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

Ergodic properties of folding maps on spheres

Almut Burchard 1,, , Gregory R. Chambers 2, and  Anne Dranovski 1,

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.

Keywords: Reflection, two-point symmetrization, piecewise isometry, Coxeter group, dense trajectory, transitive reduction, random walks on spheres, invariant measure, unique ergodicity.
Mathematics Subject Classification: 20F55, 37A45, 51F15, 60J15.
Citation: Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[7]

A. Burchard, Rate of convergence of random polarizations, preprint, arXiv: 1108.5500, 2011. Google Scholar

[8]

A. Burchard and M. Fortier, Random polarizations, Adv. Math., 234 (2013), 550-573.  doi: 10.1016/j.aim.2012.10.010.  Google Scholar

[9]

A. Burchard and M. Schmuckenschläger, Comparison theorems for exit times, Geom. Funct. Anal., 11 (2001), 651-692.  doi: 10.1007/PL00001681.  Google Scholar

[10]

H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2), 35 (1934), 588-621.  doi: 10.2307/1968753.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[13]

J. D. Dixon, The probability of generating the symmetric group, Math. Z., 110 (1969), 199-205.  doi: 10.1007/BF01110210.  Google Scholar

[14]

H. G. Eggleston, Convexity Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.   Google Scholar

[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.  Google Scholar

[19]

B. Klartag, Rate of convergence of geometric symmetrizations, Geom. Funct. Anal., 14 (2004), 1322-1338.  doi: 10.1007/s00039-004-0493-4.  Google Scholar

[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.  Google Scholar

[21]

D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. (2), 44 (1943), 454-470.  doi: 10.2307/1968975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

U. Porod, The cut-off phenomenon for random reflections, Ann. Probab., 24 (1996), 74-96.  doi: 10.1214/aop/1042644708.  Google Scholar

[25]

J. S. Rosenthal, Random rotations: Characters and random walks on SO(N), Ann. Probab., 22 (1994), 398-423.  doi: 10.1214/aop/1176988864.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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  Google Scholar

[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.  Google Scholar

[7]

A. Burchard, Rate of convergence of random polarizations, preprint, arXiv: 1108.5500, 2011. Google Scholar

[8]

A. Burchard and M. Fortier, Random polarizations, Adv. Math., 234 (2013), 550-573.  doi: 10.1016/j.aim.2012.10.010.  Google Scholar

[9]

A. Burchard and M. Schmuckenschläger, Comparison theorems for exit times, Geom. Funct. Anal., 11 (2001), 651-692.  doi: 10.1007/PL00001681.  Google Scholar

[10]

H. S. M. Coxeter, Discrete groups generated by reflections, Ann. of Math. (2), 35 (1934), 588-621.  doi: 10.2307/1968753.  Google Scholar

[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 Google Scholar

[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.  Google Scholar

[13]

J. D. Dixon, The probability of generating the symmetric group, Math. Z., 110 (1969), 199-205.  doi: 10.1007/BF01110210.  Google Scholar

[14]

H. G. Eggleston, Convexity Cambridge Tracts in Mathematics and Mathematical Physics, No. 47, Cambridge University Press, New York, 1958.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

A. Goetz, Dynamics of piecewise isometries, Illinois J. Math., 44 (2000), 465-478.   Google Scholar

[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.  Google Scholar

[19]

B. Klartag, Rate of convergence of geometric symmetrizations, Geom. Funct. Anal., 14 (2004), 1322-1338.  doi: 10.1007/s00039-004-0493-4.  Google Scholar

[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.  Google Scholar

[21]

D. Montgomery and H. Samelson, Transformation groups of spheres, Ann. of Math. (2), 44 (1943), 454-470.  doi: 10.2307/1968975.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

U. Porod, The cut-off phenomenon for random reflections, Ann. Probab., 24 (1996), 74-96.  doi: 10.1214/aop/1042644708.  Google Scholar

[25]

J. S. Rosenthal, Random rotations: Characters and random walks on SO(N), Ann. Probab., 22 (1994), 398-423.  doi: 10.1214/aop/1176988864.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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