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| THE SCENERY FLOWFOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING |
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Citation： |
P. ARNOUX,A. M. FISHER.THE SCENERY FLOWFOR GEOMETRIC STRUCTURES ON THE TORUS: THE LINEAR SETTING[J].Chinese Annals of Mathematics B,2001,22(4):427~470 |
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Net amount： 859 |
Authors： |
P. ARNOUX; A. M. FISHER |
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| Abstract： |
The authors define the scenery flow of the torus.
The flow space is the union of all flat 2-dimensional tori of area $1$
with a marked direction (or equivalently, the union of all tori
with a quadratic differential of norm 1). This is a $5$-dimensional
space, and the flow acts by following individual points under an
extremal deformation of the quadratic differential. The authors define
associated horocycle and translation flows; the latter preserve each
torus and are the horizontal and vertical flows of the corresponding
quadratic differential.
The scenery flow projects to the geodesic flow on the modular surface,
and admits, for each orientation preserving hyperbolic
toral automorphism, an invariant $3$-dimensional subset on which it is
the suspension flow of that map.
The authors first give a simple algebraic
definition in terms of the group of affine maps of the plane, and
prove that the flow is Anosov. They give an explicit formula
for the first-return map of the flow on convenient cross-sections. Then, in
the main part of the paper, the authors give several different models
for the flow and its cross-sections, in terms of:
\item{$\bullet$} stacking and rescaling periodic tilings of the plane;
\item{$\bullet$} symbolic dynamics: the natural extension of the recoding of
Sturmian sequences, or the $S$-adic system generated by two
substitutions;
\item{$\bullet$} zooming and subdividing quasi-periodic tilings of the real
line, or aperiodic quasicrystals of minimal complexity;
\item{$\bullet$} the natural extension of two-dimensional continued fractions;
\item{$\bullet$} induction on exchanges of three intervals;
\item{$\bullet$} rescaling on pairs of transverse measure foliations on the
torus,
or the Teichm\"uller flow on the twice-punctured torus. |
Keywords： |
Modular surface, Continued fractions, Sturmian sequences, Plane tilings,
Teichmuller flow, Substitution dynamical system |
Classification： |
32G15, 30F60, 53D25, 37D40 |
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