Book Review
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MathSciNet review:
3363149
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Book Information:
Author:
Tomasz Downarowicz
Title:
Entropy in dynamical systems
Additional book information:
New Mathematical Monographs, Vol. 18,
Cambridge University Press,
Cambridge,
2011,
xii$+$391 pp.,
ISBN 978-0-521-88885-1,
US $101.00
R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 175106, DOI 10.1090/S0002-9947-1965-0175106-9
Mike Boyle, Doris Fiebig, and Ulf Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math. 14 (2002), no. 5, 713–757. MR 1924775, DOI 10.1515/form.2002.031
Lewis Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), no. 1, 217–245. MR 2552252, DOI 10.1090/S0894-0347-09-00637-7
Lewis Bowen, Every countably infinite group is almost Ornstein, Dynamical systems and group actions, Contemp. Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012, pp. 67–78. MR 2931910, DOI 10.1090/conm/567/11234
Jérôme Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100 (1997), 125–161. MR 1469107, DOI 10.1007/BF02773637
David Burguet, $\scr C^2$ surface diffeomorphisms have symbolic extensions, Invent. Math. 186 (2011), no. 1, 191–236. MR 2836054, DOI 10.1007/s00222-011-0317-8
Tomasz Downarowicz and Bartosz Frej, Measure-theoretic and topological entropy of operators on function spaces, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 455–481. MR 2129106, DOI 10.1017/S014338570400032X
T. Downarowicz and Y. Lacroix, The law of series, Ergodic Theory Dynam. Systems 31 (2011), no. 2, 351–367. MR 2776379, DOI 10.1017/S0143385709001217
Tomasz Downarowicz and Alejandro Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math. 176 (2009), no. 3, 617–636. MR 2501298, DOI 10.1007/s00222-008-0172-4
Tomasz Downarowicz and Sheldon Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math. 160 (2005), no. 3, 453–499. MR 2178700, DOI 10.1007/s00222-004-0413-0
Tomasz Downarowicz, Entropy of a symbolic extension of a dynamical system, Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1051–1070. MR 1849601, DOI 10.1017/S014338570100150X
Tomasz Downarowicz, Entropy structure, J. Anal. Math. 96 (2005), 57–116. MR 2177182, DOI 10.1007/BF02787825
Tomasz Downarowicz, A mathematical approach to the law of series, Wiad. Mat. 47 (2011), no. 1, 1–16 (Polish). MR 2961805
Tomasz Downarowicz and Jacek Serafin, Possible entropy functions, Israel J. Math. 135 (2003), 221–250. MR 1997045, DOI 10.1007/BF02776059
T. Downarowicz and J. Serafin, A short proof of the Ornstein theorem, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 587–597. MR 2901361, DOI 10.1017/S0143385711000265
Matthew Foreman, Daniel J. Rudolph, and Benjamin Weiss, The conjugacy problem in ergodic theory, Ann. of Math. (2) 173 (2011), no. 3, 1529–1586. MR 2800720, DOI 10.4007/annals.2011.173.3.7
Matthew Foreman and Benjamin Weiss, An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 277–292. MR 2060477
Anatole Katok, Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn. 1 (2007), no. 4, 545–596. MR 2342699, DOI 10.3934/jmd.2007.1.545
A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861–864 (Russian). MR 0103254
David Kerr and Hanfeng Li, Entropy and the variational principle for actions of sofic groups, Invent. Math. 186 (2011), no. 3, 501–558. MR 2854085, DOI 10.1007/s00222-011-0324-9
Michael Keane and Meir Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2) 109 (1979), no. 2, 397–406. MR 528969, DOI 10.2307/1971117
Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352. MR 257322, DOI 10.1016/0001-8708(70)90029-0
Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348 (1970). MR 274716, DOI 10.1016/0001-8708(70)90008-3
Donald S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math. 10 (1973), 49–62. MR 316682, DOI 10.1016/0001-8708(73)90097-2
Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753, DOI 10.1090/S0273-0979-1980-14702-3
Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005, DOI 10.1007/BF02790325
Jacek Serafin, Finitary codes, a short survey, Dynamics & stochastics, IMS Lecture Notes Monogr. Ser., vol. 48, Inst. Math. Statist., Beachwood, OH, 2006, pp. 262–273. MR 2306207, DOI 10.1214/lnms/1196285827
A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR 223 (1975), no. 2, 300–302 (Russian). MR 0409769
References
- R. L. Adler, A. G. Konheim, and M. H. McAndrew, Topological entropy, Trans. Amer. Math. Soc. 114 (1965), 309–319. MR 0175106 (30 \#5291)
- Mike Boyle, Doris Fiebig, and Ulf Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math. 14 (2002), no. 5, 713–757. MR 1924775 (2003g:37024), DOI 10.1515/form.2002.031
- Lewis Bowen, Measure conjugacy invariants for actions of countable sofic groups, J. Amer. Math. Soc. 23 (2010), no. 1, 217–245. MR 2552252 (2011b:37010), DOI 10.1090/S0894-0347-09-00637-7
- Lewis Bowen, Every countably infinite group is almost Ornstein, Dynamical systems and group actions, Contemp. Math., vol. 567, Amer. Math. Soc., Providence, RI, 2012, pp. 67–78. MR 2931910, DOI 10.1090/conm/567/11234
- Jérôme Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math. 100 (1997), 125–161. MR 1469107 (99g:58071), DOI 10.1007/BF02773637
- David Burguet, $\mathcal {C}^2$ surface diffeomorphisms have symbolic extensions, Invent. Math. 186 (2011), no. 1, 191–236. MR 2836054 (2012k:37049), DOI 10.1007/s00222-011-0317-8
- Tomasz Downarowicz and Bartosz Frej, Measure-theoretic and topological entropy of operators on function spaces, Ergodic Theory Dynam. Systems 25 (2005), no. 2, 455–481. MR 2129106 (2006k:47024), DOI 10.1017/S014338570400032X
- T. Downarowicz and Y. Lacroix, The law of series, Ergodic Theory Dynam. Systems 31 (2011), no. 2, 351–367. MR 2776379 (2012h:37009), DOI 10.1017/S0143385709001217
- Tomasz Downarowicz and Alejandro Maass, Smooth interval maps have symbolic extensions: the antarctic theorem, Invent. Math. 176 (2009), no. 3, 617–636. MR 2501298 (2010d:37038), DOI 10.1007/s00222-008-0172-4
- Tomasz Downarowicz and Sheldon Newhouse, Symbolic extensions and smooth dynamical systems, Invent. Math. 160 (2005), no. 3, 453–499. MR 2178700 (2006j:37021), DOI 10.1007/s00222-004-0413-0
- Tomasz Downarowicz, Entropy of a symbolic extension of a dynamical system, Ergodic Theory Dynam. Systems 21 (2001), no. 4, 1051–1070. MR 1849601 (2002e:37016), DOI 10.1017/S014338570100150X
- Tomasz Downarowicz, Entropy structure, J. Anal. Math. 96 (2005), 57–116. MR 2177182 (2006g:37016), DOI 10.1007/BF02787825
- Tomasz Downarowicz, A mathematical approach to the law of series, Wiad. Mat. 47 (2011), no. 1, 1–16 (Polish). MR 2961805
- Tomasz Downarowicz and Jacek Serafin, Possible entropy functions, Israel J. Math. 135 (2003), 221–250. MR 1997045 (2004c:37013), DOI 10.1007/BF02776059
- T. Downarowicz and J. Serafin, A short proof of the Ornstein theorem, Ergodic Theory Dynam. Systems 32 (2012), no. 2, 587–597. MR 2901361, DOI 10.1017/S0143385711000265
- Matthew Foreman, Daniel J. Rudolph, and Benjamin Weiss, The conjugacy problem in ergodic theory, Ann. of Math. (2) 173 (2011), no. 3, 1529–1586. MR 2800720 (2012k:37006), DOI 10.4007/annals.2011.173.3.7
- Matthew Foreman and Benjamin Weiss, An anti-classification theorem for ergodic measure preserving transformations, J. Eur. Math. Soc. (JEMS) 6 (2004), no. 3, 277–292. MR 2060477 (2005c:37006)
- Anatole Katok, Fifty years of entropy in dynamics: 1958–2007, J. Mod. Dyn. 1 (2007), no. 4, 545–596. MR 2342699 (2008i:37001), DOI 10.3934/jmd.2007.1.545
- A. N. Kolmogorov, A new metric invariant of transient dynamical systems and automorphisms in Lebesgue spaces, Dokl. Akad. Nauk SSSR (N.S.) 119 (1958), 861–864 (Russian). MR 0103254 (21 \#2035a)
- David Kerr and Hanfeng Li, Entropy and the variational principle for actions of sofic groups, Invent. Math. 186 (2011), no. 3, 501–558. MR 2854085, DOI 10.1007/s00222-011-0324-9
- Michael Keane and Meir Smorodinsky, Bernoulli schemes of the same entropy are finitarily isomorphic, Ann. of Math. (2) 109 (1979), no. 2, 397–406. MR 528969 (80f:28024), DOI 10.2307/1971117
- Donald Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352 (1970). MR 0257322 (41 \#1973)
- Donald Ornstein, Two Bernoulli shifts with infinite entropy are isomorphic, Advances in Math. 5 (1970), 339–348 (1970). MR 0274716 (43 \#478a)
- Donald S. Ornstein, An example of a Kolmogorov automorphism that is not a Bernoulli shift, Advances in Math. 10 (1973), 49–62. MR 0316682 (47 \#5229)
- Donald S. Ornstein and Benjamin Weiss, Ergodic theory of amenable group actions. I. The Rohlin lemma, Bull. Amer. Math. Soc. (N.S.) 2 (1980), no. 1, 161–164. MR 551753 (80j:28031), DOI 10.1090/S0273-0979-1980-14702-3
- Donald S. Ornstein and Benjamin Weiss, Entropy and isomorphism theorems for actions of amenable groups, J. Analyse Math. 48 (1987), 1–141. MR 910005 (88j:28014), DOI 10.1007/BF02790325
- Jacek Serafin, Finitary codes, a short survey, Dynamics & stochastics, IMS Lecture Notes Monogr. Ser., vol. 48, Inst. Math. Statist., Beachwood, OH, 2006, pp. 262–273. MR 2306207 (2008a:37009), DOI 10.1214/lnms/1196285827
- A. M. Stepin, Bernoulli shifts on groups, Dokl. Akad. Nauk SSSR 223 (1975), no. 2, 300–302 (Russian). MR 0409769 (53 \#13521)
Review Information:
Reviewer:
L. Bowen
Affiliation:
Department of Mathematics University of Texas at Austin
Email:
lpbowen@math.utexas.edu
Journal:
Bull. Amer. Math. Soc.
51 (2014), 669-674
DOI:
https://doi.org/10.1090/S0273-0979-2014-01445-4
Published electronically:
May 13, 2014
Review copyright:
© Copyright 2014
American Mathematical Society
