain recurrent component <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> on the projective bundle <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P upper E> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {P} E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> consider the exponential growth rates associated with (finite time) <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=left-parenthesis epsilon comma upper T right-parenthesis> <mml:semantics> <mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>(varepsilon ,T)</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-chains in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and define the Morse spectrum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript upper M o Baseline left-parenthesis script upper M comma normal upper Phi right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>M</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma _{Mo}(mathcal {M},Phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> over <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}</mml:annotation> </mml:semantics> </mml:math> </inline-formula> as the limits of these growth rates as <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=epsilon right-arrow 0> <mml:semantics> <mml:mrow> <mml:mi>ε<!-- ε --></mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding=application/x-tex>varepsilon rightarrow 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper T right-arrow normal infinity> <mml:semantics> <mml:mrow> <mml:mi>T</mml:mi> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>T rightarrow infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The Morse spectrum <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript upper M o Baseline left-parenthesis normal upper Phi right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>M</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma _{Mo}(Phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi> <mml:semantics> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:annotation encoding=application/x-tex>Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is then the union over all components <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=script upper M subset-of double-struck upper P upper E> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathcal {M}subset mathbb {P}E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. This spectrum is a synthesis of the topological approach of Selgrade and Salamon/Zehnder with the spectral concepts based on exponential growth rates, such as the Oseledec̆ spectrum or the dichotomy spectrum of Sacker/Sell. It turns out that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript upper M o Baseline left-parenthesis normal upper Phi right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>M</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma _{Mo}(Phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains all Lyapunov exponents of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi> <mml:semantics> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:annotation encoding=application/x-tex>Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for arbitrary initial values, and the <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Sigma Subscript upper M o Baseline left-parenthesis script upper M comma normal upper Phi right-parenthesis> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>Σ<!-- Σ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>M</mml:mi> <mml:mi>o</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>M</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>Sigma _{Mo}(mathcal {M},Phi )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> are closed intervals, whose boundary points are actually Lyapunov exponents. Using the fact that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Phi> <mml:semantics> <mml:mi mathvariant=normal>Φ<!-- Φ --></mml:mi> <mml:annotation encoding=application/x-tex>Phi</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is cohomologous to a subflow of a smooth linear flow on a trivial bundle, one can prove integral representations of all Morse and all Lyapunov exponents via smooth ergodic theory. A comparison with other spectral concepts shows that, in general, the Morse spectrum is contained in the topological spectrum and the dichotomy spectrum, but the spectral sets agree if the induced flow on the base space is chain recurrent. However, even in this case, the associated subbundle decompositions of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper E> <mml:semantics> <mml:mi>E</mml:mi> <mml:annotation encoding=application/x-tex>E</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may be finer for the Morse spectrum than for the dynamical spectrum. If one can show that the (closure of the) Floquet spectrum (i.e. the Lyapunov spectrum based on periodic trajectories in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper P upper E> <mml:semantics> <mml:mrow> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>P</mml:mi> </mml:mrow> <mml:mi>E</mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>mathbb {P} E</mml:annotation> </mml:semantics> </mml:math> </inline-formula>) agrees with the Morse spectrum, then one obtains equality for the Floquet, the entire Oseledeč, the Lyapunov, and the Morse spectrum. We present an example (flows induced by <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper C Superscript normal infinity> <mml:semantics> <mml:msup> <mml:mi>C</mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∞<!-- ∞ --></mml:mi> </mml:mrow> </mml:msup> <mml:annotation encoding=application/x-tex>C^{infty }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> vector fields with hyperbolic chain recurrent components on the projective bundle) where this fact can be shown using a version of Bowen’s Shadowing Lemma." @default.
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• W1882052079 date "1996-01-01" @default.
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• W1882052079 title "The Morse spectrum of linear flows on vector bundles" @default.
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