March  2022, 42(3): 1105-1126. doi: 10.3934/dcds.2021148

Zero-dimensional and symbolic extensions of topological flows

1. 

CNRS, Sorbonne Université, LPSM, 75005 Paris, France

2. 

Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warszawa, Poland

Received  April 2021 Revised  July 2021 Published  March 2022 Early access  October 2021

A zero-dimensional (resp. symbolic) flow is a suspension flow over a zero-dimensional system (resp. a subshift). We show that any topological flow admits a principal extension by a zero-dimensional flow. Following [6] we deduce that any topological flow admits an extension by a symbolic flow if and only if its time-$ t $ map admits an extension by a subshift for any $ t\neq 0 $. Moreover the existence of such an extension is preserved under orbit equivalence for regular topological flows, but this property does not hold for singular flows. Finally we investigate symbolic extensions for singular suspension flows. In particular, the suspension flow over the full shift on $ \{0,1\}^{\mathbb Z} $ with a roof function $ f $ vanishing at the zero sequence $ 0^\infty $ admits a principal symbolic extension or not depending on the smoothness of $ f $ at $ 0^\infty $.

Citation: David Burguet, Ruxi Shi. Zero-dimensional and symbolic extensions of topological flows. Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1105-1126. doi: 10.3934/dcds.2021148
References:
[1]

L. M. Abramov, On the entropy of a flow, Dokl. Akad. Nauk SSSR, 128 (1959), 873-875. 

[2]

M. Beboutoff and W. Stepanoff, Sur la mesure invariante dans les systemes dynamiques qui ne different que par le temps, Rec. Math. [Mat. Sbornik] N.S., 7 (1940), 143-166. 

[3]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[4]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161.  doi: 10.1007/s00222-003-0335-2.

[5]

M. BoyleD. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757.  doi: 10.1515/form.2002.031.

[6]

D. Burguet, Symbolic extensions and uniform generators for topological regular flows, J. Differential Equations, 267 (2019), 4320-4372.  doi: 10.1016/j.jde.2019.05.001.

[7]

D. Burguet and T. Downarowicz, Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits, J. Dynam. Differential Equations, 31 (2019), 815-852.  doi: 10.1007/s10884-018-9674-y.

[8]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.  doi: 10.1007/BF02787825.

[9] T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, 18. Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511976155.
[10]

T. Downarowicz and D. Huczek, Zero-dimensional principal extensions, Acta Appl. Math., 126 (2013), 117-129.  doi: 10.1007/s10440-013-9810-y.

[11]

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.

[12]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., (1999), 227–262.

[13]

T. Ohno, A weak equivalence and topological entropy, Publ. Res. Inst. Math. Sci., 16 (1980), 289-298.  doi: 10.2977/prims/1195187508.

[14]

W. Sun and C. Zhang, Zero entropy versus infinite entropy, Discrete Contin. Dyn. Syst., 30 (2011), 1237-1242.  doi: 10.3934/dcds.2011.30.1237.

show all references

References:
[1]

L. M. Abramov, On the entropy of a flow, Dokl. Akad. Nauk SSSR, 128 (1959), 873-875. 

[2]

M. Beboutoff and W. Stepanoff, Sur la mesure invariante dans les systemes dynamiques qui ne different que par le temps, Rec. Math. [Mat. Sbornik] N.S., 7 (1940), 143-166. 

[3]

R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.  doi: 10.1016/0022-0396(72)90013-7.

[4]

M. Boyle and T. Downarowicz, The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161.  doi: 10.1007/s00222-003-0335-2.

[5]

M. BoyleD. Fiebig and U. Fiebig, Residual entropy, conditional entropy and subshift covers, Forum Math., 14 (2002), 713-757.  doi: 10.1515/form.2002.031.

[6]

D. Burguet, Symbolic extensions and uniform generators for topological regular flows, J. Differential Equations, 267 (2019), 4320-4372.  doi: 10.1016/j.jde.2019.05.001.

[7]

D. Burguet and T. Downarowicz, Uniform generators, symbolic extensions with an embedding, and structure of periodic orbits, J. Dynam. Differential Equations, 31 (2019), 815-852.  doi: 10.1007/s10884-018-9674-y.

[8]

T. Downarowicz, Entropy structure, J. Anal. Math., 96 (2005), 57-116.  doi: 10.1007/BF02787825.

[9] T. Downarowicz, Entropy in Dynamical Systems, New Mathematical Monographs, 18. Cambridge University Press, Cambridge, 2011.  doi: 10.1017/CBO9780511976155.
[10]

T. Downarowicz and D. Huczek, Zero-dimensional principal extensions, Acta Appl. Math., 126 (2013), 117-129.  doi: 10.1007/s10440-013-9810-y.

[11]

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.

[12]

E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., (1999), 227–262.

[13]

T. Ohno, A weak equivalence and topological entropy, Publ. Res. Inst. Math. Sci., 16 (1980), 289-298.  doi: 10.2977/prims/1195187508.

[14]

W. Sun and C. Zhang, Zero entropy versus infinite entropy, Discrete Contin. Dyn. Syst., 30 (2011), 1237-1242.  doi: 10.3934/dcds.2011.30.1237.

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