2018, 13: 115-145. doi: 10.3934/jmd.2018014

The mapping class group of a shift of finite type

1. 

Department of Mathematics, University of Maryland, College Park, MD 20742-4015, USA

2. 

Algebra and Applications Research Unit, Department of Mathematics and Statistics, Prince of Songkla University, Songkhla, Thailand 90110

Dedicated to Roy Adler, in memory of his insight, humor, and kindness

Received  April 27, 2017 Revised  August 18, 2017 Published  December 2018

Let $(X_A,σ_{A})$ be a nontrivial irreducible shift of finite type (SFT), with $\mathscr{M}_A$ denoting its mapping class group: the group of flow equivalences of its mapping torus $\mathsf{S} X_A$, (i.e., self homeomorphisms of $\mathsf{S} X_A$ which respect the direction of the suspension flow) modulo the subgroup of flow equivalences of $\mathsf{S} X_A$ isotopic to the identity. We develop and apply machinery (flow codes, cohomology constraints) and provide context for the study of $\mathscr M_A$, and prove results including the following. $\mathscr{M}_A$ acts faithfully and $n$-transitively (for every $n$ in $\mathbb{N}$) by permutations on the set of circles of $\mathsf{S} X_A$. The center of $\mathscr{M}_A$ is trivial. The outer automorphism group of $\mathscr{M}_A$ is nontrivial. In many cases, $\text{Aut}(σ_{A})$ admits a nonspatial automorphism. For every SFT $(X_B,σ_B)$ flow equivalent to $(X_A,σ_{A})$, $\mathscr{M}_A$ contains embedded copies of ${\rm{Aut}}({\sigma _B})/\left\langle {{\sigma _B}} \right\rangle $, induced by return maps to invariant cross sections; but, elements of $\mathscr M_A$ not arising from flow equivalences with invariant cross sections are abundant. $\mathscr{M}_A$ is countable and has solvable word problem. $\mathscr{M}_A$ is not residually finite. Conjugacy classes of many (possibly all) involutions in $\mathscr M_A$ can be classified by the $G$-flow equivalence classes of associated $G$-SFTs, for $G = \mathbb{Z}/2\mathbb{Z}$. There are many open questions.

Citation: Mike Boyle, Sompong Chuysurichay. The mapping class group of a shift of finite type. Journal of Modern Dynamics, 2018, 13: 115-145. doi: 10.3934/jmd.2018014
References:
[1]

F. Blanchard and G. Hansel, Systèmes codés, Theoret. Comput. Sci., 44 (1986), 17-49.   Google Scholar

[2]

R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2), 106 (1977), 73-92.  doi: 10.2307/1971159.  Google Scholar

[3]

M. Boyle, Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math., 204 (2002), 273-317.  doi: 10.2140/pjm.2002.204.273.  Google Scholar

[4]

M. Boyle, Positive K-theory and symbolic dynamics, in Dynamics and Randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems, 7, Kluwer Acad. Publ., 2002, 31-52.  Google Scholar

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M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of G-SFTs, arXiv: 1512.05238, 2015. Google Scholar

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M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of sofic shifts, Israel J. Math., to appear; arXiv: 1511.03481, 2015. Google Scholar

[7]

M. BoyleT. Carlsen and S. Eilers, Flow equivalence and isotopy for subshifts, Dyn. Syst., 32 (2017), 305-325.  doi: 10.1080/14689367.2016.1207753.  Google Scholar

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M. Boyle and U.-R. Fiebig, The action of inert finite-order automorphisms on finite subsystems of the shift, Ergodic Theory Dynam. Systems, 11 (1991), 413-425.   Google Scholar

[9]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.  doi: 10.1007/BF02761039.  Google Scholar

[10]

M. Boyle and D. Huang, Poset block equivalence of integral matrices, Trans. Amer. Math. Soc., 355 (2003), 3861-3886.  doi: 10.1090/S0002-9947-03-02947-7.  Google Scholar

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M. Boyle and W. Krieger, Almost Markov and shift equivalent sofic systems, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 33-93.  Google Scholar

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M. Boyle and M. C. Sullivan, Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings, Proc. London Math. Soc. (3), 91 (2005), 184-214.  doi: 10.1112/S0024611505015285.  Google Scholar

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M. Boyle and J. B. Wagoner, Positive algebraic K-theory and shifts of finite type, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 45-66.  Google Scholar

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V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov.  Google Scholar

[18]

S. Chuysurichay, Positive Rational Strong Shift Equivalence and the Mapping Class Group of a Shift of Finite Type, Thesis (Ph.D.)-University of Maryland, College Park, 2011, 95 pp, ProQuest LLC, Ann Arbor, MI.  Google Scholar

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V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147–161. doi: 10.3934/jmd.2018015.  Google Scholar

[21]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.  doi: 10.3934/jmd.2016.10.483.  Google Scholar

[22]

S. DonosoF. DurandA. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95.  doi: 10.1017/etds.2015.70.  Google Scholar

[23]

S. EilersG. RestorffE. Ruiz and A. P. W. Sorensen, The complete classification of unital graph C*-algebras: Geometric and strong, Canad. J. Math., 70 (2018), 294-353.  doi: 10.4153/CJM-2017-016-7.  Google Scholar

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U.-R. Fiebig, Periodic points and finite group actions on shifts of finite type, Ergodic Theory Dynam. Systems, 13 (1993), 485-514.   Google Scholar

[26]

R. J. Fokkink, The Structure of Trajectories, Thesis (Ph.D.)-Technische Universiteit Delft (The Netherlands), 1991, 112pp, ProQuest LLC, Ann Arbor, MI.  Google Scholar

[27]

J. Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems, 4 (1984), 53-66.   Google Scholar

[28]

T. GiordanoI. F. Putnam and C. F. Skau, Topological orbit equivalence and $ C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111.   Google Scholar

[29]

T. GiordanoI. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.  Google Scholar

[30]

R. I. Grigorchuk and K. S. Medinets, On the algebraic properties of topological full groups, Mat. Sb., 205 (2014), 87-108.   Google Scholar

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G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.  Google Scholar

[32]

M. Hochman, On the automorphism groups of multidimensional shifts of finite type, Ergodic Theory Dynam. Systems, 30 (2010), 809-840.  doi: 10.1017/S0143385709000248.  Google Scholar

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K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787.  doi: 10.4007/annals.2013.178.2.7.  Google Scholar

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K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230 (1991).   Google Scholar

[35]

K. H. Kim and F. W. Roush, Free $ Z_p$ actions on subshifts, Pure Math. Appl., 8 (1997), 293-322.   Google Scholar

[36]

K. H. KimF. W. Roush and J. B. Wagoner, Automorphisms of the dimension group and gyration numbers, J. Amer. Math. Soc., 5 (1992), 191-212.  doi: 10.1090/S0894-0347-1992-1124983-3.  Google Scholar

[37]

K. H. Kim, F. W. Roush and S. G. Williams, Duality and its consequences for ordered cohomology of finite type subshifts, in Combinatorial & Computational Mathematics (Pohang, 2000), World Sci. Publ., River Edge, NJ, 2001, 243-265.  Google Scholar

[38]

Y.-O. KimJ. Lee and K. K. Park, A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.  doi: 10.2140/pjm.2003.209.289.  Google Scholar

[39]

D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.   Google Scholar

[40]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  Google Scholar

[41]

N. Long, Fixed point shifts of inert involutions, Discrete Contin. Dyn. Syst., 25 (2009), 1297-1317.  doi: 10.3934/dcds.2009.25.1297.  Google Scholar

[42]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math., 54 (2014), 863-877.  doi: 10.1215/21562261-2801849.  Google Scholar

[43]

H. Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.   Google Scholar

[44]

M. Nasu, Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 564-607.  Google Scholar

[45]

W. Parry and D. Sullivan, A topological invariant of flows on 1-dimensional spaces, Topology, 14 (1975), 297-299.  doi: 10.1016/0040-9383(75)90012-9.  Google Scholar

[46]

V. G. Pestov, Hyperlinear and sofic groups: A brief guide, Bull. Symbolic Logic, 14 (2008), 449-480.  doi: 10.2178/bsl/1231081461.  Google Scholar

[47]

G. Restorff, Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math., 598 (2006), 185-210.   Google Scholar

[48]

M. Rørdam, Classification of Cuntz-Krieger algebras, K-Theory, 9 (1995), 31-58.  doi: 10.1007/BF00965458.  Google Scholar

[49]

J. Patrick Ryan, The shift and commutivity. Ⅱ, Math. Systems Theory, 8 (1974/75), 249-250.  doi: 10.1007/BF01762673.  Google Scholar

[50]

V. Salo, Groups and monoids of cellular automata, in Cellular Automata and Discrete Complex Systems, Lecture Notes in Comput. Sci., 9099, Springer, Heidelberg, 2015, 17-45.  Google Scholar

[51]

V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory Dynam. Systems, 35 (2015), 2292-2310.  doi: 10.1017/etds.2014.29.  Google Scholar

[52]

M. Schraudner, On the algebraic properties of the automorphism groups of countable-state Markov shifts, Ergodic Theory Dynam. Systems, 26 (2006), 551-583.  doi: 10.1017/S0143385705000507.  Google Scholar

[53]

S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284.  doi: 10.2307/1969999.  Google Scholar

[54]

J. B. Wagoner, Strong shift equivalence theory and the shift equivalence problem, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 271-296.  doi: 10.1090/S0273-0979-99-00798-3.  Google Scholar

[55]

J. B. Wagoner, Strong shift equivalence and $ K_2$ of the dual numbers, J. Reine Angew. Math., 521 (2000), 119-160, with an appendix by K. H. Kim and F. W. Roush.  Google Scholar

[56]

B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350-359.   Google Scholar

show all references

References:
[1]

F. Blanchard and G. Hansel, Systèmes codés, Theoret. Comput. Sci., 44 (1986), 17-49.   Google Scholar

[2]

R. Bowen and J. Franks, Homology for zero-dimensional nonwandering sets, Ann. of Math. (2), 106 (1977), 73-92.  doi: 10.2307/1971159.  Google Scholar

[3]

M. Boyle, Flow equivalence of shifts of finite type via positive factorizations, Pacific J. Math., 204 (2002), 273-317.  doi: 10.2140/pjm.2002.204.273.  Google Scholar

[4]

M. Boyle, Positive K-theory and symbolic dynamics, in Dynamics and Randomness (Santiago, 2000), Nonlinear Phenom. Complex Systems, 7, Kluwer Acad. Publ., 2002, 31-52.  Google Scholar

[5]

M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of G-SFTs, arXiv: 1512.05238, 2015. Google Scholar

[6]

M. Boyle, T. Carlsen and S. Eilers, Flow equivalence of sofic shifts, Israel J. Math., to appear; arXiv: 1511.03481, 2015. Google Scholar

[7]

M. BoyleT. Carlsen and S. Eilers, Flow equivalence and isotopy for subshifts, Dyn. Syst., 32 (2017), 305-325.  doi: 10.1080/14689367.2016.1207753.  Google Scholar

[8]

M. Boyle and U.-R. Fiebig, The action of inert finite-order automorphisms on finite subsystems of the shift, Ergodic Theory Dynam. Systems, 11 (1991), 413-425.   Google Scholar

[9]

M. Boyle and D. Handelman, Orbit equivalence, flow equivalence and ordered cohomology, Israel J. Math., 95 (1996), 169-210.  doi: 10.1007/BF02761039.  Google Scholar

[10]

M. Boyle and D. Huang, Poset block equivalence of integral matrices, Trans. Amer. Math. Soc., 355 (2003), 3861-3886.  doi: 10.1090/S0002-9947-03-02947-7.  Google Scholar

[11]

M. Boyle and W. Krieger, Periodic points and automorphisms of the shift, Trans. Amer. Math. Soc., 302 (1987), 125-149.  doi: 10.1090/S0002-9947-1987-0887501-5.  Google Scholar

[12]

M. Boyle and W. Krieger, Almost Markov and shift equivalent sofic systems, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 33-93.  Google Scholar

[13]

M. BoyleD. Lind and D. Rudolph, The automorphism group of a shift of finite type, Trans. Amer. Math. Soc., 306 (1988), 71-114.  doi: 10.1090/S0002-9947-1988-0927684-2.  Google Scholar

[14]

M. Boyle and S. Schmieding, Finite group extensions of shifts of finite type: K-theory, Parry and Livšic, Ergodic Theory Dynam. Systems, 37 (2017), 1026-1059.  doi: 10.1017/etds.2015.87.  Google Scholar

[15]

M. Boyle and M. C. Sullivan, Equivariant flow equivalence for shifts of finite type, by matrix equivalence over group rings, Proc. London Math. Soc. (3), 91 (2005), 184-214.  doi: 10.1112/S0024611505015285.  Google Scholar

[16]

M. Boyle and J. B. Wagoner, Positive algebraic K-theory and shifts of finite type, in Modern Dynamical Systems and Applications, Cambridge Univ. Press, Cambridge, 2004, 45-66.  Google Scholar

[17]

V. Capraro and M. Lupini, Introduction to Sofic and Hyperlinear Groups and Connes' Embedding Conjecture, Lecture Notes in Mathematics, 2136, Springer, Cham, 2015. With an appendix by Vladimir Pestov.  Google Scholar

[18]

S. Chuysurichay, Positive Rational Strong Shift Equivalence and the Mapping Class Group of a Shift of Finite Type, Thesis (Ph.D.)-University of Maryland, College Park, 2011, 95 pp, ProQuest LLC, Ann Arbor, MI.  Google Scholar

[19]

E. M. Coven, A. Quas and R. Yassawi, Computing automorphism groups of shifts using atypical equivalence classes, Discrete Anal., (2016), Paper No. 3, 28pp.  Google Scholar

[20]

V. Cyr, J. Franks, B. Kra and S. Petite, Distortion and the automorphism group of a shift, J. Mod. Dyn., 13 (2018), 147–161. doi: 10.3934/jmd.2018015.  Google Scholar

[21]

V. Cyr and B. Kra, The automorphism group of a minimal shift of stretched exponential growth, J. Mod. Dyn., 10 (2016), 483-495.  doi: 10.3934/jmd.2016.10.483.  Google Scholar

[22]

S. DonosoF. DurandA. Maass and S. Petite, On automorphism groups of low complexity subshifts, Ergodic Theory Dynam. Systems, 36 (2016), 64-95.  doi: 10.1017/etds.2015.70.  Google Scholar

[23]

S. EilersG. RestorffE. Ruiz and A. P. W. Sorensen, The complete classification of unital graph C*-algebras: Geometric and strong, Canad. J. Math., 70 (2018), 294-353.  doi: 10.4153/CJM-2017-016-7.  Google Scholar

[24]

B. Farb and D. Margalit, A Primer on Mapping Class Groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.  Google Scholar

[25]

U.-R. Fiebig, Periodic points and finite group actions on shifts of finite type, Ergodic Theory Dynam. Systems, 13 (1993), 485-514.   Google Scholar

[26]

R. J. Fokkink, The Structure of Trajectories, Thesis (Ph.D.)-Technische Universiteit Delft (The Netherlands), 1991, 112pp, ProQuest LLC, Ann Arbor, MI.  Google Scholar

[27]

J. Franks, Flow equivalence of subshifts of finite type, Ergodic Theory Dynam. Systems, 4 (1984), 53-66.   Google Scholar

[28]

T. GiordanoI. F. Putnam and C. F. Skau, Topological orbit equivalence and $ C^*$-crossed products, J. Reine Angew. Math., 469 (1995), 51-111.   Google Scholar

[29]

T. GiordanoI. F. Putnam and C. F. Skau, Full groups of Cantor minimal systems, Israel J. Math., 111 (1999), 285-320.  doi: 10.1007/BF02810689.  Google Scholar

[30]

R. I. Grigorchuk and K. S. Medinets, On the algebraic properties of topological full groups, Mat. Sb., 205 (2014), 87-108.   Google Scholar

[31]

G. A. Hedlund, Endomorphisms and automorphisms of the shift dynamical system, Math. Systems Theory, 3 (1969), 320-375. doi: 10.1007/BF01691062.  Google Scholar

[32]

M. Hochman, On the automorphism groups of multidimensional shifts of finite type, Ergodic Theory Dynam. Systems, 30 (2010), 809-840.  doi: 10.1017/S0143385709000248.  Google Scholar

[33]

K. Juschenko and N. Monod, Cantor systems, piecewise translations and simple amenable groups, Ann. of Math. (2), 178 (2013), 775-787.  doi: 10.4007/annals.2013.178.2.7.  Google Scholar

[34]

K. H. Kim and F. W. Roush, On the automorphism groups of subshifts, Pure Math. Appl. Ser. B, 1 (1990), 203-230 (1991).   Google Scholar

[35]

K. H. Kim and F. W. Roush, Free $ Z_p$ actions on subshifts, Pure Math. Appl., 8 (1997), 293-322.   Google Scholar

[36]

K. H. KimF. W. Roush and J. B. Wagoner, Automorphisms of the dimension group and gyration numbers, J. Amer. Math. Soc., 5 (1992), 191-212.  doi: 10.1090/S0894-0347-1992-1124983-3.  Google Scholar

[37]

K. H. Kim, F. W. Roush and S. G. Williams, Duality and its consequences for ordered cohomology of finite type subshifts, in Combinatorial & Computational Mathematics (Pohang, 2000), World Sci. Publ., River Edge, NJ, 2001, 243-265.  Google Scholar

[38]

Y.-O. KimJ. Lee and K. K. Park, A zeta function for flip systems, Pacific J. Math., 209 (2003), 289-301.  doi: 10.2140/pjm.2003.209.289.  Google Scholar

[39]

D. A. Lind, The entropies of topological Markov shifts and a related class of algebraic integers, Ergodic Theory Dynam. Systems, 4 (1984), 283-300.   Google Scholar

[40]

D. Lind and B. Marcus, An Introduction to Symbolic Dynamics and Coding, Cambridge University Press, Cambridge, 1995.  Google Scholar

[41]

N. Long, Fixed point shifts of inert involutions, Discrete Contin. Dyn. Syst., 25 (2009), 1297-1317.  doi: 10.3934/dcds.2009.25.1297.  Google Scholar

[42]

K. Matsumoto and H. Matui, Continuous orbit equivalence of topological Markov shifts and Cuntz-Krieger algebras, Kyoto J. Math., 54 (2014), 863-877.  doi: 10.1215/21562261-2801849.  Google Scholar

[43]

H. Matui, Topological full groups of one-sided shifts of finite type, J. Reine Angew. Math., 705 (2015), 35-84.   Google Scholar

[44]

M. Nasu, Topological conjugacy for sofic systems and extensions of automorphisms of finite subsystems of topological Markov shifts, in Dynamical Systems (College Park, MD, 1986-87), Lecture Notes in Math., 1342, Springer, Berlin, 1988, 564-607.  Google Scholar

[45]

W. Parry and D. Sullivan, A topological invariant of flows on 1-dimensional spaces, Topology, 14 (1975), 297-299.  doi: 10.1016/0040-9383(75)90012-9.  Google Scholar

[46]

V. G. Pestov, Hyperlinear and sofic groups: A brief guide, Bull. Symbolic Logic, 14 (2008), 449-480.  doi: 10.2178/bsl/1231081461.  Google Scholar

[47]

G. Restorff, Classification of Cuntz-Krieger algebras up to stable isomorphism, J. Reine Angew. Math., 598 (2006), 185-210.   Google Scholar

[48]

M. Rørdam, Classification of Cuntz-Krieger algebras, K-Theory, 9 (1995), 31-58.  doi: 10.1007/BF00965458.  Google Scholar

[49]

J. Patrick Ryan, The shift and commutivity. Ⅱ, Math. Systems Theory, 8 (1974/75), 249-250.  doi: 10.1007/BF01762673.  Google Scholar

[50]

V. Salo, Groups and monoids of cellular automata, in Cellular Automata and Discrete Complex Systems, Lecture Notes in Comput. Sci., 9099, Springer, Heidelberg, 2015, 17-45.  Google Scholar

[51]

V. Salo and I. Törmä, Block maps between primitive uniform and Pisot substitutions, Ergodic Theory Dynam. Systems, 35 (2015), 2292-2310.  doi: 10.1017/etds.2014.29.  Google Scholar

[52]

M. Schraudner, On the algebraic properties of the automorphism groups of countable-state Markov shifts, Ergodic Theory Dynam. Systems, 26 (2006), 551-583.  doi: 10.1017/S0143385705000507.  Google Scholar

[53]

S. Schwartzman, Asymptotic cycles, Ann. of Math. (2), 66 (1957), 270-284.  doi: 10.2307/1969999.  Google Scholar

[54]

J. B. Wagoner, Strong shift equivalence theory and the shift equivalence problem, Bull. Amer. Math. Soc. (N.S.), 36 (1999), 271-296.  doi: 10.1090/S0273-0979-99-00798-3.  Google Scholar

[55]

J. B. Wagoner, Strong shift equivalence and $ K_2$ of the dual numbers, J. Reine Angew. Math., 521 (2000), 119-160, with an appendix by K. H. Kim and F. W. Roush.  Google Scholar

[56]

B. Weiss, Sofic groups and dynamical systems, Ergodic Theory and Harmonic Analysis (Mumbai, 1999), Sankhyā Ser. A, 62 (2000), 350-359.   Google Scholar

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