Self-similarity and spectral theory: on the spectrum of substitutions

A. I. Bufetov ; B. Solomyak

Algebra i analiz, Tome 34 (2022) no. 3, pp. 5-50

Résumé

This survey of the spectral properties of substitution dynamical systems is devoted to primitive aperiodic substitutions and associated dynamical systems: $\mathbb{Z}$-actions and $\mathbb{R}$-actions, the latter viewed as tiling flows. The focus is on the continuous part of the spectrum. For $\mathbb{Z}$-actions the maximal spectral type can be represented in terms of matrix Riesz products, whereas for tiling flows, the local dimension of the spectral measure is governed by the spectral cocycle. References are given to complete proofs and emphasize ideas and various links.

Détail

Keywords: entropy, complexity, dynamical system, coding.
Mots-clés : substitutions

Bibliographie

[1] el Abdalaoui E. H., “A new class of rank-one transformations with singular spectrum”, Ergodic Theory Dynam. Systems, 27:5 (2007), 1541–1555 | DOI | MR | Zbl | DOI | MR | Zbl

[2] el Abdalaoui E. H., On the Mahler measure of the spectrum of rank one maps, arXiv: 2108.13416 | MR | MR

[3] el Abdalaoui E. H., Nadkarni M. G., “Calculus of generalized Riesz products. Recent trends in ergodic theory and dynamical systems”, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015, 145–180 | DOI | MR | Zbl | DOI | MR | Zbl

[4] Adamczewski B., “Symbolic discrepancy and self-similar dynamics”, Ann. Inst. Fourier (Grenoble), 54:7 (2004), 2201–2234 | DOI | MR | Zbl | DOI | MR | Zbl

[5] Akiyama S., Barge M., Berthé V., Lee J.-Y., Siegel A., “On the Pisot substitution conjecture”, Mathematics of aperiodic order, Progr. Math., 309, Birkhäuser/Springer, Basel, 2015, 33–72 | DOI | MR | Zbl | DOI | MR | Zbl

[6] Allouche J.-P., “On a Golay–Shapiro-like sequence”, Unif. Distrib. Theory, 11:2 (2016), 205–210 | DOI | MR | Zbl | DOI | MR | Zbl

[7] Allouche J.-P., Liardet P., “Generalized Rudin–Shapiro sequences”, Acta Arith., 60:1 (1991), 1–27 | DOI | MR | Zbl | DOI | MR | Zbl

[8] Allouche J.-P., Shallit J., “The ubiquitous Prouhet–Thue–Morse sequence”, Sequences and their applications (Singapore, 1998), Springer Ser. Discrete Math. Theor. Comput. Sci., Springer, London, 1999, 1–16 | MR | Zbl | MR | Zbl

[9] Allouche J.-P., Shallit J., Automatic sequences. Theory, applications, generalizations, Cambridge Univ. Press, Cambridge, 2003 | MR | Zbl | MR | Zbl

[10] Aubry S., Godrèche C., Luck J. M., “Scaling properties of a structure intermediate between quasiperiodic and random”, J. Statist. Phys., 51:5-6 (1988), 1033–1075 | DOI | MR | Zbl | DOI | MR | Zbl

[11] Avila A., Forni G., “Weak mixing for interval exchange transformations and translation flows”, Ann. Math. (2), 165:2 (2007), 637–664 | DOI | MR | Zbl | DOI | MR | Zbl

[12] Avila A., Forni G., Safaee P., Quantitative weak mixing for interval exchange transformations, arXiv: 2105.10547

[13] Baake M., Frank N. P., Grimm U., Robinson E. A., Jr., “Geometric properties of a binary non-Pisot inflation and absence of absolutely continuous diffraction”, Studia Math., 247:2 (2019), 109–154 | DOI | MR | Zbl | DOI | MR | Zbl

[14] Baake M., Gähler F., Grimm U., “Spectral and topological properties of a family of generalised Thue–Morse sequences”, J. Math. Phys., 53:3 (2012), 032701 | DOI | MR | Zbl | DOI | MR | Zbl

[15] Baake M., Grimm U., Aperiodic order, v. 1, Encyclopedia Math. Appl., 149, A mathematical invitation, Cambridge Univ. Press, Cambridge, 2013 | MR | Zbl | MR | Zbl

[16] Baake M., Grimm U., “Squirals and beyond: substitution tilings with singular continuous spectrum”, Ergodic Theory Dynam. Systems, 34 (2014), 1077–1102 | DOI | MR | Zbl | DOI | MR | Zbl

[17] Baake M., Grimm U., Mañibo N., “Spectral analysis of a family of binary inflation rules”, Lett. Math. Phys., 108:8 (2018), 1783–1805 | DOI | MR | Zbl | DOI | MR | Zbl

[18] Baake M., Gähler F., Mañibo N., “Renormalisation of pair correlation measures for primitive inflation rules and absence of absolutely continuous diffraction”, Comm. Math. Phys., 370:2 (2019), 591–635 | DOI | MR | Zbl | DOI | MR | Zbl

[19] Barge M., Diamond B., “Coincidence for substitutions of Pisot type”, Bull. Soc. Math. France, 130:4 (2002), 619–626 | DOI | MR | Zbl | DOI | MR | Zbl

[20] Berthé V., Delecroix V., “Beyond substitutive dynamical systems: $S$-adic expansions”, Numeration and substitution 2012, RIMS Kôkyûroku Bessatsu, B46, Res. Inst. Math. Sci. (RIMS), Kyoto, 2012, 81–123 | MR | MR

[21] Bartlett A., “Spectral theory of $\mathbb{Z}^d$ substitutions”, Ergodic Theory Dynam. Systems, 38:4 (2018), 1289–1341 | DOI | MR | Zbl | DOI | MR | Zbl

[22] Berend D., Radin Ch., Are there chaotic tilings?, Comm. Math. Phys., 152:2 (1993), 215–219 | DOI | MR | Zbl | DOI | MR | Zbl

[23] Berlinkov A., Solomyak B., “Singular substitutions of constant length”, Ergodic Theory Dynam. Systems, 39:9 (2019), 2384–2402 | DOI | MR | Zbl | DOI | MR | Zbl

[24] Bourgain J., “On the spectral type of Ornstein's class one transformation”, Israel J. Math., 84 (1993), 53–63 | DOI | MR | Zbl | DOI | MR | Zbl

[25] Borichev A., Sodin M., Weiss B., “Spectra of stationary processes on $\mathbb{Z}$”, $50$ years with Hardy spaces, Oper. Theory Adv. Appl., 261, Birkhäuser/Springer, Cham, 2018, 141–157 | MR | Zbl | MR | Zbl

[26] Brillhart J., Morton P., “Über Summen von Rudin–Shapiroschen Koeffizienten”, Illinois J. Math., 22:1 (1978), 126–148 | DOI | MR | Zbl | DOI | MR | Zbl

[27] Brillhart J., Morton P., “A case study in mathematical research: the Golay–Rudin–Shapiro sequence”, Amer. Math. Monthly, 103:10 (1996), 854–869 | DOI | MR | Zbl | DOI | MR | Zbl

[28] Bufetov A. I., “Predelnye teoremy dlya spetsialnykh potokov nad preobrazovaniyami Vershika”, Uspekhi mat. nauk, 68:5 (2013), 3–80 | MR | Zbl | MR | Zbl

[29] Bufetov A. I., “Limit theorems for translation flows”, Ann. Math. (2), 179:2 (2014), 431–499 | DOI | MR | Zbl | DOI | MR | Zbl

[30] Bufetov A. I., Solomyak B., “Limit theorems for self-similar tilings”, Comm. Math. Phys., 319:3 (2013), 761–789 | DOI | MR | Zbl | DOI | MR | Zbl

[31] Bufetov A. I., Solomyak B., “On the modulus of continuity for spectral measures in substitution dynamics”, Adv. Math., 260 (2014), 84–129 | DOI | MR | Zbl | DOI | MR | Zbl

[32] Bufetov A. I., Solomyak B., “The Hölder property for the spectrum of translation flows in genus two”, Israel J. Math., 223:1 (2018), 205–259 | DOI | MR | Zbl | DOI | MR | Zbl

[33] Bufetov A. I., Solomyak B., “On ergodic averages for parabolic product flows”, Bull. Soc. Math. France, 146:4 (2018), 675–690 | DOI | MR | Zbl | DOI | MR | Zbl

[34] Bufetov A. I., Solomyak B., “A spectral cocycle for substitution systems and translation flows”, J. Anal. Math., 141:1 (2020), 165–205 | DOI | MR | Zbl | DOI | MR | Zbl

[35] Bufetov A. I., Solomyak B., On singular substitution $\mathbb{Z}$-actions, arXiv: 2003.11287

[36] Bufetov A. I., Solomyak B., “Hölder regularity for the spectrum of translation flows”, J. Éc. Polytech. Math., 8 (2021), 279–310 | DOI | MR | Zbl | DOI | MR | Zbl

[37] Chan L., Grimm U., “Spectrum of a Rudin–Shapiro-like sequence”, Adv. Appl. Math., 87 (2017), 16–23 | DOI | MR | Zbl | DOI | MR | Zbl

[38] Chan L., Grimm U., “Substitution-based sequences with absolutely continuous diffraction”, J. Phys. Conf. Ser., 809:1 (2017), 012027 | DOI | DOI

[39] Choksi J. R., Nadkarni M. G., “The maximal spectral type of a rank one transformation”, Canad. Math. Bull., 37 (1994), 29–36 | DOI | MR | Zbl | DOI | MR | Zbl

[40] Clark A., Sadun L., “When size matters: subshifts and their related tiling spaces”, Ergodic Theory Dynam. Systems, 23:4 (2003), 1043–1057 | DOI | MR | Zbl | DOI | MR | Zbl

[41] Coven E. M., Keane M. S., “The structure of substitution minimal sets”, Trans. Amer. Math. Soc., 162 (1971), 89–102 | DOI | MR | DOI | MR

[42] Crisp D., Moran W., Pollington A., Shiue P., “Substitution invariant cutting sequences”, J. Théor. Nombres Bordeaux, 5:1 (1993), 123–137 | DOI | MR | Zbl | DOI | MR | Zbl

[43] Dai X.-R., Feng D.-J., Wang Y., “Refinable functions with non-integer dilations”, J. Funct. Anal., 250 (2007), 1–20 | DOI | MR | Zbl | DOI | MR | Zbl

[44] Dekking M F., “The spectrum of dynamical systems arising from substitutions of constant length”, Z. Wahr. Verw. Gebiete, 41:3 (1977/78), 221–239 | DOI | MR | DOI | MR

[45] Dekking M. F., Keane M., “Mixing properties of substitutions”, Z. Wahr. Verw. Gebiete, 42 (1978), 23–33 | DOI | MR | Zbl | DOI | MR | Zbl

[46] Dooley A. H., Eigen S. J., “A family of generalized Riesz products”, Canad. J. Math., 48:2 (1996), 302–315 | DOI | MR | Zbl | DOI | MR | Zbl

[47] Dumont J.-M., Kamae T., Takahashi S., “Minimal cocycles with the scaling property and substitutions”, Israel J. Math., 95 (1996), 393–410 | DOI | MR | Zbl | DOI | MR | Zbl

[48] Dumont J.-M., Thomas A., “Systèmes de numeration et fonctions fractales relatifs aux substitutions”, Theoret. Comput. Sci., 65:2 (1989), 153–169 | DOI | MR | Zbl | DOI | MR | Zbl

[49] Durand F., Host B., Skau C., “Substitutional dynamical systems, Bratteli diagrams and dimension groups”, Ergodic Theory Dynam. Systems, 19:4 (1999), 953–993 | DOI | MR | Zbl | DOI | MR | Zbl

[50] Dworkin S., “Spectral theory and $X$-ray diffraction”, J. Math. Phys., 34:7 (1993), 2965–2967 | DOI | MR | Zbl | DOI | MR | Zbl

[51] Einsiedler M., Ward T., Ergodic theory with a view towards number theory, Grad. Texts in Math., 259, Springer-Verlag London, Ltd., London, 2011 | MR | Zbl | MR | Zbl

[52] Erdős P., “On a family of symmetric Bernoulli convolutions”, Amer. J. Math., 61 (1939), 974–975 | DOI | MR | DOI | MR

[53] Erdős P., “On the smoothness properties of Bernoulli convolutions”, Amer. J. Math., 62 (1940), 180–186 | DOI | MR | DOI | MR

[54] Ferenczi S., Mauduit Ch., Nogueira A., “Substitution dynamical systems: algebraic characterization of eigenvalues”, Ann. Sci. École Norm. Sup. (4), 29:4 (1996), 519–533 | DOI | MR | Zbl | DOI | MR | Zbl

[55] Forni G., Twisted translation flows and effective weak mixing, arXiv: 1908.11040

[56] Forni G., Masur H., Smillie J., “Bill Veech's contributions to dynamical systems”, J. Mod. Dyn., 14 (2019), v–xxv | MR | Zbl | MR | Zbl

[57] Frank N. P., “Substitution sequences in $\mathbb{Z}^d$ with a non-simple Lebesgue component in the spectrum”, Ergodic Theory Dynam. Systems, 23:2 (2003), 519–532 | DOI | MR | Zbl | DOI | MR | Zbl

[58] Fogg N. P., Substitutions in dynamics, arithmetics and combinatorics, Lecture Notes in Math., 1794, Springer, Berlin, 2002 | DOI | MR | Zbl | DOI | MR | Zbl

[59] Falconer K. J., Techniques in fractal geometry, John Wiley Sons, Chichester, 1997 | MR | Zbl | MR | Zbl

[60] Furstenberg H., Kesten H., “Products of random matrices”, Ann. Math. Statist., 31:2 (1960), 457–469 | DOI | MR | Zbl | DOI | MR | Zbl

[61] Garsia A., “Arithmetic properties of Bernoullli convolutions”, Trans. Amer. Math. Soc., 102 (1962), 409–432 | DOI | MR | Zbl | DOI | MR | Zbl

[62] Golay M. J. E., “Statistic multislit spectrometry and its application to the panoramic display of infrared spectra”, J. Optical Soc. Amer., 41 (1951), 468–472 | DOI | DOI

[63] Goldberg R., “Restrictions of Fourier transforms and extension of Fourier sequences”, J. Approx. Theory, 3 (1970), 149–155 | DOI | MR | Zbl | DOI | MR | Zbl

[64] Gottschalk W. H., “Substitution minimal sets”, Trans. Amer. Math. Soc., 109 (1963), 467–491 | DOI | MR | Zbl | DOI | MR | Zbl

[65] Guinier A., X-Ray diffraction in crystals, imperfect crystals, and amorphous bodies, W. H. Freeman and Co., San-Francisco, 1963

[66] Hof A., “On diffraction by aperiodic structures”, Comm. Math. Phys., 169:1 (1995), 25–43 | DOI | MR | Zbl | DOI | MR | Zbl

[67] Hof A., “Diffraction by aperiodic structures”, The mathematics of long-range aperiodic order (Waterloo, ON, 1995), NATO Adv. Sci. Inst. Ser. C. Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997, 239–268 | MR | Zbl | MR | Zbl

[68] Hof A., “On scaling in relation to singular spectrum”, Comm. Math. Phys., 184:3 (1997), 567–577 | DOI | MR | Zbl | DOI | MR | Zbl

[69] Hollander M., Solomyak B., “Two-symbol Pisot substitutions have pure purely discrete spectrum”, Ergodic Theory Dynam. Systems, 23:2 (2003), 533–540 | DOI | MR | Zbl | DOI | MR | Zbl

[70] Host B., “Valeurs propres des systémes dynamiques définis par des substitutions de longueur variable”, Ergodic Theory Dynam. Systems, 6:4 (1986), 529–540 | DOI | MR | Zbl | DOI | MR | Zbl

[71] Host B., “Some results on uniform distribution in the multidimensional torus”, Ergodis Theory Dynam. Systems, 20:2 (2000), 439–452 | DOI | MR | Zbl | DOI | MR | Zbl

[72] Ito Sh., “A construction of transversal flows for maximal Markov automorphisms”, Tokyo J. Math., 1:2 (1978), 305–324 | DOI | MR | Zbl | DOI | MR | Zbl

[73] Kahane J.-P., “Sur la distribution de certaines séries aléatoires”, Colloq. Theor. Nombres (1969, Bordeaux), Bull. Soc. Math. France, Mém., 25, Soc. Math. France, Paris, 1971, 119–122 | MR | Zbl | MR | Zbl

[74] Kakutani Sh., “Strictly ergodic symbolic dynamical systems”, Proc. 6th Berkeley Symp. Mathematical Statistics and Probability (Univ. California, Berkeley, 1970/1971), v. II, Univ. California Press, Berkeley, 1972, 319–326 | MR | Zbl | MR | Zbl

[75] Kamae T., “Spectrum of a substitution minimal set”, J. Math. Soc. Japan, 22 (1970), 567–578 | DOI | MR | Zbl | DOI | MR | Zbl

[76] Katznelson Yi., An introduction to harmonic analysis, John Wiley Sons, New York, 1968 | MR | Zbl | MR | Zbl

[77] Keane M., “Generalized Morse sequences”, Z. Wahr. Verw. Gebiete, 10 (1968), 335–353 | DOI | MR | Zbl | DOI | MR | Zbl

[78] Klemes I., Reinhold K., “Rank one transformations with singular spectral type”, Israel J. Math., 98 (1997), 1–14 | DOI | MR | Zbl | DOI | MR | Zbl

[79] Knill O., “Singular continuous spectrum and quantitative rates of mixing”, Discrete Cont. Dynam. Systems, 4 (1998), 33–42 | DOI | MR | Zbl | DOI | MR | Zbl

[80] Kornfeld I. P., Sinai Ya. G., Fomin S. V., Ergodicheskaya teoriya, Nauka, M., 1980 | MR | MR

[81] Kwiatkowski J., Sikorski A., “Spectral properties of $G$-symbolic Morse shifts”, Bull. Soc. Math. France, 115:1 (1987), 19–33 | DOI | MR | Zbl | DOI | MR | Zbl

[82] Ledrappier F., “Des produits de Riesz comme mesures spectrales”, Ann. Inst. H. Poincaré Sect. B (N.S.), 6 (1970), 335–344 | MR | Zbl | MR | Zbl

[83] Livshits A. N., “O spektrakh adicheskikh preobrazovanii markovskikh kompaktov”, Uspekhi mat nauk, 42:3 (1987), 189–190 | MR | Zbl | MR | Zbl

[84] Livshits A. N., “Dostatochnoe uslovie slabogo peremeshivaniya podstanovok i statsionarnykh adicheskikh preobrazovanii”, Mat. zametki, 44:6 (1988), 785–793

[85] Livshits A. N., “Some examples of adic transformations and substitutions”, Selecta Math. Soviet., 11 (1992), 83–104 | MR | MR

[86] Mahler K., “The spectrum of an array and its application to the study of the translation properties of a simple class of arithmetical functions. Pt. II. On the translation properties of a simple class of arithmetical functions”, J. Math. Massachusetts, 6 (1927), 158–163 | Zbl | Zbl

[87] Martin J. C., “Substitution minimal flows”, Amer. J. Math., 93 (1971), 503–526 | DOI | MR | Zbl | DOI | MR | Zbl

[88] Martin J. C., “Minimal flows arising from substitutions of non-constant length”, Math. Systems Theory, 7 (1973), 72–82 | MR | MR

[89] Mendès F. M., Tenenbaum G., “Dimension des courbes planes, papiers pliés et suites de Rudin–Shapiro”, Bull. Soc. Math. France, 109:2 (1981), 207–215 | MR | Zbl | MR | Zbl

[90] Michel P., “Stricte ergodicité d'ensembles minimaux de substitutions”, C. R. Acad. Sci. Paris Sér. A, 278 (1974), 811–813 | MR | Zbl | MR | Zbl

[91] Michel P., “Coincidence values and spectra of substitutions”, Z. Wahr. Verw. Gebiete, 42:3 (1978), 205–227 | DOI | MR | Zbl | DOI | MR | Zbl

[92] Miro E., Rust D., Sadun L., Tadeo G. S., Topological mixing of random substitutions, arXiv: 2103.02361

[93] Morse M., “Recurrent geodesics on a surface of negative curvature”, Trans. Amer. Math. Soc., 22 (1921), 84–100 | DOI | MR | Zbl | DOI | MR | Zbl

[94] Morse M., Hedlund G. A., “Symbolic dynamics”, Amer. J. Math., 60:4 (1938), 815–866 | DOI | MR | DOI | MR

[95] Morse M., Hedlund G. A., “Symbolic dynamics. II. Sturmian trajectories”, Amer. J. Math., 62 (1940), 1–42 | DOI | MR | DOI | MR

[96] Morse M., Hedlund G. A., “Unending chess, symbolic dynamics and a problem in semigroups”, Duke Math. J., 11 (1944), 1–7 | DOI | MR | Zbl | DOI | MR | Zbl

[97] Nadkarni M., Spectral theory of dynamical systems, Texts Readings Math., Second ed., Springer, Singapore, 2020 | DOI | MR | Zbl | DOI | MR | Zbl

[98] Pansiot J.-J., “Decidability of periodicity for infinite words”, RAIRO Inform. Théor. Appl., 20:1 (1986), 43–46 | DOI | MR | Zbl | DOI | MR | Zbl

[99] Peres Yu., Schlag W., Solomyak B., “Sixty years of Bernoulli convolutions”, Fractal geometry and stochastics, v. II, Progr. Probab., 46, Birkhäuser, Basel, 2000, 39–65 | MR | Zbl | MR | Zbl

[100] Prouhet E., “Mémoire sur quelques relations entre les puissances des nombres”, C. R. Acad. Sci. Paris Sér. I, 33 (1851), 225

[101] Queffelec M., Substitution dynamical systems — spectral analysis, Lecture Notes in Math., 1294, Second ed., Springer, Berlin, 2010 | DOI | MR | Zbl | DOI | MR | Zbl

[102] Radin Ch., Miles of tiles, Student Math. Library, I, Amer. Math. Soc., Providence, RI, 1999 | DOI | MR | Zbl | DOI | MR | Zbl

[103] Radin Ch., Wolff M., “Space tilings and local isomorphism”, Geom. Dedicata, 42:3 (1992), 355–360 | DOI | MR | Zbl | DOI | MR | Zbl

[104] Rauzy G., “Échanges d'intervalles et transformations induites”, Acta Arith., 34:4 (1979), 315–328 | DOI | MR | Zbl | DOI | MR | Zbl

[105] Rauzy G., “Nombres algébriques et substitutions”, Bull. Soc. Math. France, 110:2 (1982), 147–178 | DOI | MR | Zbl | DOI | MR | Zbl

[106] Reed M., Simon B., Methods of modern mathematical physics, v. I, Functional analysis, Second ed., Acad. Press, New York, 1980 | MR | Zbl | MR | Zbl

[107] Robinson E. A., Jr., “The dynamical theory of tilings and quasicrystallography”, Ergodic theory of $\mathbb{Z}^d$-actions (Warwick, 1993–1994), London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996, 451–473 | MR | Zbl | MR | Zbl

[108] Robinson E. A., Jr., “Symbolic dynamics and tilings of $\mathbb{R}^d$”, Symbolic dynamics and its applications, Proc. Sympos. Appl. Math., 60, Amer. Math. Soc., Providence, RI, 2004, 81–119 | DOI | MR | Zbl | DOI | MR | Zbl

[109] Rudin W., “Some theorems on Fourier coefficients”, Proc. Amer. Math. Soc., 10 (1959), 855–859 | DOI | MR | Zbl | DOI | MR | Zbl

[110] Saffari B., “Une fonction extrémale liée á la suite de Rudin–Shapiro”, C. R. Acad. Sci. Paris Sér. I Math., 303:4 (1986), 97–100 | MR | Zbl | MR | Zbl

[111] Salem R., “A remarkable class of algebraic integers. Proof of a conjecture by Vijayaraghavan”, Duke Math. J., 11 (1944), 103–108 | DOI | MR | Zbl | DOI | MR | Zbl

[112] Salem R., Algebraic numbers and Fourier analysis, D. C. Heath and Co., Boston, Mass., 1963 | MR | Zbl | MR | Zbl

[113] Senechal M., Quasicrystals and geometry, Cambridge Univ. Press, Cambridge, 1995 | MR | Zbl | MR | Zbl

[114] Shapiro H. S., Extremal problems for polynomials and power series, Ph.D. Thesis, Massachusetts Inst. Technology, Ann Arbor, MI, 1953 | MR | MR

[115] Shmerkin P., “On Furstenberg's intersection conjecture, self-similar measures, and the $L^q$ norms of convolutions”, Ann. of Math. (2), 189:2 (2019), 319–391 | DOI | MR | Zbl | DOI | MR | Zbl

[116] Solomyak B., “O kratnosti spektra analiticheskikh operatorov Teplitsa”, Uspekhi mat. nauk, 41:2 (1986), 209–210 | MR | Zbl | MR | Zbl

[117] Solomyak B., “Substitutions, adic transformations and beta-expansions”, Contemp. Math., 135, Amer. Math. Soc., Providence, RI, 1992, 361–372 | DOI | MR | DOI | MR

[118] Solomyak B., “Dynamics of self-similar tilings”, Ergodic Theory Dynam. Systems, 17:3 (1997), 695–738 | DOI | MR | Zbl | DOI | MR | Zbl

[119] Thue A., “Über unendliche Zeichenreihen”, Norske vid. Selsk. Skr. Mat. Nat. Kl., 7 (1906), 1–22

[120] Varjú P. P., “Recent progress on Bernoulli convolutions”, European Congress of Math., Eur. Math. Soc., Zürich, 2018, 847–867 | DOI | MR | Zbl | DOI | MR | Zbl

[121] Veech W. A., “Gauss measures for transformations on the space of interval exchange maps”, Ann. of Math. (2), 115:1 (1982), 201–242 | DOI | MR | Zbl | DOI | MR | Zbl

[122] Vershik A. M., “Teorema o markovskoi periodicheskoi approksimatsii v ergodicheskoi teorii”, Zap. nauch. semin. LOMI, 115, 1982, 72–82 | Zbl | Zbl

[123] Vershik A. M., Livshits A. N., “Adic models of ergodic transformations, spectral theory, substitutions, and related topics”, Representation theory and dynamical systems, Adv. Soviet Math., 9, Amer. Math. Soc., Providence, RI, 1992, 185–204 | MR | MR

[124] Viana M., Lectures on interval exchange transformations and teichmüller flows, preprint IMPA, 2008

[125] Walters P., An introduction to ergodic theory, Springer Grad. Texts in Math., 79, Springer-Verlag, New York, 1982 | DOI | MR | Zbl | DOI | MR | Zbl

[126] Yaari R., Uniformly distributed orbits in $\mathbb{T}^d$ and singular substitution dynamical systems, arXiv: 2108.13882

[127] Yoccoz J.-Ch., “Interval exchange maps and translation surfaces”, Homogeneous flows, moduli spaces and arithmetic, Clay Math. Proc., 10, Amer. Math. Soc., Providence, RI, 2010, 1–69 | MR | Zbl | MR | Zbl

[128] Yu H., Bernoulli convolutions with Garsia parameters in $(1, \sqrt{2}]$ have continuous density functions, arXiv: 2108.01008

[129] Zorich A., “Flat surfaces”, Frontiers in number theory, physics, and geometry, v. I, Springer, Berlin, 2006, 437–583 | DOI | MR | Zbl | DOI | MR | Zbl