March  2020, 40(3): 1257-1281. doi: 10.3934/dcds.2020077

Hausdorff dimension of a class of three-interval exchange maps

University of Maryland, Department of Mathematics, College Park, MD 20742, USA

Received  March 2018 Revised  August 2019 Published  December 2019

In [5] Bourgain proves that Sarnak's disjointness conjecture holds for a certain class of three-interval exchange maps. In the present paper we slightly improve the Diophantine condition of Bourgain and estimate the constants in the proof. We further show that the new parameter set has positive, but not full Hausdorff dimension. This, in particular, implies that the Lebesgue measure of this set is zero.

Citation: Davit Karagulyan. Hausdorff dimension of a class of three-interval exchange maps. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1257-1281. doi: 10.3934/dcds.2020077
References:
[1]

H. El AbdalaouiM. Lemanczyk and T. De La Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN, 2017 (2017), 4350-4368.  doi: 10.1093/imrn/rnw146.  Google Scholar

[2]

H. El AbdalaouiM. Lemanczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.  doi: 10.1016/j.jfa.2013.09.005.  Google Scholar

[3]

H. El AbdalaouiS. Kasjan and M. Lemanczyk, 0-1 sequences of the Thue-Morse type and Sarnak's conjecture, Proc. Amer. Math. Soc., 144 (2016), 161-176.  doi: 10.1090/proc/12683.  Google Scholar

[4]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[5]

J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.  doi: 10.1007/s11854-013-0016-z.  Google Scholar

[6]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.  Google Scholar

[7]

J. Chaika and A. Eskin, Möbius disjointness for interval exchange transformations on three intervals, J. Mod. Dyn., 14 (2019), 55-86.  doi: 10.3934/jmd.2019003.  Google Scholar

[8]

H. Davenport, On some infinite series involving arithmetical functions (Ⅱ), Quart. J. Math., 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.  Google Scholar

[9]

A. FanL. LiaoB. Wang and J. Wu, On Khintchine exponents and Lyapunov exponents of continued fractions, Ergodic Theory Dynam. Systems, 29 (2009), 73-109.  doi: 10.1017/S0143385708000138.  Google Scholar

[10]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅰ: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), 861-901.  doi: 10.5802/aif.1839.  Google Scholar

[11]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅱ: A combinatorial description of the trajectories, J. Anal. Math., 89 (2003), 239-276.  doi: 10.1007/BF02893083.  Google Scholar

[12]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅲ: Ergodic and spectral properties, J. Anal. Math., 93 (2004), 103-138.  doi: 10.1007/BF02789305.  Google Scholar

[13]

S. FerencziC. Holton and L. Zamboni, Joinings of three-interval exchange transformations, Ergodic Theory Dynam. Systems, 25 (2005), 483-502.  doi: 10.1017/S0143385704000811.  Google Scholar

[14]

S. Ferenczi and C. Mauduit, On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math, 134 (2018), 545-573.  doi: 10.1007/s11854-018-0017-z.  Google Scholar

[15]

S. Ferenczi, A generalization of the self-dual induction to every interval exchange transformation, Ann. Inst. Fourier (Grenoble), 64 (2014), 1947-2002.  doi: 10.5802/aif.2901.  Google Scholar

[16]

I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc., 37 (1941), 199-228.  doi: 10.1017/S030500410002171X.  Google Scholar

[17]

B. Green and T. Tao, The M¨obius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541–566. doi: 10.4007/annals.2012.175.2.3.  Google Scholar

[18]

M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications, 547, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9940-5.  Google Scholar

[19]

I. Jarnik, Zur metrischen theorie der diopahantischen approximationen, Proc. Mat. Fyz., 36 (1928), 91-106.   Google Scholar

[20]

D. Karagulyan, On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms, Ark. Mat., 53 (2015), 317-327.  doi: 10.1007/s11512-014-0208-5.  Google Scholar

[21]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47, (1986), 223–225. doi: 10.1007/BF01949145.  Google Scholar

[22]

A. Katok and A. Stepin, Approximations in ergodic theory, Uspehi Math. Nauk, 22 (1967), 81-106.  doi: 10.1070/RM1967v022n05ABEH001227.  Google Scholar

[23]

M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.  Google Scholar

[24]

A. Y. Khinchin, Three Pearls of Number Theory, Graylock Press, Rochester, NY, 1952, 184–185.  Google Scholar

[25]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[26]

P. Sarnak, Three Lectures on the Möbius Function Randomness and Dynamics., Available from: http://www.math.ias.edu/files/wam/2011/PSMobius.pdf.  Google Scholar

[27]

D. Schleicher, Hausdorff dimension, its properties, and its surprises, Amer. Math. Monthly, 114 (2007), 509-528.  doi: 10.1080/00029890.2007.11920440.  Google Scholar

[28]

I. M. Vinogradov, Some theorems concerning the theory of primes, Math. Sb. N. S., 2 (1937), 179-195.   Google Scholar

[29]

A. Zygmund, Trigonometric Series. Vol. I, II, Cambridge University Press, Cambridge, 1988.  Google Scholar

show all references

References:
[1]

H. El AbdalaouiM. Lemanczyk and T. De La Rue, Automorphisms with quasi-discrete spectrum, multiplicative functions and average orthogonality along short intervals, Int. Math. Res. Not. IMRN, 2017 (2017), 4350-4368.  doi: 10.1093/imrn/rnw146.  Google Scholar

[2]

H. El AbdalaouiM. Lemanczyk and T. de la Rue, On spectral disjointness of powers for rank-one transformations and Möbius orthogonality, J. Funct. Anal., 266 (2014), 284-317.  doi: 10.1016/j.jfa.2013.09.005.  Google Scholar

[3]

H. El AbdalaouiS. Kasjan and M. Lemanczyk, 0-1 sequences of the Thue-Morse type and Sarnak's conjecture, Proc. Amer. Math. Soc., 144 (2016), 161-176.  doi: 10.1090/proc/12683.  Google Scholar

[4]

J. Bourgain, P. Sarnak and T. Ziegler, Disjointness of Möbius from horocycle flows, in From Fourier Analysis and Number Theory to Radon Transforms and Geometry, Dev. Math., 28, Springer, New York, 2013, 67–83. doi: 10.1007/978-1-4614-4075-8_5.  Google Scholar

[5]

J. Bourgain, On the correlation of the Moebius function with rank-one systems, J. Anal. Math., 120 (2013), 105-130.  doi: 10.1007/s11854-013-0016-z.  Google Scholar

[6]

F. Cellarosi and Y. G. Sinai, Ergodic properties of square-free numbers, J. Eur. Math. Soc., 15 (2013), 1343-1374.  doi: 10.4171/JEMS/394.  Google Scholar

[7]

J. Chaika and A. Eskin, Möbius disjointness for interval exchange transformations on three intervals, J. Mod. Dyn., 14 (2019), 55-86.  doi: 10.3934/jmd.2019003.  Google Scholar

[8]

H. Davenport, On some infinite series involving arithmetical functions (Ⅱ), Quart. J. Math., 8 (1937), 313-320.  doi: 10.1093/qmath/os-8.1.313.  Google Scholar

[9]

A. FanL. LiaoB. Wang and J. Wu, On Khintchine exponents and Lyapunov exponents of continued fractions, Ergodic Theory Dynam. Systems, 29 (2009), 73-109.  doi: 10.1017/S0143385708000138.  Google Scholar

[10]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅰ: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), 861-901.  doi: 10.5802/aif.1839.  Google Scholar

[11]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅱ: A combinatorial description of the trajectories, J. Anal. Math., 89 (2003), 239-276.  doi: 10.1007/BF02893083.  Google Scholar

[12]

S. FerencziC. Holton and L. Zamboni, Structure of three-interval exchange transformations. Ⅲ: Ergodic and spectral properties, J. Anal. Math., 93 (2004), 103-138.  doi: 10.1007/BF02789305.  Google Scholar

[13]

S. FerencziC. Holton and L. Zamboni, Joinings of three-interval exchange transformations, Ergodic Theory Dynam. Systems, 25 (2005), 483-502.  doi: 10.1017/S0143385704000811.  Google Scholar

[14]

S. Ferenczi and C. Mauduit, On Sarnak's conjecture and Veech's question for interval exchanges, J. Anal. Math, 134 (2018), 545-573.  doi: 10.1007/s11854-018-0017-z.  Google Scholar

[15]

S. Ferenczi, A generalization of the self-dual induction to every interval exchange transformation, Ann. Inst. Fourier (Grenoble), 64 (2014), 1947-2002.  doi: 10.5802/aif.2901.  Google Scholar

[16]

I. J. Good, The fractional dimensional theory of continued fractions, Proc. Cambridge Philos. Soc., 37 (1941), 199-228.  doi: 10.1017/S030500410002171X.  Google Scholar

[17]

B. Green and T. Tao, The M¨obius function is strongly orthogonal to nilsequences, Ann. of Math. (2), 175 (2012), 541–566. doi: 10.4007/annals.2012.175.2.3.  Google Scholar

[18]

M. Iosifescu and C. Kraaikamp, Metrical Theory of Continued Fractions, Mathematics and Its Applications, 547, Kluwer Academic Publishers, Dordrecht, 2002. doi: 10.1007/978-94-015-9940-5.  Google Scholar

[19]

I. Jarnik, Zur metrischen theorie der diopahantischen approximationen, Proc. Mat. Fyz., 36 (1928), 91-106.   Google Scholar

[20]

D. Karagulyan, On Möbius orthogonality for interval maps of zero entropy and orientation-preserving circle homeomorphisms, Ark. Mat., 53 (2015), 317-327.  doi: 10.1007/s11512-014-0208-5.  Google Scholar

[21]

I. Kátai, A remark on a theorem of H. Daboussi, Acta Math. Hungar., 47, (1986), 223–225. doi: 10.1007/BF01949145.  Google Scholar

[22]

A. Katok and A. Stepin, Approximations in ergodic theory, Uspehi Math. Nauk, 22 (1967), 81-106.  doi: 10.1070/RM1967v022n05ABEH001227.  Google Scholar

[23]

M. S. Keane, Interval exchange transformations, Math. Z., 141 (1975), 25-31.  doi: 10.1007/BF01236981.  Google Scholar

[24]

A. Y. Khinchin, Three Pearls of Number Theory, Graylock Press, Rochester, NY, 1952, 184–185.  Google Scholar

[25]

P. Sarnak, Möbius randomness and dynamics, Not. S. Afr. Math. Soc., 43 (2012), 89-97.   Google Scholar

[26]

P. Sarnak, Three Lectures on the Möbius Function Randomness and Dynamics., Available from: http://www.math.ias.edu/files/wam/2011/PSMobius.pdf.  Google Scholar

[27]

D. Schleicher, Hausdorff dimension, its properties, and its surprises, Amer. Math. Monthly, 114 (2007), 509-528.  doi: 10.1080/00029890.2007.11920440.  Google Scholar

[28]

I. M. Vinogradov, Some theorems concerning the theory of primes, Math. Sb. N. S., 2 (1937), 179-195.   Google Scholar

[29]

A. Zygmund, Trigonometric Series. Vol. I, II, Cambridge University Press, Cambridge, 1988.  Google Scholar

Figure 1.  Khintchine spectrum
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