{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,9,21]],"date-time":"2025-09-21T17:39:40Z","timestamp":1758476380847},"reference-count":34,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2016,12,19]],"date-time":"2016-12-19T00:00:00Z","timestamp":1482105600000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2017,3]]},"DOI":"10.1007\/s00454-016-9849-4","type":"journal-article","created":{"date-parts":[[2016,12,19]],"date-time":"2016-12-19T17:14:06Z","timestamp":1482167646000},"page":"357-370","update-policy":"http:\/\/dx.doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Invariant Measure of Rotational Beta Expansion and Tarski\u2019s Plank Problem"],"prefix":"10.1007","volume":"57","author":[{"given":"Shigeki","family":"Akiyama","sequence":"first","affiliation":[]},{"given":"Jonathan","family":"Caalim","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2016,12,19]]},"reference":[{"key":"9849_CR1","first-page":"1","volume":"69","author":"S Akiyama","year":"2016","unstructured":"Akiyama, S., Caalim, J.: Rotational beta expansion: ergodicity and soficness. J. Math. Soc. Japan 69, 1\u201319 (2016)","journal-title":"J. Math. Soc. Japan"},{"issue":"1\u20132","key":"9849_CR2","doi-asserted-by":"crossref","first-page":"921","DOI":"10.1016\/S0304-3975(01)00103-7","volume":"270","author":"S Akiyama","year":"2002","unstructured":"Akiyama, S., Peth\u0151, A.: On canonical number systems. Theor. Comput. Sci. 270(1\u20132), 921\u2013933 (2002)","journal-title":"Theor. Comput. Sci."},{"issue":"1","key":"9849_CR3","first-page":"101","volume":"18","author":"S Akiyama","year":"2007","unstructured":"Akiyama, S., Scheicher, K.: Symmetric shift radix systems and finite expansions. Math. Pannon 18(1), 101\u2013124 (2007)","journal-title":"Math. Pannon"},{"issue":"3","key":"9849_CR4","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1007\/BF01245089","volume":"104","author":"K Ball","year":"1991","unstructured":"Ball, K.: The plank problem for symmetric bodies. Invent. Math. 104(3), 535\u2013543 (1991)","journal-title":"Invent. Math."},{"key":"9849_CR5","first-page":"990","volume":"2","author":"T Bang","year":"1951","unstructured":"Bang, T.: A solution of the \u201cplank problem\u201d. Proc. Am. Math. Soc. 2, 990\u2013993 (1951)","journal-title":"Proc. Am. Math. Soc."},{"issue":"3","key":"9849_CR6","doi-asserted-by":"crossref","first-page":"689","DOI":"10.1017\/S0143385701001341","volume":"21","author":"J Buzzi","year":"2001","unstructured":"Buzzi, J., Keller, G.: Zeta functions and transfer operators for multidimensional piecewise affine and expanding maps. Ergodic Theory Dyn. Syst. 21(3), 689\u2013716 (2001)","journal-title":"Ergodic Theory Dyn. Syst."},{"issue":"1","key":"9849_CR7","doi-asserted-by":"crossref","first-page":"157","DOI":"10.4171\/JEMS\/76","volume":"9","author":"K Dajani","year":"2007","unstructured":"Dajani, K., de Vries, M.: Invariant densities for random $$\\beta $$ \u03b2 -expansions. J. Eur. Math. Soc. 9(1), 157\u2013176 (2007)","journal-title":"J. Eur. Math. Soc."},{"key":"9849_CR8","doi-asserted-by":"crossref","first-page":"1335","DOI":"10.4153\/CJM-1982-093-4","volume":"34","author":"WJ Gilbert","year":"1982","unstructured":"Gilbert, W.J.: Complex numbers with three radix representations. Can. J. Math. 34, 1335\u20131348 (1982)","journal-title":"Can. J. Math."},{"issue":"5","key":"9849_CR9","doi-asserted-by":"crossref","first-page":"1583","DOI":"10.1017\/S0143385707000053","volume":"27","author":"P G\u00f3ra","year":"2007","unstructured":"G\u00f3ra, P.: Invariant densities for generalized $$\\beta $$ \u03b2 -maps. Ergodic Theory Dyn. Syst. 27(5), 1583\u20131598 (2007)","journal-title":"Ergodic Theory Dyn. Syst."},{"issue":"3","key":"9849_CR10","doi-asserted-by":"crossref","first-page":"272","DOI":"10.1007\/BF02764946","volume":"67","author":"P G\u00f3ra","year":"1989","unstructured":"G\u00f3ra, P., Boyarsky, A.: Absolutely continuous invariant measures for piecewise expanding $$C^2$$ C 2 transformation in $${{\\bf R}}^N$$ R N . Isr. J. Math. 67(3), 272\u2013286 (1989)","journal-title":"Isr. J. Math."},{"issue":"A22","key":"9849_CR11","doi-asserted-by":"crossref","first-page":"239","DOI":"10.1515\/INTEG.2009.023","volume":"9","author":"Sh Ito","year":"2009","unstructured":"Ito, Sh, Sadahiro, T.: Beta-expansions with negative bases. Integers 9(A22), 239\u2013259 (2009)","journal-title":"Integers"},{"issue":"1","key":"9849_CR12","doi-asserted-by":"crossref","first-page":"33","DOI":"10.2969\/jmsj\/02610033","volume":"26","author":"Sh Ito","year":"1974","unstructured":"Ito, Sh, Takahashi, Y.: Markov subshifts and realization of $$\\beta $$ \u03b2 -expansions. J. Math. Soc. Japan 26(1), 33\u201355 (1974)","journal-title":"J. Math. Soc. Japan"},{"issue":"1","key":"9849_CR13","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1017\/etds.2012.127","volume":"34","author":"C Kalle","year":"2014","unstructured":"Kalle, C.: Isomorphisms between positive and negative $$\\beta $$ \u03b2 -transformations. Ergodic Theory Dyn. Syst. 34(1), 153\u2013170 (2014)","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"9849_CR14","first-page":"255","volume":"37","author":"I K\u00e1tai","year":"1975","unstructured":"K\u00e1tai, I., Szab\u00f3, J.: Canonical number systems for complex integers. Acta Sci. Math. (Szeged) 37, 255\u2013260 (1975)","journal-title":"Acta Sci. Math. (Szeged)"},{"issue":"12","key":"9849_CR15","first-page":"A625","volume":"289","author":"G Keller","year":"1979","unstructured":"Keller, G.: Ergodicit\u00e9 et mesures invariantes pour les transformations dilatantes par morceaux d\u2019une r\u00e9gion born\u00e9e du plan, C. C.R. Acad. Sci. Paris S\u00e9r. A-B 289(12), A625\u2013A627 (1979)","journal-title":"C.R. Acad. Sci. Paris S\u00e9r. A-B"},{"issue":"3","key":"9849_CR16","doi-asserted-by":"crossref","first-page":"461","DOI":"10.1007\/BF00532744","volume":"69","author":"G Keller","year":"1985","unstructured":"Keller, G.: Generalized bounded variation and applications to piecewise monotonic transformations. Z. Wahrscheinlichkeitstheor. Verw. Geb. 69(3), 461\u2013478 (1985)","journal-title":"Z. Wahrscheinlichkeitstheor. Verw. Geb."},{"issue":"2","key":"9849_CR17","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1007\/s10474-013-0377-x","volume":"142","author":"T Kempton","year":"2014","unstructured":"Kempton, T.: On the invariant density of the random $$\\beta $$ \u03b2 -transformation. Acta Math. Hung. 142(2), 403\u2013419 (2014)","journal-title":"Acta Math. Hung."},{"key":"9849_CR18","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1007\/BF02249260","volume":"6","author":"R Kenyon","year":"1996","unstructured":"Kenyon, R.: The construction of self-similar tilings. Geom. Funct. Anal. 6, 471\u2013488 (1996)","journal-title":"Geom. Funct. Anal."},{"issue":"3","key":"9849_CR19","doi-asserted-by":"crossref","first-page":"577","DOI":"10.1007\/s00454-009-9199-6","volume":"43","author":"R Kenyon","year":"2010","unstructured":"Kenyon, R., Solomyak, B.: On the characterization of expansion maps for self-affine tilings. Discrete Comput. Geom. 43(3), 577\u2013593 (2010)","journal-title":"Discrete Comput. Geom."},{"issue":"3","key":"9849_CR20","doi-asserted-by":"crossref","first-page":"399","DOI":"10.4153\/CMB-2007-038-5","volume":"50","author":"V Komornik","year":"2007","unstructured":"Komornik, V., Loreti, P.: Expansions in complex bases. Can. Math. Bull. 50(3), 399\u2013408 (2007)","journal-title":"Can. Math. Bull."},{"issue":"2","key":"9849_CR21","doi-asserted-by":"crossref","first-page":"161","DOI":"10.1112\/jlms\/54.1.161","volume":"54","author":"JC Lagarias","year":"1996","unstructured":"Lagarias, J.C., Wang, Y.: Integral self-affine tiles in $${\\mathbb{R}}^n$$ R n I. Standard and nonstandard digit sets. J. Lond. Math. Soc. 54(2), 161\u2013179 (1996)","journal-title":"J. Lond. Math. Soc."},{"key":"9849_CR22","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1090\/S0002-9947-1978-0457679-0","volume":"235","author":"TY Li","year":"1978","unstructured":"Li, T.Y., Yorke, J.A.: Ergodic transformations from an interval into itself. Trans. Am. Math. Soc. 235, 183\u2013192 (1978)","journal-title":"Trans. Am. Math. Soc."},{"issue":"5","key":"9849_CR23","doi-asserted-by":"crossref","first-page":"1673","DOI":"10.1017\/S0143385711000514","volume":"32","author":"L Liao","year":"2012","unstructured":"Liao, L., Steiner, W.: Dynamical properties of the negative beta-transformation. Ergodic Theory Dyn. Syst. 32(5), 1673\u20131690 (2012)","journal-title":"Ergodic Theory Dyn. Syst."},{"key":"9849_CR24","doi-asserted-by":"crossref","first-page":"401","DOI":"10.1007\/BF02020954","volume":"11","author":"W Parry","year":"1960","unstructured":"Parry, W.: On the $$\\beta $$ \u03b2 -expansions of real numbers. Acta Math. Acad. Sci. Hung. 11, 401\u2013416 (1960)","journal-title":"Acta Math. Acad. Sci. Hung."},{"key":"9849_CR25","doi-asserted-by":"crossref","first-page":"95","DOI":"10.1007\/BF01897025","volume":"15","author":"W Parry","year":"1964","unstructured":"Parry, W.: Representations for real numbers. Acta Math. Acad. Sci. Hung. 15, 95\u2013105 (1964)","journal-title":"Acta Math. Acad. Sci. Hung."},{"key":"9849_CR26","doi-asserted-by":"crossref","unstructured":"Parry, W.: The Lorenz attractor and a related population model, Ergodic theory. In: Proceedings Conference Mathematics Forschungsinstitut Oberwolfach, 1978. Lecture Notes in Mathematics, vol. 729, pp. 169\u2013187. Springer, Berlin (1979)","DOI":"10.1007\/BFb0063293"},{"key":"9849_CR27","doi-asserted-by":"crossref","first-page":"477","DOI":"10.1007\/BF02020331","volume":"8","author":"A R\u00e9nyi","year":"1957","unstructured":"R\u00e9nyi, A.: Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hung. 8, 477\u2013493 (1957)","journal-title":"Acta Math. Acad. Sci. Hung."},{"issue":"1","key":"9849_CR28","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/S0304-3975(98)00100-5","volume":"210","author":"T Safer","year":"1999","unstructured":"Safer, T.: Polygonal radix representations of complex numbers. Theor. Comput. Sci. 210(1), 159\u2013171 (1999)","journal-title":"Theor. Comput. Sci."},{"key":"9849_CR29","doi-asserted-by":"crossref","first-page":"223","DOI":"10.1007\/BF02773219","volume":"116","author":"B Saussol","year":"2000","unstructured":"Saussol, B.: Absolutely continuous invariant measures for multidimensional expanding maps. Isr. J. Math. 116, 223\u2013248 (2000)","journal-title":"Isr. J. Math."},{"issue":"1","key":"9849_CR30","doi-asserted-by":"crossref","first-page":"163","DOI":"10.1017\/S0305004102005856","volume":"133","author":"K Scheicher","year":"2002","unstructured":"Scheicher, K., Thuswaldner, J.M.: Canonical number systems, counting automata and fractals. Math. Proc. Camb. Philos. Soc. 133(1), 163\u2013182 (2002)","journal-title":"Math. Proc. Camb. Philos. Soc."},{"key":"9849_CR31","unstructured":"Tarski, A.: Further remarks about degree of equivalence on polygons (English translation by I. Wirszup), Collected Papers, vol.\u00a01, pp.\u00a0597\u2013611. Birkh\u00e4user, Basel, Boston, Stuttgart (1986)"},{"issue":"3","key":"9849_CR32","doi-asserted-by":"crossref","first-page":"605","DOI":"10.1007\/s002200050003","volume":"208","author":"M Tsujii","year":"2000","unstructured":"Tsujii, M.: Absolutely continuous invariant measures for piecewise real-analytic expanding maps on the plane. Commun. Math. Phys. 208(3), 605\u2013622 (2000)","journal-title":"Commun. Math. Phys."},{"issue":"2","key":"9849_CR33","doi-asserted-by":"crossref","first-page":"349","DOI":"10.1007\/PL00005797","volume":"143","author":"M Tsujii","year":"2001","unstructured":"Tsujii, M.: Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143(2), 349\u2013373 (2001)","journal-title":"Invent. Math."},{"key":"9849_CR34","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/PL00009433","volume":"21","author":"A Vince","year":"1999","unstructured":"Vince, A.: Self-replicating tiles and their boundary. Discrete Comput. Geom. 21, 463\u2013476 (1999)","journal-title":"Discrete Comput. Geom."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9849-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-016-9849-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-016-9849-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,8,21]],"date-time":"2023-08-21T10:34:34Z","timestamp":1692614074000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-016-9849-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,12,19]]},"references-count":34,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2017,3]]}},"alternative-id":["9849"],"URL":"https:\/\/doi.org\/10.1007\/s00454-016-9849-4","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,12,19]]}}}