2018, 13: ⅴ-ⅹ. doi: 10.3934/jmd.2018v

Roy Adler and the lasting impact of his work

1. 

Department of Mathematical Sciences, Indiana University Purdue University Indianapolis, Indianapolis, IN 46202, USA

2. 

Department of Mathematics, University of British Columbia, Vancouver, BC, V6T 1Z2, Canada

3. 

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem 91904, Israel

A more detailed account of Roy's earlier work and its impact can be found in the expositions [45], [35]

Received  September 10, 2018 Published  December 2018

Citation: Bruce Kitchens, Brian Marcus, Benjamin Weiss. Roy Adler and the lasting impact of his work. Journal of Modern Dynamics, 2018, 13: ⅴ-ⅹ. doi: 10.3934/jmd.2018v
References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.

[2]

R. L. Adler, f -expansions revisited, in Recent Advances in Topological Dynamics, 318, Springer LNM, 1973, 1–5.

[3]

R. L. Adler, A selection of problems in topological and symbolic dynamics, in Lecture Notes in Math., 729, Springer-Verlag, Berlin, Heidelherg, New York, 1979, 8–12.

[4]

R. L. Adler, The torus and the disk, IBM J. Res. & Dev., 13 (1987), 224-234. doi: 10.1147/rd.312.0224.

[5]

R. L. AdlerD. Coppersmith and M. Hassner, Algorithms for sliding block codes – an application of symbolic dynamics to information theory, IEEE Trans. Inform. Theory, 29 (1983), 5-22. doi: 10.1109/TIT.1983.1056597.

[6]

R. L. AdlerJ.-P. DedieuJ. Y. MarguliesM. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numerical Analysis, 22 (2002), 359-390. doi: 10.1093/imanum/22.3.359.

[7]

R. L. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.

[8]

R. L. AdlerW. Goodwyn and B. Weiss, Equivalence of topological shifts, Israel J. Math, 27 (1977), 48-63. doi: 10.1007/BF02761605.

[9]

R. L. Adler, M. Hassner and J. P. Moussouris, Method and Apparatus for Generating a Noiseless Sliding Block Code for a (1, 7) Channel with Rate 2/3, United States Patent 4,413,251. November 1, 1983.

[10]

R. L. AdlerA. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.

[11]

R. L. AdlerB. P. KitchensM. MartensC. P. Tresser and C. W. Wu, The mathematics of halftoning, IBM J. Res. & Dev., 27 (2003), 5-15. doi: 10.1147/rd.471.0005.

[12]

R. L. AdlerB. P. KitchensM. MartensC. PughM. Shub and C. P. Tresser, Convex dynamics and applications, Erg. Theory & Dynam. Sys., 25 (2005), 321-352. doi: 10.1017/S0143385704000537.

[13]

R. L. Adler and B. Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc., 20 (1979), iv+84 pp. doi: 10.1090/memo/0219.

[14]

R. L. AdlerT. NowickiG. SwirszczC. P. Tresser and S. Winograd, Error diffusion on acute simplices: Invariant tiles, Israel J. Math., 221 (2017), 445-469. doi: 10.1007/s11856-017-1550-7.

[15]

R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225. doi: 10.1090/S0002-9939-1965-0193181-8.

[16]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56. doi: 10.1090/S0273-0979-98-00737-X.

[17]

R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A., 57 (1967), 1573-1576. doi: 10.1073/pnas.57.6.1573.

[18]

R. L. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math., 16 (1973), 263-278. doi: 10.1007/BF02756706.

[19]

J. Ashley, G. Jaquette, B. Marcus and P. Seger, Runlength Limited Encoding/Decoding with Robust Resync, US Patent 5969649.

[20]

A. BarberoE. RosnesG. Yang and O. Ytrehus, Near-field passive RFID communication: Channel model and code design, IEEE Trans. Communications, 62 (2014), 1716-1726.

[21]

K. Berg, On the Conjugacy Problem for K-Systems, Ph.D. thesis, University of Minnesota, 1967.

[22]

R. Bowen, Markov partitions for Axiom A diffeomorphisms, American Journal of Mathematics, 92 (1970), 725-747. doi: 10.2307/2373370.

[23]

R. Bowen, Topological entropy for noncompact sets, Transactions of the AMS, 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[24]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, JEMS, 19 (2017), 2739-2782. doi: 10.4171/JEMS/727.

[25]

M. Coornaert, Topological Dimension and Dynamical Systems, Springer, Universitext, 2015. doi: 10.1007/978-3-319-19794-4.

[26]

O. ElishcoT. Meyerovitch and M. Schwartz, On encoding semiconstrained systems, IEEE Trans. Inform. Theory, 64 (2018), 2474-2484. doi: 10.1109/TIT.2017.2771743.

[27]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems, Math USSR Isvestia, 5 (1971), 337-378.

[28]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180. doi: 10.1112/blms/3.2.176.

[29]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proceedings AMS, 23 (1969), 679-688. doi: 10.1090/S0002-9939-1969-0247030-3.

[30]

M. Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math., 126 (2013), 187-201. doi: 10.1007/s10440-013-9813-8.

[31]

K. A. S. Immink, EFMPlus, 8-16 Modulation Code, US Patent 5696505.

[32]

K. A. S. Immink, Codes for Mass Data Storage Systems, 2nd edition, Shannon Foundation Publishers, 2004.

[33]

A. N. Kolmogorov, New metric invariants of transitive dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.

[34]

B. Marcus, Factors and extensions of full shifts, Monats. Math., 82 (1979), 239-247. doi: 10.1007/BF01295238.

[35]

B. Marcus, The impact of Roy Adler's work on symbolic dynamics and applications to data storage, in Symbolic Dynamics and Its Applications (New Haven, CT, 1991), Contemporary Mathematics, 135, Amer. Math. Soc., Providence, RI, 1992, 33–56. doi: 10.1090/conm/135/1185079.

[36]

W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.

[37]

W. Parry, A finitary classification of topological Markov chains and sofic shifts, Bull. LMS, 9 (1977), 86-92. doi: 10.1112/blms/9.1.86.

[38]

Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics University of Chicago Press, Chicago, IL 1997. doi: 10.7208/chicago/9780226662237.001.0001.

[39]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607. doi: 10.2307/2944357.

[40]

R. Roth and P. Siegel, On parity-preserving constrained coding, Procedings of the International Symposium in Information Theory, (2018), 1804-1808.

[41]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, Journal of the AMS, 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.

[42]

C. Shannon, A mathematical theory of communication, Bell. Sys. Tech. J., 27 (1948), 379-423,623-656. doi: 10.1002/j.1538-7305.1948.tb01338.x.

[43]

J. G. Sinai, Markov partitions and $U$-diffeomorphisms, Funkcional. Anal. i Prilozen, 2 (1968), 64-89.

[44]

A. N. Trahtman, The road coloring problem, Israel J. Math., 172 (2009), 51-60. doi: 10.1007/s11856-009-0062-5.

[45]

B. Weiss, On the work of Roy Adler in ergodic theory and dynamical systems, in Symbolic Dynamics and Its Applications (New Haven, CT, 1991), Contemporary Mathematics, 135, Amer. Math. Soc., Providence, RI, 1992, 19–32. doi: 10.1090/conm/135/1185078.

[46]

R. F. Williams, Classification of subshifts of finite type, Annals of Math., 98 (1973), 120–153; Erratum, 99 (1974), 380–381. doi: 10.2307/1970908.

show all references

References:
[1]

J. Aaronson, An Introduction to Infinite Ergodic Theory, American Mathematical Society, Providence, RI, 1997. doi: 10.1090/surv/050.

[2]

R. L. Adler, f -expansions revisited, in Recent Advances in Topological Dynamics, 318, Springer LNM, 1973, 1–5.

[3]

R. L. Adler, A selection of problems in topological and symbolic dynamics, in Lecture Notes in Math., 729, Springer-Verlag, Berlin, Heidelherg, New York, 1979, 8–12.

[4]

R. L. Adler, The torus and the disk, IBM J. Res. & Dev., 13 (1987), 224-234. doi: 10.1147/rd.312.0224.

[5]

R. L. AdlerD. Coppersmith and M. Hassner, Algorithms for sliding block codes – an application of symbolic dynamics to information theory, IEEE Trans. Inform. Theory, 29 (1983), 5-22. doi: 10.1109/TIT.1983.1056597.

[6]

R. L. AdlerJ.-P. DedieuJ. Y. MarguliesM. Martens and M. Shub, Newton's method on Riemannian manifolds and a geometric model for the human spine, IMA J. Numerical Analysis, 22 (2002), 359-390. doi: 10.1093/imanum/22.3.359.

[7]

R. L. Adler and L. Flatto, Geodesic flows, interval maps, and symbolic dynamics, Bull. Amer. Math. Soc., 25 (1991), 229-334. doi: 10.1090/S0273-0979-1991-16076-3.

[8]

R. L. AdlerW. Goodwyn and B. Weiss, Equivalence of topological shifts, Israel J. Math, 27 (1977), 48-63. doi: 10.1007/BF02761605.

[9]

R. L. Adler, M. Hassner and J. P. Moussouris, Method and Apparatus for Generating a Noiseless Sliding Block Code for a (1, 7) Channel with Rate 2/3, United States Patent 4,413,251. November 1, 1983.

[10]

R. L. AdlerA. G. Konheim and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc., 114 (1965), 309-319. doi: 10.1090/S0002-9947-1965-0175106-9.

[11]

R. L. AdlerB. P. KitchensM. MartensC. P. Tresser and C. W. Wu, The mathematics of halftoning, IBM J. Res. & Dev., 27 (2003), 5-15. doi: 10.1147/rd.471.0005.

[12]

R. L. AdlerB. P. KitchensM. MartensC. PughM. Shub and C. P. Tresser, Convex dynamics and applications, Erg. Theory & Dynam. Sys., 25 (2005), 321-352. doi: 10.1017/S0143385704000537.

[13]

R. L. Adler and B. Marcus, Topological entropy and equivalence of dynamical systems, Mem. Amer. Math. Soc., 20 (1979), iv+84 pp. doi: 10.1090/memo/0219.

[14]

R. L. AdlerT. NowickiG. SwirszczC. P. Tresser and S. Winograd, Error diffusion on acute simplices: Invariant tiles, Israel J. Math., 221 (2017), 445-469. doi: 10.1007/s11856-017-1550-7.

[15]

R. L. Adler and R. Palais, Homeomorphic conjugacy of automorphisms on the torus, Proc. Amer. Math. Soc., 16 (1965), 1222-1225. doi: 10.1090/S0002-9939-1965-0193181-8.

[16]

R. L. Adler, Symbolic dynamics and Markov partitions, Bull. Amer. Math. Soc. (N.S.), 35 (1998), 1-56. doi: 10.1090/S0273-0979-98-00737-X.

[17]

R. L. Adler and B. Weiss, Entropy, a complete metric invariant for automorphisms of the torus, Proc. Nat. Acad. Sci. U.S.A., 57 (1967), 1573-1576. doi: 10.1073/pnas.57.6.1573.

[18]

R. L. Adler and B. Weiss, The ergodic infinite measure preserving transformation of Boole, Israel J. Math., 16 (1973), 263-278. doi: 10.1007/BF02756706.

[19]

J. Ashley, G. Jaquette, B. Marcus and P. Seger, Runlength Limited Encoding/Decoding with Robust Resync, US Patent 5969649.

[20]

A. BarberoE. RosnesG. Yang and O. Ytrehus, Near-field passive RFID communication: Channel model and code design, IEEE Trans. Communications, 62 (2014), 1716-1726.

[21]

K. Berg, On the Conjugacy Problem for K-Systems, Ph.D. thesis, University of Minnesota, 1967.

[22]

R. Bowen, Markov partitions for Axiom A diffeomorphisms, American Journal of Mathematics, 92 (1970), 725-747. doi: 10.2307/2373370.

[23]

R. Bowen, Topological entropy for noncompact sets, Transactions of the AMS, 184 (1973), 125-136. doi: 10.1090/S0002-9947-1973-0338317-X.

[24]

M. Boyle and J. Buzzi, The almost Borel structure of surface diffeomorphisms, Markov shifts and their factors, JEMS, 19 (2017), 2739-2782. doi: 10.4171/JEMS/727.

[25]

M. Coornaert, Topological Dimension and Dynamical Systems, Springer, Universitext, 2015. doi: 10.1007/978-3-319-19794-4.

[26]

O. ElishcoT. Meyerovitch and M. Schwartz, On encoding semiconstrained systems, IEEE Trans. Inform. Theory, 64 (2018), 2474-2484. doi: 10.1109/TIT.2017.2771743.

[27]

E. I. Dinaburg, On the relations among various entropy characteristics of dynamical systems, Math USSR Isvestia, 5 (1971), 337-378.

[28]

T. N. T. Goodman, Relating topological entropy and measure entropy, Bull. London Math. Soc., 3 (1971), 176-180. doi: 10.1112/blms/3.2.176.

[29]

L. W. Goodwyn, Topological entropy bounds measure-theoretic entropy, Proceedings AMS, 23 (1969), 679-688. doi: 10.1090/S0002-9939-1969-0247030-3.

[30]

M. Hochman, Isomorphism and embedding of Borel systems on full sets, Acta Appl. Math., 126 (2013), 187-201. doi: 10.1007/s10440-013-9813-8.

[31]

K. A. S. Immink, EFMPlus, 8-16 Modulation Code, US Patent 5696505.

[32]

K. A. S. Immink, Codes for Mass Data Storage Systems, 2nd edition, Shannon Foundation Publishers, 2004.

[33]

A. N. Kolmogorov, New metric invariants of transitive dynamical systems and automorphisms of Lebesgue spaces, Dokl. Akad. Nauk SSSR, 119 (1958), 861-864.

[34]

B. Marcus, Factors and extensions of full shifts, Monats. Math., 82 (1979), 239-247. doi: 10.1007/BF01295238.

[35]

B. Marcus, The impact of Roy Adler's work on symbolic dynamics and applications to data storage, in Symbolic Dynamics and Its Applications (New Haven, CT, 1991), Contemporary Mathematics, 135, Amer. Math. Soc., Providence, RI, 1992, 33–56. doi: 10.1090/conm/135/1185079.

[36]

W. Parry, Intrinsic Markov chains, Trans. AMS, 112 (1964), 55-66. doi: 10.1090/S0002-9947-1964-0161372-1.

[37]

W. Parry, A finitary classification of topological Markov chains and sofic shifts, Bull. LMS, 9 (1977), 86-92. doi: 10.1112/blms/9.1.86.

[38]

Y. B. Pesin, Dimension Theory in Dynamical Systems, Chicago Lectures in Mathematics University of Chicago Press, Chicago, IL 1997. doi: 10.7208/chicago/9780226662237.001.0001.

[39]

M. Ratner, On Raghunathan's measure conjecture, Ann. of Math., 134 (1991), 545-607. doi: 10.2307/2944357.

[40]

R. Roth and P. Siegel, On parity-preserving constrained coding, Procedings of the International Symposium in Information Theory, (2018), 1804-1808.

[41]

O. Sarig, Symbolic dynamics for surface diffeomorphisms with positive entropy, Journal of the AMS, 26 (2013), 341-426. doi: 10.1090/S0894-0347-2012-00758-9.

[42]

C. Shannon, A mathematical theory of communication, Bell. Sys. Tech. J., 27 (1948), 379-423,623-656. doi: 10.1002/j.1538-7305.1948.tb01338.x.

[43]

J. G. Sinai, Markov partitions and $U$-diffeomorphisms, Funkcional. Anal. i Prilozen, 2 (1968), 64-89.

[44]

A. N. Trahtman, The road coloring problem, Israel J. Math., 172 (2009), 51-60. doi: 10.1007/s11856-009-0062-5.

[45]

B. Weiss, On the work of Roy Adler in ergodic theory and dynamical systems, in Symbolic Dynamics and Its Applications (New Haven, CT, 1991), Contemporary Mathematics, 135, Amer. Math. Soc., Providence, RI, 1992, 19–32. doi: 10.1090/conm/135/1185078.

[46]

R. F. Williams, Classification of subshifts of finite type, Annals of Math., 98 (1973), 120–153; Erratum, 99 (1974), 380–381. doi: 10.2307/1970908.

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