# On the finite blocking property

Thierry Monteil^{[1]}

- [1] Institut de Mathématiques de Luminy, CNRS UMR 6206, Case 907, 163 Avenue de Luminy, 13288 Marseille cedex 09 (France)

Annales de l’institut Fourier (2005)

- Volume: 55, Issue: 4, page 1195-1217
- ISSN: 0373-0956

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topMonteil, Thierry. "On the finite blocking property." Annales de l’institut Fourier 55.4 (2005): 1195-1217. <http://eudml.org/doc/116217>.

@article{Monteil2005,

abstract = {A planar polygonal billiard $\{\mathcal \{P\}\}$ is said to have the finite blocking property if
for every pair $(O,A)$ of points in $\{\mathcal \{P\}\}$ there exists a finite number of
“blocking” points $B_1, \dots , B_n$ such that every billiard trajectory from $O$ to
$A$ meets one of the $B_i$’s. Generalizing our construction of a counter-example to a
theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the
finite blocking property are the square, the equilateral triangle and the hexagon. Then
we extend this result to translation surfaces. We prove that the only Veech surfaces with
the finite blocking property are the torus branched coverings. We also provide a local
sufficient condition for a translation surface to fail the finite blocking property. This
enables us to give a complete classification for the L-shaped surfaces as well as to
obtain a density result in the space of translation surfaces in every genus $g\ge 2$.},

affiliation = {Institut de Mathématiques de Luminy, CNRS UMR 6206, Case 907, 163 Avenue de Luminy, 13288 Marseille cedex 09 (France)},

author = {Monteil, Thierry},

journal = {Annales de l’institut Fourier},

keywords = {Blocking property; polygonal billiards; regular polygons; translation surfaces; Veech surfaces; torus branched covering; illumination; quadratic differentials; blocking property},

language = {eng},

number = {4},

pages = {1195-1217},

publisher = {Association des Annales de l'Institut Fourier},

title = {On the finite blocking property},

url = {http://eudml.org/doc/116217},

volume = {55},

year = {2005},

}

TY - JOUR

AU - Monteil, Thierry

TI - On the finite blocking property

JO - Annales de l’institut Fourier

PY - 2005

PB - Association des Annales de l'Institut Fourier

VL - 55

IS - 4

SP - 1195

EP - 1217

AB - A planar polygonal billiard ${\mathcal {P}}$ is said to have the finite blocking property if
for every pair $(O,A)$ of points in ${\mathcal {P}}$ there exists a finite number of
“blocking” points $B_1, \dots , B_n$ such that every billiard trajectory from $O$ to
$A$ meets one of the $B_i$’s. Generalizing our construction of a counter-example to a
theorem of Hiemer and Snurnikov, we show that the only regular polygons that have the
finite blocking property are the square, the equilateral triangle and the hexagon. Then
we extend this result to translation surfaces. We prove that the only Veech surfaces with
the finite blocking property are the torus branched coverings. We also provide a local
sufficient condition for a translation surface to fail the finite blocking property. This
enables us to give a complete classification for the L-shaped surfaces as well as to
obtain a density result in the space of translation surfaces in every genus $g\ge 2$.

LA - eng

KW - Blocking property; polygonal billiards; regular polygons; translation surfaces; Veech surfaces; torus branched covering; illumination; quadratic differentials; blocking property

UR - http://eudml.org/doc/116217

ER -

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