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Local well-posedness for the Zakharov system in dimension d ≤ 3
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 |
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 [
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L. M. Abramov,
On the entropy of a flow, Dokl. Akad. Nauk SSSR, 128 (1959), 873-875.
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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.
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Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
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M. Boyle and T. Downarowicz,
The entropy theory of symbolic extensions, Invent. Math., 156 (2004), 119-161.
doi: 10.1007/s00222-003-0335-2. |
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M. Boyle, D. 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. |
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E. Lindenstrauss, Mean dimension, small entropy factors and an embedding theorem, Inst. Hautes Études Sci. Publ. Math., (1999), 227–262. |
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T. Ohno,
A weak equivalence and topological entropy, Publ. Res. Inst. Math. Sci., 16 (1980), 289-298.
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[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. Boyle, D. 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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