doi: 10.3934/era.2021015

Ergodic measures of intermediate entropy for affine transformations of nilmanifolds

CAS Wu Wen-Tsun Key Laboratory of Mathematics, and Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

Received  August 2020 Revised  January 2021 Published  February 2021

In this paper we study ergodic measures of intermediate entropy for affine transformations of nilmanifolds. We prove that if an affine transformation $ \tau $ of nilmanifold has a periodic point, then for every $ a\in[0, h_{top}(\tau)] $ there exists an ergodic measure $ \mu_a $ of $ \tau $ such that $ h_{\mu_a}(\tau) = a $.

Citation: Wen Huang, Leiye Xu, Shengnan Xu. Ergodic measures of intermediate entropy for affine transformations of nilmanifolds. Electronic Research Archive, doi: 10.3934/era.2021015
References:
[1]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[2]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, Ergodic Theory Dynam. Systems, 40 (2020), 2098-2115.  doi: 10.1017/etds.2019.7.  Google Scholar

[3]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125–161. doi: 10.1007/BF02773637.  Google Scholar

[4]

N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb{Z}^d$-systems via specification and beyond, arXiv: 1903.05716. Google Scholar

[5]

M.-R. Herman, Construction d'un difféomorphisme minimal d'entropie topologique non nulle, Ergodic Theory Dynam. Systems, 1 (1981), 65-76.  doi: 10.1017/S0143385700001164.  Google Scholar

[6]

W. HuangX. Ye and G. Zhang, Relative entropy tuples, relative U.P.E. and C.P.E. extensions, Israel J. Math., 158 (2007), 249-283.  doi: 10.1007/s11856-007-0013-y.  Google Scholar

[7]

A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proc. Int. Congress Math., 2 (1983), 1245-1253.   Google Scholar

[8]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.  doi: 10.1112/jlms/s2-16.3.568.  Google Scholar

[9]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.  Google Scholar

[10]

H. Li, Compact group automorphisms, addition formulas and Fuglede-Kadison determinants, Ann. of Math., 176 (2012), 303-347.  doi: 10.4007/annals.2012.176.1.5.  Google Scholar

[11]

D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, 2 (1982), 49-68.  doi: 10.1017/S0143385700009573.  Google Scholar

[12]

M. Misiurewicz, A short proof of the variational principle for a Z+N action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 147–157. Astérisque, No. 40, Soc. Math. France, Paris, 1976.  Google Scholar

[13]

A. Quas and T. Soo, Weak mixing suspension flows over shifts of finite type are universal,, J. Mod. Dyn., 6 (2012), 427-449.  doi: 10.3934/jmd.2012.6.427.  Google Scholar

[14]

A. Quas and T. Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc., 368 (2016), 4137-4170.  doi: 10.1090/tran/6489.  Google Scholar

show all references

References:
[1]

R. Bowen, Entropy for group endomorphisms and homogeneous spaces, Trans. Amer. Math. Soc., 153 (1971), 401-414.  doi: 10.1090/S0002-9947-1971-0274707-X.  Google Scholar

[2]

D. Burguet, Topological and almost Borel universality for systems with the weak specification property, Ergodic Theory Dynam. Systems, 40 (2020), 2098-2115.  doi: 10.1017/etds.2019.7.  Google Scholar

[3]

J. Buzzi, Intrinsic ergodicity of smooth interval maps, Israel J. Math., 100 (1997), 125–161. doi: 10.1007/BF02773637.  Google Scholar

[4]

N. Chandgotia and T. Meyerovitch, Borel subsystems and ergodic universality for compact $\mathbb{Z}^d$-systems via specification and beyond, arXiv: 1903.05716. Google Scholar

[5]

M.-R. Herman, Construction d'un difféomorphisme minimal d'entropie topologique non nulle, Ergodic Theory Dynam. Systems, 1 (1981), 65-76.  doi: 10.1017/S0143385700001164.  Google Scholar

[6]

W. HuangX. Ye and G. Zhang, Relative entropy tuples, relative U.P.E. and C.P.E. extensions, Israel J. Math., 158 (2007), 249-283.  doi: 10.1007/s11856-007-0013-y.  Google Scholar

[7]

A. Katok, Nonuniform hyperbolicity and structure of smooth dynamical systems, Proc. Int. Congress Math., 2 (1983), 1245-1253.   Google Scholar

[8]

F. Ledrappier and P. Walters, A relativised variational principle for continuous transformations, J. London Math. Soc., 16 (1977), 568-576.  doi: 10.1112/jlms/s2-16.3.568.  Google Scholar

[9]

A. Leibman, Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold, Ergodic Theory Dynam. Systems, 25 (2005), 201-213.  doi: 10.1017/S0143385704000215.  Google Scholar

[10]

H. Li, Compact group automorphisms, addition formulas and Fuglede-Kadison determinants, Ann. of Math., 176 (2012), 303-347.  doi: 10.4007/annals.2012.176.1.5.  Google Scholar

[11]

D. A. Lind, Dynamical properties of quasihyperbolic toral automorphisms, Ergodic Theory Dynam. Systems, 2 (1982), 49-68.  doi: 10.1017/S0143385700009573.  Google Scholar

[12]

M. Misiurewicz, A short proof of the variational principle for a Z+N action on a compact space, International Conference on Dynamical Systems in Mathematical Physics (Rennes, 1975), pp. 147–157. Astérisque, No. 40, Soc. Math. France, Paris, 1976.  Google Scholar

[13]

A. Quas and T. Soo, Weak mixing suspension flows over shifts of finite type are universal,, J. Mod. Dyn., 6 (2012), 427-449.  doi: 10.3934/jmd.2012.6.427.  Google Scholar

[14]

A. Quas and T. Soo, Ergodic universality of some topological dynamical systems, Trans. Amer. Math. Soc., 368 (2016), 4137-4170.  doi: 10.1090/tran/6489.  Google Scholar

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