March 2017, 37(3): 1183-1200. doi: 10.3934/dcds.2017049

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

* Corresponding author: A. Burchard

Received  May 2016 Revised  November 2016 Published  December 2016

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.

Citation: Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems - A, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049
References:
[1]

A. V. AhoM. 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. AhoM. 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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