Abstract:
We study “typical” metric (ergodic) properties of measure preserving homeomorphisms of regularly connected cellular polyhedra and of some other spaces. In 1941 Oxtoby and Ulam proved (for a narrower class of spaces) that ergodicity is such a property. Using a modification of their construction and the method of approximating metric automorphisms by periodic ones, we prove in this paper that almost all properties that are “typical' for the metric automorphisms of the Lebesgue spaces are also "typical” for the situation under discussion.
This publication is cited in the following 22 articles:
Jozef Bobok, Jernej Činč, Piotr Oprocha, Serge Troubetzkoy, “Continuous Lebesgue measure-preserving maps on one-dimensional manifolds: A survey”, Topology and its Applications, 2024, 109101
Gabriel Lacerda, Sergio Romaña, “Typical Conservative Homeomorphisms Have Total Metric Mean Dimension”, IEEE Trans. Inform. Theory, 70:11 (2024), 7664
Stefano Bianchini, Martina Zizza, “Properties of Mixing BV Vector Fields”, Commun. Math. Phys., 402:2 (2023), 1953
Aldo Filomeno, Synthese Library, 477, Current Debates in Philosophy of Science, 2023, 391
V. V. Ryzhikov, “Compact families and typical entropy invariants of measure-preserving actions”, Trans. Moscow Math. Soc., 82 (2021), 117–123
Marlies Gerber, Philipp Kunde, “A smooth zero-entropy diffeomorphism whose product with itself is loosely Bernoulli”, JAMA, 141:2 (2020), 521
Schnurr M., “Generic Properties of Extensions”, Ergod. Theory Dyn. Syst., 39:11 (2019), PII S0143385717001456, 3144–3168
Philipp Kunde, “Smooth diffeomorphisms with homogeneous spectrum and disjointness of convolutions”, JMD, 10:02 (2016), 439
PIERRE-ANTOINE GUIHÉNEUF, “Dynamical properties of spatial discretizations of a generic homeomorphism”, Ergod. Th. Dynam. Sys, 2014, 1
V. V. Ryzhikov, “Spectral multiplicities and asymptotic operator properties of actions with invariant measure”, Sb. Math., 200:12 (2009), 1833–1845
V. V. Ryzhikov, “Weak limits of powers, simple spectrum of
symmetric products, and rank-one mixing constructions”, Sb. Math., 198:5 (2007), 733–754
J.N. Qiang, S.W. Zhong, “Clarifications on the Integration Path of Transient Energy Function”, IEEE Trans Power Syst, 20:2 (2005), 883
Bezuglyi S., Kwiatkowski J., Medynets K., “Approximation in Ergodic Theory, Borel, and Cantor Dynamics”, Algebraic and Topological Dynamics, Contemporary Mathematics Series, 385, eds. Kolyada S., Manin Y., Ward T., Amer Mathematical Soc, 2005, 39–64
Alpern S. Prasad V., “Properties Generic for Lebesgue Space Automorphisms Are Generic for Measure-Preserving Manifold Homeomorphisms”, Ergod. Theory Dyn. Syst., 22:Part 6 (2002), 1587–1620
V. I. Bogachev, “Measures on topological spaces”, Journal of Mathematical Sciences (New York), 91:4 (1998), 3033
G. R. Goodson, V. V. Ryzhikov, “Conjugations, joinings, and direct products of locally rank one dynamical systems”, J Dyn Control Syst, 3:3 (1997), 321
I. A. Vinogradova, A. G. El'kin, Yu. V. Prokhorov, B. A. Efimov, L. P. Kuptsov, N. Kh. Rozov, V. A. Oskolkov, L. D. Kudryavtsev, B. V. Khvedelidze, A. A. Zakharov, M. Sh. Tsalenko, E. D. Solomentsev, Yu. L. Ershov, I. V. Dolgachev, B. B. Venkov, A. N. Parshin, A. I. Kostrikin, A. B. Ivanov, A. P. Terekhin, V. F. Emelyanov, V. V. Sazonov, M. I. Voǐtsekhovskiǐ, I. I. Volkov, P. S. Aleksandrov, A. V. Prokhorov, A. M. Zubkov, V. N. Grishin, A. A. Danilevich, N. M. Nagornyǐ, E. G. D'yakonov, Kh. D. Ikramov, N. S. Bakhvalov, A. V. Arkhangel'skiǐ, V. V. Rumyantsev, A. V. Zarelua, A. A. Mal'tsev, O. A. Ivanova, V. P. Fedotov, I. P. Kubilyus, B. M. Bredikhin, P. L. Dobrushin, V. V. Prelov, A. V. Mikhalev, V. A. Andrunakievich, V. V. Fedorchuk, V. P. Platonov, A. P. Favorskiǐ, D. V. Anosov, V. I. Danilov, E. L. Tonkov, A. L. Onishchik, T. S. Pigolkina, T. S. Pogolkina, L. A. Skornyakov, V. I. Sobolev, I. Kh. Sabitov, V. I. Lebedev, A. V. Lykov, A., Encyclopaedia of Mathematics, 1995, 1
Steve Alpern, V. S. Prasad, “Dynamics induced on the ends of non-compact manifold”, Ergod Th Dynam Sys, 8:1 (1988)
M. Hazewinkel, Encyclopaedia of Mathematics, 1988, 1
Citation in format AMSBIB
Contact us: