American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

[1]

Jean-Pierre Conze, Y. Guivarc'h. Ergodicity of group actions and spectral gap, applications to random walks and Markov shifts. Discrete and Continuous Dynamical Systems, 2013, 33 (9) : 4239-4269. doi: 10.3934/dcds.2013.33.4239
[2]

Wen-Chiao Cheng. Two-point pre-image entropy. Discrete and Continuous Dynamical Systems, 2007, 17 (1) : 107-119. doi: 10.3934/dcds.2007.17.107
[3]

Feliz Minhós, A. I. Santos. Higher order two-point boundary value problems with asymmetric growth. Discrete and Continuous Dynamical Systems - S, 2008, 1 (1) : 127-137. doi: 10.3934/dcdss.2008.1.127
[4]

Wenming Zou. Multiple solutions results for two-point boundary value problem with resonance. Discrete and Continuous Dynamical Systems, 1998, 4 (3) : 485-496. doi: 10.3934/dcds.1998.4.485
[5]

Jon Chaika, Rodrigo Treviño. Logarithmic laws and unique ergodicity. Journal of Modern Dynamics, 2017, 11: 563-588. doi: 10.3934/jmd.2017022
[6]

Xiankun Ren. Periodic measures are dense in invariant measures for residually finite amenable group actions with specification. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1657-1667. doi: 10.3934/dcds.2018068
[7]

Kesong Yan, Qian Liu, Fanping Zeng. Classification of transitive group actions. Discrete and Continuous Dynamical Systems, 2021, 41 (12) : 5579-5607. doi: 10.3934/dcds.2021089
[8]

Charles Pugh, Michael Shub, Alexander Starkov. Unique ergodicity, stable ergodicity, and the Mautner phenomenon for diffeomorphisms. Discrete and Continuous Dynamical Systems, 2006, 14 (4) : 845-855. doi: 10.3934/dcds.2006.14.845
[9]

Jerry L. Bona, Hongqiu Chen, Shu-Ming Sun, Bing-Yu Zhang. Comparison of quarter-plane and two-point boundary value problems: The KdV-equation. Discrete and Continuous Dynamical Systems - B, 2007, 7 (3) : 465-495. doi: 10.3934/dcdsb.2007.7.465
[10]

Xiao-Yu Zhang, Qing Fang. A sixth order numerical method for a class of nonlinear two-point boundary value problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 31-43. doi: 10.3934/naco.2012.2.31
[11]

Jerry Bona, Hongqiu Chen, Shu Ming Sun, B.-Y. Zhang. Comparison of quarter-plane and two-point boundary value problems: the BBM-equation. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 921-940. doi: 10.3934/dcds.2005.13.921
[12]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete and Continuous Dynamical Systems, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[13]

Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence, Part 1: Isotropic turbulence. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 155-168. doi: 10.3934/dcdss.2011.4.155
[14]

Marcel Lesieur. Two-point closure based large-eddy simulations in turbulence. Part 2: Inhomogeneous cases. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 227-241. doi: 10.3934/dcds.2010.28.227
[15]

Henk Bruin, Gregory Clack. Inducing and unique ergodicity of double rotations. Discrete and Continuous Dynamical Systems, 2012, 32 (12) : 4133-4147. doi: 10.3934/dcds.2012.32.4133
[16]

Hans Koch. A renormalization group fixed point associated with the breakup of golden invariant tori. Discrete and Continuous Dynamical Systems, 2004, 11 (4) : 881-909. doi: 10.3934/dcds.2004.11.881
[17]

George W. Patrick, Tyler Helmuth. Two rolling disks or spheres. Discrete and Continuous Dynamical Systems - S, 2010, 3 (1) : 129-140. doi: 10.3934/dcdss.2010.3.129
[18]

Terasan Niyomsataya, Ali Miri, Monica Nevins. Decoding affine reflection group codes with trellises. Advances in Mathematics of Communications, 2012, 6 (4) : 385-400. doi: 10.3934/amc.2012.6.385
[19]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185
[20]

Michał Misiurewicz, Peter Raith. Strict inequalities for the entropy of transitive piecewise monotone maps. Discrete and Continuous Dynamical Systems, 2005, 13 (2) : 451-468. doi: 10.3934/dcds.2005.13.451

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (116)
  • HTML views (69)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy