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Asymptotic analysis of the scattering problem for the Helmholtz equations with high wave numbers
Ergodic properties of folding maps on spheres
1. | University of Toronto, Department of Mathematics, 40 St. George St., Room 6290, Toronto, ON M5S 2E4, Canada |
2. | University of Chicago, Department of Mathematics, 5734 S. University Avenue, Room 208C, Chicago, IL 60637, USA |
We consider the trajectories of points on $ \mathbb{S}^{d-1} $ under sequences of certain folding maps associated with reflections. The main result characterizes collections of folding maps that produce dense trajectories. The minimal number of maps in such a collection is d+1.
References:
[1] |
A. V. Aho, M. R. Garey and J. D. Ullman,
The transitive reduction of a directed graph, SIAM J. Comput., 1 (1972), 131-137.
doi: 10.1137/0201008. |
[2] |
G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999.
doi: 10.1017/CBO9781107325937. |
[3] |
A. Baernstein Ⅱ and B. A. Taylor,
Spherical rearrangements, subharmonic functions, and *-functions in n-space, Duke Math. J., 43 (1976), 245-268.
doi: 10.1215/S0012-7094-76-04322-2. |
[4] |
W. Beckner,
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242.
doi: 10.2307/2946638. |
[5] |
Y. Benyamini, Two-point symmetrization, the isoperimetric inequality on the sphere, and
some applications, in Texas Functional Analysis Seminar 1983-1984, Longhorn Notes, University of Texas Press, Austin, (1984), 53-76 |
[6] |
F. Brock and A. Yu. Solynin,
An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352 (2000), 1759-1796.
doi: 10.1090/S0002-9947-99-02558-1. |
[7] |
A. Burchard, Rate of convergence of random polarizations, preprint, arXiv: 1108.5500, 2011. |
[8] |
A. Burchard and M. Fortier,
Random polarizations, Adv. Math., 234 (2013), 550-573.
doi: 10.1016/j.aim.2012.10.010. |
[9] |
A. Burchard and M. Schmuckenschläger,
Comparison theorems for exit times, Geom. Funct. Anal., 11 (2001), 651-692.
doi: 10.1007/PL00001681. |
[10] |
H. S. M. Coxeter,
Discrete groups generated by reflections, Ann. of Math. (2), 35 (1934), 588-621.
doi: 10.2307/1968753. |
[11] |
J. De Keyser and J. Van Schaftingen, Approximation of symmetrizations by Markov processes, Indiana Univ. Math. J. to appear (2016); preprint, arXiv: 1508.00464 |
[12] |
P. Diaconis and M. Shahshahani,
Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete, 57 (1981), 159-179.
doi: 10.1007/BF00535487. |
[13] |
J. D. Dixon,
The probability of generating the symmetric group, Math. Z., 110 (1969), 199-205.
doi: 10.1007/BF01110210. |
[14] |
H. G. Eggleston,
Convexity Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. |
[15] |
M. Einsiedler and T. Ward,
Ergodic Theory, with a View towards Number Theory Graduate Texts in Mathematics, No. 259, Springer Verlag, London, 2011.
doi: 10.1007/978-0-85729-021-2. |
[16] |
A. A. Felikson, Spherical simplexes that generate discrete reflection groups, Mat. Sb. , 195 (2004), 127-142; translation in Sb. Math. , 195 (2004), 585-598, arXiv: math.MG/0212244.
doi: 10.1070/SM2004v195n04ABEH000816. |
[17] |
A. Goetz,
Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.
|
[18] |
D. A. Klain,
Steiner symmetrization using a finite set of directions, Adv. Appl. Math., 48 (2012), 340-353.
doi: 10.1016/j.aam.2011.09.004. |
[19] |
B. Klartag,
Rate of convergence of geometric symmetrizations, Geom. Funct. Anal., 14 (2004), 1322-1338.
doi: 10.1007/s00039-004-0493-4. |
[20] |
N. Levitt and H. J. Sussmann,
On controllability by means of two vector fields, SIAM J. Control, 13 (1975), 1271-1281.
doi: 10.1137/0313079. |
[21] |
D. Montgomery and H. Samelson,
Transformation groups of spheres, Ann. of Math. (2), 44 (1943), 454-470.
doi: 10.2307/1968975. |
[22] |
C. Morpurgo,
Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Math. J., 114 (2002), 477-553.
doi: 10.1215/S0012-7094-02-11433-1. |
[23] |
Y. Peres and P. Sousi,
An isoperimetric inequality for the Wiener sausage, Geom. Funct. Anal., 22 (2012), 1000-1014.
doi: 10.1007/s00039-012-0184-5. |
[24] |
U. Porod,
The cut-off phenomenon for random reflections, Ann. Probab., 24 (1996), 74-96.
doi: 10.1214/aop/1042644708. |
[25] |
J. S. Rosenthal,
Random rotations: Characters and random walks on SO(N), Ann. Probab., 22 (1994), 398-423.
doi: 10.1214/aop/1176988864. |
[26] |
F. Silva Leite and P. Crouch,
Closed forms for the exponential mapping on matrix Lie groups based on Putzer's method, J. Math. Phys., 40 (1999), 3561-3568.
doi: 10.1063/1.532908. |
[27] |
S. R. S. Varadhan,
Probability Theory Courant Lecture Notes in Mathematics, vol. 7, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/cln/007. |
show all references
References:
[1] |
A. V. Aho, M. R. Garey and J. D. Ullman,
The transitive reduction of a directed graph, SIAM J. Comput., 1 (1972), 131-137.
doi: 10.1137/0201008. |
[2] |
G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71, Cambridge University Press, 1999.
doi: 10.1017/CBO9781107325937. |
[3] |
A. Baernstein Ⅱ and B. A. Taylor,
Spherical rearrangements, subharmonic functions, and *-functions in n-space, Duke Math. J., 43 (1976), 245-268.
doi: 10.1215/S0012-7094-76-04322-2. |
[4] |
W. Beckner,
Sharp Sobolev inequalities on the sphere and the Moser-Trudinger inequality, Ann. of Math. (2), 138 (1993), 213-242.
doi: 10.2307/2946638. |
[5] |
Y. Benyamini, Two-point symmetrization, the isoperimetric inequality on the sphere, and
some applications, in Texas Functional Analysis Seminar 1983-1984, Longhorn Notes, University of Texas Press, Austin, (1984), 53-76 |
[6] |
F. Brock and A. Yu. Solynin,
An approach to symmetrization via polarization, Trans. Amer. Math. Soc., 352 (2000), 1759-1796.
doi: 10.1090/S0002-9947-99-02558-1. |
[7] |
A. Burchard, Rate of convergence of random polarizations, preprint, arXiv: 1108.5500, 2011. |
[8] |
A. Burchard and M. Fortier,
Random polarizations, Adv. Math., 234 (2013), 550-573.
doi: 10.1016/j.aim.2012.10.010. |
[9] |
A. Burchard and M. Schmuckenschläger,
Comparison theorems for exit times, Geom. Funct. Anal., 11 (2001), 651-692.
doi: 10.1007/PL00001681. |
[10] |
H. S. M. Coxeter,
Discrete groups generated by reflections, Ann. of Math. (2), 35 (1934), 588-621.
doi: 10.2307/1968753. |
[11] |
J. De Keyser and J. Van Schaftingen, Approximation of symmetrizations by Markov processes, Indiana Univ. Math. J. to appear (2016); preprint, arXiv: 1508.00464 |
[12] |
P. Diaconis and M. Shahshahani,
Generating a random permutation with random transpositions, Z. Wahrsch. Verw. Gebiete, 57 (1981), 159-179.
doi: 10.1007/BF00535487. |
[13] |
J. D. Dixon,
The probability of generating the symmetric group, Math. Z., 110 (1969), 199-205.
doi: 10.1007/BF01110210. |
[14] |
H. G. Eggleston,
Convexity Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958. |
[15] |
M. Einsiedler and T. Ward,
Ergodic Theory, with a View towards Number Theory Graduate Texts in Mathematics, No. 259, Springer Verlag, London, 2011.
doi: 10.1007/978-0-85729-021-2. |
[16] |
A. A. Felikson, Spherical simplexes that generate discrete reflection groups, Mat. Sb. , 195 (2004), 127-142; translation in Sb. Math. , 195 (2004), 585-598, arXiv: math.MG/0212244.
doi: 10.1070/SM2004v195n04ABEH000816. |
[17] |
A. Goetz,
Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.
|
[18] |
D. A. Klain,
Steiner symmetrization using a finite set of directions, Adv. Appl. Math., 48 (2012), 340-353.
doi: 10.1016/j.aam.2011.09.004. |
[19] |
B. Klartag,
Rate of convergence of geometric symmetrizations, Geom. Funct. Anal., 14 (2004), 1322-1338.
doi: 10.1007/s00039-004-0493-4. |
[20] |
N. Levitt and H. J. Sussmann,
On controllability by means of two vector fields, SIAM J. Control, 13 (1975), 1271-1281.
doi: 10.1137/0313079. |
[21] |
D. Montgomery and H. Samelson,
Transformation groups of spheres, Ann. of Math. (2), 44 (1943), 454-470.
doi: 10.2307/1968975. |
[22] |
C. Morpurgo,
Sharp inequalities for functional integrals and traces of conformally invariant operators, Duke Math. J., 114 (2002), 477-553.
doi: 10.1215/S0012-7094-02-11433-1. |
[23] |
Y. Peres and P. Sousi,
An isoperimetric inequality for the Wiener sausage, Geom. Funct. Anal., 22 (2012), 1000-1014.
doi: 10.1007/s00039-012-0184-5. |
[24] |
U. Porod,
The cut-off phenomenon for random reflections, Ann. Probab., 24 (1996), 74-96.
doi: 10.1214/aop/1042644708. |
[25] |
J. S. Rosenthal,
Random rotations: Characters and random walks on SO(N), Ann. Probab., 22 (1994), 398-423.
doi: 10.1214/aop/1176988864. |
[26] |
F. Silva Leite and P. Crouch,
Closed forms for the exponential mapping on matrix Lie groups based on Putzer's method, J. Math. Phys., 40 (1999), 3561-3568.
doi: 10.1063/1.532908. |
[27] |
S. R. S. Varadhan,
Probability Theory Courant Lecture Notes in Mathematics, vol. 7, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2001.
doi: 10.1090/cln/007. |
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