Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T02:48:54.827Z Has data issue: false hasContentIssue false

Informal research statement

Published online by Cambridge University Press:  20 February 2012

Abstract

The editors are reproducing here an informal research statement that Dan wrote in one of his last years at the University of Maryland. It provides a rare insight into how he viewed his own work and as usual he did an excellent job in setting forth the main points in a clear way. The summary was retyped, lightly edited to correct typos, and provided with references by Ayşe Şahin. One of the editors (B.W.) added a brief appendix to cover some of the highlights of Dan’s work during the last years of his life. Following the research summary is a complete list of Dan’s publications as well as a second list for the other references cited.

Type
Research Article
Copyright
Copyright © Cambridge University Press 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References

[ALR95]Assani, I., Lesigne, E. and Rudolph, D.. Wiener–Wintner return-times ergodic theorem. Israel J. Math. 92(1–3) (1995), 375395.CrossRefGoogle Scholar
[BR88]Bergelson, V. and Rudolph, D. J.. Weakly mixing actions of F have infinite subgroup actions which are Bernoulli. Dynamical Systems (College Park, MD, 1986–1987) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 722.Google Scholar
[BHM+95]Blanchard, F., Host, B., Maass, A., Martinez, S. and Rudolph, D. J.. Entropy pairs for a measure. Ergod. Th. & Dynam. Sys. 15(4) (1995), 621632.CrossRefGoogle Scholar
[BLR88]Boyle, M., Lind, D. and Rudolph, D.. The automorphism group of a shift of finite type. Trans. Amer. Math. Soc. 306(1) (1988), 71114.CrossRefGoogle Scholar
[BLR97]Bułatek, W., Lemańczyk, M. and Rudolph, D.. Constructions of cocycles over irrational rotations. Studia Math. 125(1) (1997), 111.Google Scholar
[DR09]Danilenko, A. I. and Rudolph, D. J.. Conditional entropy theory in infinite measure and a question of Krengel. Israel J. Math. 172 (2009), 93117.CrossRefGoogle Scholar
[dJR84a]del Junco, A. and Rudolph, D. J.. On ergodic actions whose self-joinings are graphs. CWI Rep. PM R8408, Stichting Mathematisch Centrum Centrum voor Wiskunde en Informatica, Amsterdam, 1984.Google Scholar
[dJR84b]del Junco, A. and Rudolph, D. J.. Kakutani equivalence of ergodic Z n actions. Ergod. Th. & Dynam. Sys. 4(1) (1984), 89104.CrossRefGoogle Scholar
[dJR87a]del Junco, A. and Rudolph, D.. On ergodic actions whose self-joinings are graphs. Ergod. Th. & Dynam. Sys. 7(4) (1987), 531557.CrossRefGoogle Scholar
[dJR87b]del Junco, A. and Rudolph, D. J.. A rank-one, rigid, simple, prime map. Ergod. Th. & Dynam. Sys. 7(2) (1987), 229247.CrossRefGoogle Scholar
[dJR96]del Junco, A. and Rudolph, D. J.. Residual behavior of induced maps. Israel J. Math. 93 (1996), 387398.CrossRefGoogle Scholar
[dJRW09]del Junco, A., Rudolph, D. J. and Weiss, B.. Measured topological orbit and Kakutani equivalence. Discrete Contin. Dyn. Syst. Ser. S 2(2) (2009), 221238.Google Scholar
[DGRS08]Dooley, A. H., Golodets, V. Ya., Rudolph, D. J. and Sinel’shchikov, S. D.. Non-Bernoulli systems with completely positive entropy. Ergod. Th. & Dynam. Sys. 28(1) (2008), 87124.CrossRefGoogle Scholar
[DR]Dykstra, A. and Rudolph, D. J.. Nearly continuous Kakutani equivalence for the Morse minimal system. Preprint.Google Scholar
[DR10]Dykstra, A. and Rudolph, D. J.. Any two irrational rotations are nearly continuously Kakutani equivalent. J. Anal. Math. 110 (2010), 339384.CrossRefGoogle Scholar
[FR98]Feldman, J. and Rudolph, D. J.. Standardness of sequences of σ-fields given by certain endomorphisms. Fund. Math. 157(2–3) (1998), 175189 Dedicated to the memory of Wiesław Szlenk.CrossRefGoogle Scholar
[FRM80]Feldman, J., Rudolph, D. J. and Moore, C. C.. Affine extensions of a Bernoulli shift. Trans. Amer. Math. Soc. 257(1) (1980), 171191.CrossRefGoogle Scholar
[FdJR94]Fieldsteel, A., del Junco, A. and Rudolph, D. J.. α-equivalence: a refinement of Kakutani equivalence. Ergod. Th. & Dynam. Sys. 14(1) (1994), 69102.CrossRefGoogle Scholar
[FR90]Fieldsteel, A. and Rudolph, D. J.. Stability of m-equivalence to the weak Pinsker property. Ergod. Th. & Dynam. Sys. 10(1) (1990), 119129.CrossRefGoogle Scholar
[FR92]Fieldsteel, A. and Rudolph, D. J.. An ergodic transformation with trivial Kakutani centralizer. Ergod. Th. & Dynam. Sys. 12(3) (1992), 459478.CrossRefGoogle Scholar
[FRW06]Foreman, M. D., Rudolph, D. J. and Weiss, B.. On the conjugacy relation in ergodic theory. C. R. Math. Acad. Sci. Paris 343(10) (2006), 653656.CrossRefGoogle Scholar
[FRW11]Foreman, M., Rudolph, D. J. and Weiss, B.. The conjugacy problem in ergodic theory. Ann. of Math. (2) 173(3) (2011), 15291586.CrossRefGoogle Scholar
[GR84]Glasner, S. and Rudolph, D.. Uncountably many topological models for ergodic transformations. Ergod. Th. & Dynam. Sys. 4(2) (1984), 233236.CrossRefGoogle Scholar
[GHR92]Glasner, E., Host, B. and Rudolph, D.. Simple systems and their higher order self-joinings. Israel J. Math. 78(1) (1992), 131142.CrossRefGoogle Scholar
[GdJLR96]Goodson, G. R., del Junco, A., Lemańczyk, M. and Rudolph, D. J.. Ergodic transformations conjugate to their inverses by involutions. Ergod. Th. & Dynam. Sys. 16(1) (1996), 97124.CrossRefGoogle Scholar
[HHR00]Heicklen, D., Hoffman, C. and Rudolph, D. J.. Entropy and dyadic equivalence of random walks on a random scenery. Adv. Math. 156(2) (2000), 157179.CrossRefGoogle Scholar
[HR02a]Hoffman, C. and Rudolph, D.. A dyadic endomorphism which is Bernoulli but not standard. Israel J. Math. 130 (2002), 365379.CrossRefGoogle Scholar
[HR02b]Hoffman, C. and Rudolph, D.. Uniform endomorphisms which are isomorphic to a Bernoulli shift. Ann. of Math. (2) 156(1) (2002), 79101.CrossRefGoogle Scholar
[HR03]Hoffman, C. and Rudolph, D.. If the [T,Id] automorphism is Bernoulli then the [T,Id] endomorphism is standard. Studia Math. 155(3) (2003), 195206.CrossRefGoogle Scholar
[ILR93]Iwanik, A., Lemańczyk, M. and Rudolph, D.. Absolutely continuous cocycles over irrational rotations. Israel J. Math. 83(1–2) (1993), 7395.CrossRefGoogle Scholar
[JR92]Johnson, A. S. A. and Rudolph, D. J.. Commuting endomorphisms of the circle. Ergod. Th. & Dynam. Sys. 12(4) (1992), 743748.CrossRefGoogle Scholar
[JR95]Johnson, A. and Rudolph, D. J.. Convergence under ×q of ×p invariant measures on the circle. Adv. Math. 115(1) (1995), 117140.CrossRefGoogle Scholar
[KR97]Kammeyer, J. W. and Rudolph, D. J.. Restricted orbit equivalence for ergodic Z d actions. I. Ergod. Th. & Dynam. Sys. 17(5) (1997), 10831129.CrossRefGoogle Scholar
[KR02]Kammeyer, J. W. and Rudolph, D. J.. Restricted Orbit Equivalence for Actions of Discrete Amenable Groups (Cambridge Tracts in Mathematics, 146). Cambridge University Press, Cambridge, 2002.CrossRefGoogle Scholar
[KOR08]Kosek, W., Ormes, N. and Rudolph, D. J.. Flow–orbit equivalence for minimal Cantor systems. Ergod. Th. & Dynam. Sys. 28(2) (2008), 481500.CrossRefGoogle Scholar
[KLR92]Kwiatkowski, J., Lemańczyk, M. and Rudolph, D.. Weak isomorphism of measure-preserving diffeomorphisms. Israel J. Math. 80(1–2) (1992), 3364.CrossRefGoogle Scholar
[KLR94]Kwiatkowski, J., Lemańczyk, M. and Rudolph, D.. A class of real cocycles having an analytic coboundary modification. Israel J. Math. 87(1–3) (1994), 337360.CrossRefGoogle Scholar
[LPWR94]Lacey, M., Petersen, K., Wierdl, M. and Rudolph, D.. Random ergodic theorems with universally representative sequences. Ann. Inst. H. Poincaré Probab. Stat. 30(3) (1994), 353395.Google Scholar
[LR04]Lin, C.-H. and Rudolph, D.. Sections for semiflows and Kakutani shift equivalence. Modern Dynamical Systems and Applications. Cambridge University Press, Cambridge, 2004, pp. 145161.Google Scholar
[NR97]Nogueira, A. and Rudolph, D.. Topological weak-mixing of interval exchange maps. Ergod. Th. & Dynam. Sys. 17(5) (1997), 11831209.CrossRefGoogle Scholar
[ORW82]Ornstein, D. S., Rudolph, D. J. and Weiss, B.. Equivalence of measure preserving transformations. Mem. Amer. Math. Soc. 37(262) (1982).Google Scholar
[RR87]Rahe, M. H. and Rudolph, D. J.. Loose Bernoullicity is preserved under exponentiation by integrable functions. Ergod. Th. & Dynam. Sys. 7(2) (1987), 263265.CrossRefGoogle Scholar
[RR08a]Roychowdhury, M. K. and Rudolph, D. J.. Any two irreducible Markov chains of equal entropy are finitarily Kakutani equivalent. Israel J. Math. 165 (2008), 2941.CrossRefGoogle Scholar
[RR08b]Roychowdhury, M. K. and Rudolph, D. J.. The Morse minimal system is finitarily Kakutani equivalent to the binary odometer. Fund. Math. 198(2) (2008), 149163.CrossRefGoogle Scholar
[RR09a]Roychowdhury, M. K. and Rudolph, D. J.. Any two irreducible Markov chains are finitarily orbit equivalent. Israel J. Math. 174 (2009), 349368.CrossRefGoogle Scholar
[RR09b]Roychowdhury, M. K. and Rudolph, D. J.. Nearly continuous Kakutani equivalence of adding machines. J. Mod. Dyn. 3(1) (2009), 103119.CrossRefGoogle Scholar
[RS76]Rudolph, D. J. and Schwarz, G.. On attaining . Israel J. Math. 24(3–4) (1976), 185190.CrossRefGoogle Scholar
[RS77]Rudolph, D. J. and Schwarz, G.. The limits in of multi-step Markov chains. Israel J. Math. 28(1–2) (1977), 103109.CrossRefGoogle Scholar
[RS80]Rudolph, D. and M. Steele, J.. Sizes of order statistical events of stationary processes. Ann. Probab. 8(6) (1980), 10791084.CrossRefGoogle Scholar
[RS89]Rudolph, D. J. and Silva, C. E.. Minimal self-joinings for nonsingular transformations. Ergod. Th. & Dynam. Sys. 9(4) (1989), 759800.CrossRefGoogle Scholar
[RS95]Rudolph, D. J. and Schmidt, K.. Almost block independence and Bernoullicity of Z d-actions by automorphisms of compact abelian groups. Invent. Math. 120(3) (1995), 455488.CrossRefGoogle Scholar
[Rud75]Rudolph, D. J.. Non-Bernoulli Behavior of the Roots of K-Automorphisms. ProQuest LLC, Ann Arbor, MI, 1975. PhD Thesis, Stanford University.Google Scholar
[Rud76a]Rudolph, D. J.. A two-valued step coding for ergodic flows. Math. Z. 150(3) (1976), 201220.CrossRefGoogle Scholar
[Rud76b]Rudolph, D. J.. Two nonisomorphic K-automorphisms with isomorphic squares. Israel J. Math. 23(3–4) (1976), 274287.CrossRefGoogle Scholar
[Rud77]Rudolph, D.. Two nonisomorphic K-automorphisms all of whose powers beyond one are isomorphic. Israel J. Math. 27(3–4) (1977), 277298.CrossRefGoogle Scholar
[Rud78a]Rudolph, D. J.. Classifying the isometric extensions of a Bernoulli shift. J. Anal. Math. 34(1979) (1978), 3660.CrossRefGoogle Scholar
[Rud78b]Rudolph, D. J.. Counting the relatively finite factors of a Bernoulli shift. Israel J. Math. 30(3) (1978), 255263.CrossRefGoogle Scholar
[Rud78c]Rudolph, D. J.. If a finite extension of a Bernoulli shift has no finite rotation factors, it is Bernoulli. Israel J. Math. 30(3) (1978), 193206.CrossRefGoogle Scholar
[Rud78d]Rudolph, D. J.. If a two-point extension of a Bernoulli shift has an ergodic square, then it is Bernoulli. Israel J. Math. 30(1–2) (1978), 159180.CrossRefGoogle Scholar
[Rud78e]Rudolph, D. J.. The second centralizer of a Bernoulli shift is just its powers. Israel J. Math. 29(2–3) (1978), 167178.CrossRefGoogle Scholar
[Rud79a]Rudolph, D.. Smooth orbit equivalence of ergodic R d actions, d≥2. Trans. Amer. Math. Soc. 253 (1979), 291302.Google Scholar
[Rud79b]Rudolph, D. J.. An example of a measure preserving map with minimal self-joinings, and applications. J. Anal. Math. 35 (1979), 97122.CrossRefGoogle Scholar
[Rud81]Rudolph, D. J.. A characterization of those processes finitarily isomorphic to a Bernoulli shift. Ergodic Theory and Dynamical Systems, I (College Park, MD, 1979–1980) (Progress in Mathematics, 10). Birkhäuser, Boston, MA, 1981, pp. 164.Google Scholar
[Rud82a]Rudolph, D. J.. Ergodic behaviour of Sullivan’s geometric measure on a geometrically finite hyperbolic manifold. Ergod. Th. & Dynam. Sys. 2(3–4 (1983)) (1982), 491512.CrossRefGoogle Scholar
[Rud82b]Rudolph, D. J.. A mixing Markov chain with exponentially decaying return times is finitarily Bernoulli. Ergod. Th. & Dynam. Sys. 2(1) (1982), 8597.CrossRefGoogle Scholar
[Rud83]Rudolph, D. J.. An isomorphism theory for Bernoulli free Z-skew-compact group actions. Adv. Math. 47(3) (1983), 241257.CrossRefGoogle Scholar
[Rud84]Rudolph, D. J.. Inner and barely linear time changes of ergodic R k actions. Conference in Modern Analysis and Probability (New Haven, CT, 1982) (Contemporary Mathematics, 26). American Mathematical Society, Providence, RI, 1984, pp. 351371.CrossRefGoogle Scholar
[Rud85a]Rudolph, D. J.. k-fold mixing lifts to weakly mixing isometric extensions. Ergod. Th. & Dynam. Sys. 5(3) (1985), 445447.CrossRefGoogle Scholar
[Rud85b]Rudolph, D. J.. Restricted orbit equivalence. Mem. Amer. Math. Soc. 54(323) (1985).Google Scholar
[Rud86]Rudolph, D. J.. Z n and R n cocycle extensions and complementary algebras. Ergod. Th. & Dynam. Sys. 6(4) (1986), 583599.CrossRefGoogle Scholar
[Rud88a]Rudolph, D. J.. Asymptotically Brownian skew products give non-loosely Bernoulli K-automorphisms. Invent. Math. 91(1) (1988), 105128.CrossRefGoogle Scholar
[Rud88b]Rudolph, D. J.. Rectangular tilings of Rn and free Rn-actions. Dynamical Systems (College Park, MD, 1986–1987) (Lecture Notes in Mathematics, 1342). Springer, Berlin, 1988, pp. 653688.Google Scholar
[Rud89]Rudolph, D. J.. Markov tilings of ℝn and representations of ℝn actions. Contemp. Math. 94 (1989), 271290.CrossRefGoogle Scholar
[Rud90a]Rudolph, D. J.. Fundamentals of Measurable Dynamics (Oxford Science Publications). The Clarendon Press–Oxford University Press, New York, 1990.Google Scholar
[Rud90b]Rudolph, D. J.. ×2 and ×3 invariant measures and entropy. Ergod. Th. & Dynam. Sys. 10(2) (1990), 395406.CrossRefGoogle Scholar
[Rud94]Rudolph, D. J.. A joinings proof of Bourgain’s return time theorem. Ergod. Th. & Dynam. Sys. 14(1) (1994), 197203.CrossRefGoogle Scholar
[Rud95]Rudolph, D. J.. Eigenfunctions of T×S and the Conze–Lesigne algebra. Ergodic Theory and its Connections with Harmonic Analysis (Alexandria, 1993) (London Mathematical Society Lecture Note Series, 205). Cambridge University Press, Cambridge, 1995, pp. 369432.CrossRefGoogle Scholar
[Rud98a]Rudolph, D.. Residuality and orbit equivalence. Topological Dynamics and Applications (Minneapolis, MN, 1995) (Contemporary Mathematics, 215). American Mathematical Society, Providence, RI, 1998, pp. 243254.CrossRefGoogle Scholar
[Rud98b]Rudolph, D. J.. Fully generic sequences and a multiple-term return-times theorem. Invent. Math. 131(1) (1998), 199228.CrossRefGoogle Scholar
[Rud02]Rudolph, D. J.. Applications of orbit equivalence to actions of discrete amenable groups. Proceedings of the International Congress of Mathematicians, Vol. III (Beijing, 2002). Higher Education Press, Beijing, 2002, pp. 339347.Google Scholar
[Rud04]Rudolph, D. J.. Pointwise and L 1 mixing relative to a sub-sigma algebra. Illinois J. Math. 48(2) (2004), 505517.CrossRefGoogle Scholar
[Rud05]Rudolph, D.. An entropy-preserving Dye’s theorem for ergodic actions. J. Anal. Math. 95 (2005), 144.CrossRefGoogle Scholar
[Rud07]Rudolph, D. J.. Ergodic theory on Borel foliations by and . Topics in Harmonic Analysis and Ergodic Theory (Contemporary Mathematics, 444). American Mathematical Society, Providence, RI, 2007, pp. 89113.CrossRefGoogle Scholar
[RW00]Rudolph, D. J. and Weiss, B.. Entropy and mixing for amenable group actions. Ann. of Math. (2) 151(3) (2000), 11191150.CrossRefGoogle Scholar

Other References

[85]Bourgain, J.. Pointwise ergodic theorems for arithmetic sets. Inst. Hautes Études Sci. Publ. Math. (69) (1989), 545 With an appendix by the author, Harry Furstenberg, Yitzhak Katznelson, and Donald S. Ornstein.CrossRefGoogle Scholar
[86]Burton, R. M. and Rothstein, A.. Isomorphism theorems in ergodic theory. Tech. Rep. No. 54, Oregon State University, 1986.Google Scholar
[3]Connes, A., Feldman, J. and Weiss, B.. An amenable equivalence relation is generated by a single transformation. Ergod. Th. & Dynam. Sys. 1(4 (1981)) (1982), 431450.CrossRefGoogle Scholar
[4]Dooley, A. H. and Golodets, V. Ya.. The spectrum of completely positive entropy actions of countable amenable groups. J. Funct. Anal. 196(1) (2002), 118.CrossRefGoogle Scholar
[5]Dye, H. A.. On groups of measure preserving transformations. I. Amer. J. Math. 81 (1959), 119159.CrossRefGoogle Scholar
[6]Hamachi, T. and Keane, M. S.. Finitary orbit equivalence of odometers. Bull. Lond. Math. Soc. 38(3) (2006), 450458.CrossRefGoogle Scholar
[7]Heicklen, D.. Bernoullis are standard when entropy is not an obstruction. Israel J. Math. 107 (1998), 141155.CrossRefGoogle Scholar
[8]Heicklen, D. and Hoffman, C.. T,T −1 is not standard. Ergod. Th. & Dynam. Sys. 18(4) (1998), 875878.CrossRefGoogle Scholar
[9]Heicklen, D. and Hoffman, C.. Rational maps are d-adic Bernoulli. Ann. of Math. (2) 156(1) (2002), 103114.CrossRefGoogle Scholar
[10]Holt, E. N.. A ratio ergodic theorem on Borel actions of ℤd and ℝd. ProQuest LLC, Ann Arbor, MI, 2009. PhD Thesis, Colorado State University.Google Scholar
[11]Johnson, A. S. A.. Measures on the circle invariant under multiplication by a nonlacunary subsemigroup of the integers. Israel J. Math. 77(1–2) (1992), 211240.CrossRefGoogle Scholar
[12]Kalikow, S. A.. T,T −1 transformation is not loosely Bernoulli. Ann. of Math. (2) 115(2) (1982), 393409.CrossRefGoogle Scholar
[13]Kamiński, B. and Liardet, P.. Spectrum of multidimensional dynamical systems with positive entropy. Studia Math. 108(1) (1994), 7785.CrossRefGoogle Scholar
[14]Katok, A. B.. Monotone equivalence in ergodic theory. Izv. Akad. Nauk SSSR Ser. Mat. 41(1) (1977), 104157.Google Scholar
[15]Katok, A. B.. The special representation theorem for multi-dimensional group actions. Dynamical Systems (Vol. I – Warsaw) (Astérisque, 49). Société Mathématique, Paris, 1977, pp. 117140.Google Scholar
[16]Keane, M. and Smorodinsky, M.. Bernoulli schemes of the same entropy are finitarily isomorphic. Ann. of Math. (2) 109(2) (1979), 397406.CrossRefGoogle Scholar
[17]Krengel, U.. On Rudolph’s representation of aperiodic flows. Ann. Inst. H. Poincaré Sect. B (N.S.) 12(4) (1976), 319338.Google Scholar
[18]Kubo, I.. Quasi-flows. Nagoya Math. J. 35 (1969), 130.CrossRefGoogle Scholar
[19]Lindenstrauss, E.. Invariant measures and arithmetic quantum unique ergodicity. Ann. of Math. (2) 163(1) (2006), 165219.CrossRefGoogle Scholar
[20]McClendon, D. M.. Continuity of conditional measures associated to measure-preserving semiflows. Trans. Amer. Math. Soc. 361(1) (2009), 331341.CrossRefGoogle Scholar
[21]Newberger, F. A.. The ergodic theory of Bowen–Margulis measure. ProQuest LLC, Ann Arbor, MI, 1998. PhD Thesis, University of Maryland.Google Scholar
[22]Ornstein, D.. Bernoulli shifts with the same entropy are isomorphic. Adv. Math. 4 (1970), 337352.CrossRefGoogle Scholar
[23]Ornstein, D. S. and Weiss, B.. Entropy and isomorphism theorems for actions of amenable groups. J. Anal. Math. 48 (1987), 1141.CrossRefGoogle Scholar
[24]Patterson, S. J.. The limit set of a Fuchsian group. Acta Math. 136(3–4) (1976), 241273.CrossRefGoogle Scholar
[25]Şahin, A. A.. Tiling representations of r 2 actions and α-equivalence in two dimensions. Ergod. Th. & Dynam. Sys. 18(5) (1998), 12111255.CrossRefGoogle Scholar
[26]Sullivan, D.. The density at infinity of a discrete group of hyperbolic motions. Publ. Math. Inst. Hautes Études Sci. 50 (1979), 171202.CrossRefGoogle Scholar
[27]Thouvenot, J. P.. Personal communication with Rudolph.Google Scholar