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. Google Scholar

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[44]

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

[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. Google Scholar

[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. Google Scholar

show all references

References:
[1]

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

[2]

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

[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. Google Scholar

[4]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

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

[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.Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

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

[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. Google Scholar

[21]

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

[22]

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

[23]

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

[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. Google Scholar

[25]

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

[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. Google Scholar

[27]

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

[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. Google Scholar

[29]

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

[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. Google Scholar

[31]

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

[32]

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

[33]

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

[34]

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

[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. Google Scholar

[36]

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

[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. Google Scholar

[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. Google Scholar

[39]

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

[40]

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

[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. Google Scholar

[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. Google Scholar

[43]

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

[44]

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

[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. Google Scholar

[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. Google Scholar

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