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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[19]

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

[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.

[21]

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

[22]

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

[23]

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

[24]

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

[25]

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

[26]

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

[27]

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

[28]

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

[29]

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

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