Define the recurrence set of Gauss transformation $T$ on the field of Laurent series as following $$E(x_0)=\{x\in I: T^n(x)\in I_{t_n}(x_0) for infinitely many $n$\},$$ where $I_{t_n}(x_0)$ denotes $t_n$-th order cylinder of $x_0$. In this paper, the Hausdorff dimension of the set $E(x_0)$ is determined.
References
E. Artin, Quadratische Köper im Gebiete der höeren Kongruenzen, I-II, Math. Z. 19 (1924), 153--46. MR1544651 10.1007/BF01181074 E. Artin, Quadratische Köper im Gebiete der höeren Kongruenzen, I-II, Math. Z. 19 (1924), 153--46. MR1544651 10.1007/BF01181074
V. Berthé, H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Exposition Math. 4 (2000), 257--284. MR1788323 1024.11050 V. Berthé, H. Nakada, On continued fraction expansions in positive characteristic: equivalence relations and some metric properties, Exposition Math. 4 (2000), 257--284. MR1788323 1024.11050
M. Fuchs, On metric Diophantine approximation in the field of formal Laurent series, Finite Fields Appl. 8 (2002), 343--368. MR1910397 1013.11034 10.1006/ffta.2001.0346 M. Fuchs, On metric Diophantine approximation in the field of formal Laurent series, Finite Fields Appl. 8 (2002), 343--368. MR1910397 1013.11034 10.1006/ffta.2001.0346
X. H. Hu et al. Cantor sets determined by partial quotients of continued fractions of Laurent series, Finite Fields Appl. 14 (2008), 417--437. MR2401985 1137.11052 10.1016/j.ffa.2007.04.002 X. H. Hu et al. Cantor sets determined by partial quotients of continued fractions of Laurent series, Finite Fields Appl. 14 (2008), 417--437. MR2401985 1137.11052 10.1016/j.ffa.2007.04.002
S. Kristensen, On well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 2 (2003), 255--268. MR2006063 1088.11056 10.1017/S0305004103006911 S. Kristensen, On well-approximable matrices over a field of formal series, Math. Proc. Cambridge Philos. Soc. 2 (2003), 255--268. MR2006063 1088.11056 10.1017/S0305004103006911
H. Niederreiter, The probabilistic theory of linear complexity, in: C.G. Gnther (Ed.), Advances in CryptologyEUROCRYPT88, Lecture Notes in Computer Science, 330 (1988), 191--209. MR994663 H. Niederreiter, The probabilistic theory of linear complexity, in: C.G. Gnther (Ed.), Advances in CryptologyEUROCRYPT88, Lecture Notes in Computer Science, 330 (1988), 191--209. MR994663
H. Niederreiter, M. Vielhaber, Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. Complexity, 3 (1997), 353--383. MR1475570 0934.94013 10.1006/jcom.1997.0451 H. Niederreiter, M. Vielhaber, Linear complexity profiles: Hausdorff dimensions for almost perfect profiles and measures for general profiles, J. Complexity, 3 (1997), 353--383. MR1475570 0934.94013 10.1006/jcom.1997.0451
B. Saussol, S. Troubetzkoy, S. Vaienti, Recurrence, dimensions and Lyapunov exponents, J. Stat. Phys. 106 (2002), 623--634. MR1884547 1138.37300 10.1023/A:1013710422755 B. Saussol, S. Troubetzkoy, S. Vaienti, Recurrence, dimensions and Lyapunov exponents, J. Stat. Phys. 106 (2002), 623--634. MR1884547 1138.37300 10.1023/A:1013710422755
W.M. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith. 95 (2000), 139--166. MR1785412 0987.11041 W.M. Schmidt, On continued fractions and Diophantine approximation in power series fields, Acta Arith. 95 (2000), 139--166. MR1785412 0987.11041
J. Wu, On the sum of degrees of digits occurring in continued fraction expansions of Laurent series, Math. Proc. Cambridge Philos. Soc. 138 (2005), 9--20. MR2127223 1062.11054 10.1017/S0305004104008163 J. Wu, On the sum of degrees of digits occurring in continued fraction expansions of Laurent series, Math. Proc. Cambridge Philos. Soc. 138 (2005), 9--20. MR2127223 1062.11054 10.1017/S0305004104008163
