Inverting the Furstenberg correspondence

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October 2012, 32(10): 3421-3431. doi: 10.3934/dcds.2012.32.3421

Inverting the Furstenberg correspondence

Jeremy Avigad ^1,

1.	Departments of Philosophy and Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, United States

Received May 2011 Revised February 2012 Published May 2012

Abstract
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Given a sequence of sets $A_n \subseteq \{0,\ldots,n-1\}$, the Furstenberg correspondence principle provides a shift-invariant measure on $2^N$ that encodes combinatorial information about infinitely many of the $A_n$'s. Here it is shown that this process can be inverted, so that for any such measure, ergodic or not, there are finite sets whose combinatorial properties approximate it arbitarily well. The finite approximations are obtained from the measure by an explicit construction, with an explicit upper bound on how large $n$ has to be to yield a sufficiently good approximation.
We draw conclusions for computable measure theory, and show, in particular, that given any computable shift-invariant measure on $2^N$, there is a computable element of $2^N$ that is generic for the measure. We also consider a generalization of the correspondence principle to countable discrete amenable groups, and once again provide an effective inverse.

Keywords: Furstenberg correspondence, computable measure theory, generic points..

Mathematics Subject Classification: Primary: 37A45; Secondary: 03F6.

Citation: Jeremy Avigad. Inverting the Furstenberg correspondence. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3421-3431. doi: 10.3934/dcds.2012.32.3421

References:

[1]	Jeremy Avigad, Uncomputably noisy ergodic limits, Notre Dame Journal of Formal Logic, to appear.
[2]	Jeremy Avigad, Philipp Gerhardy and Henry Towsner, Local stability of ergodic averages, Trans. Amer. Math. Soc., 362 (2010), 261-288. doi: 10.1090/S0002-9947-09-04814-4.
[3]	Vitaly Bergelson, Ergodic theory and Diophantine problems, in "Topics in Symbolic Dynamics and Applications" (eds. F. Blanchard, A. Maass and A. Nogueira), London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, (2000), 167-205.
[4]	Vitaly Bergelson and Hillel Furstenberg, WM groups and Ramsey theory, Topology Appl., 156 (2009), 2572-2580. doi: 10.1016/j.topol.2009.04.007.
[5]	Vitaly Bergelson, Hillel Furstenberg and Benjamin Weiss, Piecewise-Bohr sets of integers and combinatorial number theory, in "Topics in Discrete Mathematics" (eds. M. Klazar, et al.), Algorithms Combin., 26, Springer, Berlin, (2006), 13-37.
[6]	Vitaly Bergelson, Alexander Leibman and Emmanuel Lesigne, Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory, J. Anal. Math., 103 (2007), 47-92. doi: 10.1007/s11854-008-0002-z.
[7]	Vitaly Bergelson and Randall McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in "Topological Dynamics and Applications" (eds. M. G. Nerurkar, D. P. Dokken and D. B. Ellis) (Minneapolis, MN, 1995), Contemp. Math., 215, Amer. Math. Soc., Providence, RI, (1998), 205-222.
[8]	Vasco Brattka, Peter Hertling and Klaus Weihrauch, A tutorial on computable analysis, in "New Computational Paradigms: Changing Conceptions of What is Computable" (eds. S. Barry Cooper, Benedikt Löwe and Andrea Sorbi), Springer, New York, (2008), 425-491.
[9]	C. M. Colebrook, The Hausdorff dimension of certain sets of nonnormal numbers, Michigan Math. J., 17 (1970), 103-116. doi: 10.1307/mmj/1029000420.
[10]	H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256. doi: 10.1007/BF02813304.
[11]	H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.
[12]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties, Theor. Comput. Sci., 410 (2009), 2207-2222. doi: 10.1016/j.tcs.2009.02.010.
[13]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems, in "Proceedings of Computability and Complexity in Analysis (CCA)'' (eds. Xizhong Zheng and Ning Zhong), Electronic Proceedings in Theoretical Computer Science, arXiv:1006.0392.
[14]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, Dynamical systems, simulation, abstract computation, 2011, arXiv:1101.0833.
[15]	Andrzej Grzegorczyk, On the definitions of computable real continuous functions, Fundamenta Mathematicae, 44 (1957), 61-71.
[16]	Bernard Host and Bryna Kra, Uniformity seminorms on ℓ$^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276. doi: 10.1007/s11854-009-0024-1.
[17]	Mathieu Hoyrup, Randomness and the ergodic decomposition, in "Computability in Europe (CiE) 2011" (eds. Benedikt Löwe, et al.), Springer, Berlin, (2011), 122-131.
[18]	Mathieu Hoyrup and Cristóbal Rojas, Computability of probability measures and Martin-Löf randomness over metric spaces, Inform. and Comput., 207 (2009), 830-847. doi: 10.1016/j.ic.2008.12.009.
[19]	Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.
[20]	Alexander P. Kreuzer, The cohesive principle and the Bolzano-Weierstraß principle, Math. Log. Q., 57 (2011), 292-298. doi: 10.1002/malq.201010008.
[21]	Pavol Safarik and Ulrich Kohlenbach, On the computational content of the Bolzano-Weierstraß principle, Math. Log. Q., 56 (2010), 508-532. doi: 10.1002/malq.200910106.
[22]	Karl Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.
[23]	Karl Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X.
[24]	Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.
[25]	Terence Tao, "Poincaré's Legacies, Pages from Year Two of a Mathematical Blog," Part I, Amer. Math. Soc., Providence, RI, 2009.
[26]	Henry Townser, A general correspondence between averages and integrals, arXiv:0804.2773.
[27]	V. V. V'yugin, Ergodic convergence in probability, and an ergodic theorem for individual random sequences, Teor. Veroyatnost. i Primenen., 42 (1997), 35-50; translation in Theory Probab. Appl. 42 (1997), 39-50.
[28]	V. V. V'yugin, Ergodic theorems for individual random sequences, Theoret. Comput. Sci., 207 (1998), 343-361. doi: 10.1016/S0304-3975(98)00072-3.
[29]	Klaus Weihrauch, Computability on the probability measures on the Borel sets of the unit interval, Theoret. Comput. Sci., 219 (1999), 421-437. doi: 10.1016/S0304-3975(98)00298-9.

show all references

References:

[1]	Jeremy Avigad, Uncomputably noisy ergodic limits, Notre Dame Journal of Formal Logic, to appear.
[2]	Jeremy Avigad, Philipp Gerhardy and Henry Towsner, Local stability of ergodic averages, Trans. Amer. Math. Soc., 362 (2010), 261-288. doi: 10.1090/S0002-9947-09-04814-4.
[3]	Vitaly Bergelson, Ergodic theory and Diophantine problems, in "Topics in Symbolic Dynamics and Applications" (eds. F. Blanchard, A. Maass and A. Nogueira), London Math. Soc. Lecture Note Ser., 279, Cambridge Univ. Press, Cambridge, (2000), 167-205.
[4]	Vitaly Bergelson and Hillel Furstenberg, WM groups and Ramsey theory, Topology Appl., 156 (2009), 2572-2580. doi: 10.1016/j.topol.2009.04.007.
[5]	Vitaly Bergelson, Hillel Furstenberg and Benjamin Weiss, Piecewise-Bohr sets of integers and combinatorial number theory, in "Topics in Discrete Mathematics" (eds. M. Klazar, et al.), Algorithms Combin., 26, Springer, Berlin, (2006), 13-37.
[6]	Vitaly Bergelson, Alexander Leibman and Emmanuel Lesigne, Complexities of finite families of polynomials, Weyl systems, and constructions in combinatorial number theory, J. Anal. Math., 103 (2007), 47-92. doi: 10.1007/s11854-008-0002-z.
[7]	Vitaly Bergelson and Randall McCutcheon, Recurrence for semigroup actions and a non-commutative Schur theorem, in "Topological Dynamics and Applications" (eds. M. G. Nerurkar, D. P. Dokken and D. B. Ellis) (Minneapolis, MN, 1995), Contemp. Math., 215, Amer. Math. Soc., Providence, RI, (1998), 205-222.
[8]	Vasco Brattka, Peter Hertling and Klaus Weihrauch, A tutorial on computable analysis, in "New Computational Paradigms: Changing Conceptions of What is Computable" (eds. S. Barry Cooper, Benedikt Löwe and Andrea Sorbi), Springer, New York, (2008), 425-491.
[9]	C. M. Colebrook, The Hausdorff dimension of certain sets of nonnormal numbers, Michigan Math. J., 17 (1970), 103-116. doi: 10.1307/mmj/1029000420.
[10]	H. Furstenberg, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions, J. Analyse Math., 31 (1977), 204-256. doi: 10.1007/BF02813304.
[11]	H. Furstenberg, "Recurrence in Ergodic Theory and Combinatorial Number Theory," M. B. Porter Lectures, Princeton University Press, Princeton, N.J., 1981.
[12]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, A constructive Borel-Cantelli lemma. Constructing orbits with required statistical properties, Theor. Comput. Sci., 410 (2009), 2207-2222. doi: 10.1016/j.tcs.2009.02.010.
[13]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, Computing the speed of convergence of ergodic averages and pseudorandom points in computable dynamical systems, in "Proceedings of Computability and Complexity in Analysis (CCA)'' (eds. Xizhong Zheng and Ning Zhong), Electronic Proceedings in Theoretical Computer Science, arXiv:1006.0392.
[14]	Stefano Galatolo, Mathieu Hoyrup and Cristóbal Rojas, Dynamical systems, simulation, abstract computation, 2011, arXiv:1101.0833.
[15]	Andrzej Grzegorczyk, On the definitions of computable real continuous functions, Fundamenta Mathematicae, 44 (1957), 61-71.
[16]	Bernard Host and Bryna Kra, Uniformity seminorms on ℓ$^\infty$ and applications, J. Anal. Math., 108 (2009), 219-276. doi: 10.1007/s11854-009-0024-1.
[17]	Mathieu Hoyrup, Randomness and the ergodic decomposition, in "Computability in Europe (CiE) 2011" (eds. Benedikt Löwe, et al.), Springer, Berlin, (2011), 122-131.
[18]	Mathieu Hoyrup and Cristóbal Rojas, Computability of probability measures and Martin-Löf randomness over metric spaces, Inform. and Comput., 207 (2009), 830-847. doi: 10.1016/j.ic.2008.12.009.
[19]	Anatole Katok and Boris Hasselblatt, "Introduction to the Modern Theory of Dynamical Systems," With a supplementary chapter by Katok and Leonardo Mendoza, Encyclopedia of Mathematics and its Applications, 54, Cambridge Univ. Press, Cambridge, 1995.
[20]	Alexander P. Kreuzer, The cohesive principle and the Bolzano-Weierstraß principle, Math. Log. Q., 57 (2011), 292-298. doi: 10.1002/malq.201010008.
[21]	Pavol Safarik and Ulrich Kohlenbach, On the computational content of the Bolzano-Weierstraß principle, Math. Log. Q., 56 (2010), 508-532. doi: 10.1002/malq.200910106.
[22]	Karl Sigmund, Generic properties of invariant measures for Axiom A diffeomorphisms, Invent. Math., 11 (1970), 99-109. doi: 10.1007/BF01404606.
[23]	Karl Sigmund, On dynamical systems with the specification property, Trans. Amer. Math. Soc., 190 (1974), 285-299. doi: 10.1090/S0002-9947-1974-0352411-X.
[24]	Terence Tao, Norm convergence of multiple ergodic averages for commuting transformations, Ergodic Theory Dynam. Systems, 28 (2008), 657-688. doi: 10.1017/S0143385708000011.
[25]	Terence Tao, "Poincaré's Legacies, Pages from Year Two of a Mathematical Blog," Part I, Amer. Math. Soc., Providence, RI, 2009.
[26]	Henry Townser, A general correspondence between averages and integrals, arXiv:0804.2773.
[27]	V. V. V'yugin, Ergodic convergence in probability, and an ergodic theorem for individual random sequences, Teor. Veroyatnost. i Primenen., 42 (1997), 35-50; translation in Theory Probab. Appl. 42 (1997), 39-50.
[28]	V. V. V'yugin, Ergodic theorems for individual random sequences, Theoret. Comput. Sci., 207 (1998), 343-361. doi: 10.1016/S0304-3975(98)00072-3.
[29]	Klaus Weihrauch, Computability on the probability measures on the Borel sets of the unit interval, Theoret. Comput. Sci., 219 (1999), 421-437. doi: 10.1016/S0304-3975(98)00298-9.

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