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223 results on '"Bernoulli shift"'

1. On discrete properties of Bernoulli shift.

Author

Halušková, Emília and Schwartzová, Radka

Subjects
*DYNAMICAL systems, *DIRECTED graphs, *ALGEBRA
Abstract

Monounary algebras are the most simple type of an algebraic structure. Oriented graphs with one outgoing arrow from every vertex represent them. The aim of this paper is to point out the interdisciplinary relationships concerning this structure. Bernoulli shift is a paradigmatic mapping in dynamical systems. It is also called dyadic, bit shift, doubling or sawtooth. We offer a look at the properties of this mapping via monounary algebras. [ABSTRACT FROM AUTHOR]

Published
2024
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2. Equivariant maps to subshifts whose points have small stabilizers.

Author

Bernshteyn, Anton

Subjects
INFINITE groups, BOREL sets, STABBINGS (Crime)
Abstract

Let $ \Gamma $ be a countably infinite group. Given $ k \in {\mathbb{N}} $, we use $ \textsf{Free}\left({k^ \Gamma}\right) $ to denote the free part of the shift action $ \Gamma \curvearrowright k^ \Gamma $. Seward and Tucker-Drob showed that there exists a free subshift $ \mathscr{S} \subseteq \textsf{Free}\left({2^ \Gamma}\right) $ such that every free Borel action of $ \Gamma $ on a Polish space admits a Borel $ \Gamma $-equivariant map to $ \mathscr{S} $. Here we generalize this result as follows. Let $ \mathscr{S} $ be a subshift of finite type. Suppose that $ \pi \colon \textsf{Free}\left({k^ \Gamma}\right) \to \mathscr{S} $ is a continuous $ \Gamma $-equivariant map and let $ \mathrm{Stab}(\pi) $ be the set of all group elements that fix every point in the image of $ \pi $. Unless $ \pi $ is constant, $ \mathrm{Stab}(\pi) $ is a finite normal subgroup of $ \Gamma $. We prove that there exists a subshift $ \mathscr{S}' \subseteq \mathscr{S} $ such that:● the stabilizer of every point in $ \mathscr{S}' $ is $ \mathrm{Stab}(\pi) $ and● every free Borel action of $ \Gamma $ on a Polish space admits a Borel $ \Gamma $-equivariant map to $ \mathscr{S}' $.In particular, if the shift action of $ \Gamma $ on the image of $ \pi $ is faithful (i.e., if $ \mathrm{Stab}(\pi) $ is trivial), then the subshift $ \mathscr{S}' $ is free. [ABSTRACT FROM AUTHOR]

Published
2023
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3. Bernoulli shifts with bases of equal entropy are isomorphic.

Author

Seward, Brandon

Subjects
UNCERTAINTY (Information theory), ENTROPY, CARDINAL numbers, PROBABILITY theory
Abstract

We prove that if $ G $ is a countably infinite group and $ (L, \lambda) $ and $ (K, \kappa) $ are probability spaces having equal Shannon entropy, then the Bernoulli shifts $ G \curvearrowright (L^G, \lambda^G) $ and $ G \curvearrowright (K^G, \kappa^G) $ are isomorphic. This extends Ornstein's famous isomorphism theorem to all countably infinite groups. Our proof builds on a slightly weaker theorem by Lewis Bowen in 2011 that required both $ \lambda $ and $ \kappa $ have at least $ 3 $ points in their support. We furthermore produce finitary isomorphisms in the case where both $ L $ and $ K $ are finite. [ABSTRACT FROM AUTHOR]

Published
2022
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4. Harmonic models and Bernoullicity.

Author

Hayes, Ben

Abstract

We give many examples of algebraic actions which are factors of Bernoulli shifts. These include certain harmonic models over left-orderable groups of large enough growth, as well as algebraic actions associated to certain lopsided elements in any left-orderable group. For many of our examples, the acting group is amenable so these actions are Bernoulli (and not just a factor of a Bernoulli), but there is no obvious Bernoulli partition. [ABSTRACT FROM AUTHOR]

Published
2021
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5. Complicated behavior in cubic Hénon maps.

Author

Anastassiou, S.

Subjects
*PHASE space, *COUPLES
Abstract

We study a generalized Hénon map in two-dimensional space. We find a region of the phase space where the nonwandering set exists, specify parameter values for which this nonwandering set is hyperbolic, and prove that our map when restricted to a specific invariant subset is topologically conjugate to the Bernoulli three-shift. Coupling two such maps, as a result, we obtain a map in four-dimensional space and show that Bernoulli shifts also exist in this map. [ABSTRACT FROM AUTHOR]

Published
2021
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6. Non-commutative Brillouin Torus

Author

Prodan, Emil, Berestycki, Nathanaël, Series editor, Dafermos, Mihalis, Series editor, Eguchi, Tohru, Series editor, Kuniba, Atsuo, Series editor, Marcolli, Matilde, Series editor, Nachtergaele, Bruno, Series editor, and Prodan, Emil

Published
2017
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8. Strong ergodicity phenomena for Bernoulli shifts of bounded algebraic dimension.

Author

Panagiotopoulos, Aristotelis and Shani, Assaf

Subjects
*PERMUTATION groups, *MODEL theory, *ORBITS (Astronomy)
Abstract

The algebraic dimension of a Polish permutation group Q ≤ Sym (N) is the size of the largest A ⊆ N with the property that the orbit of every a ∈ A under the pointwise stabilizer of A ∖ { a } is infinite. We study the Bernoulli shift P ↷ R N for various Polish permutation groups P and we provide criteria under which the P -shift is generically ergodic relative to the injective part of the Q -shift, when Q has algebraic dimension ≤ n. We use this to show that the sequence of pairwise ⁎-reduction-incomparable equivalence relations defined in [18] is a strictly increasing sequence in the Borel reduction hierarchy. We also use our main theorem to exhibit an equivalence relation of pinned cardinal ℵ 1 + which strongly resembles the equivalence relation of pinned cardinal ℵ 1 + from [25] , but which does not Borel reduce to the latter. It remains open whether they are actually incomparable under Borel reductions. Our proofs rely on the study of symmetric models whose symmetries come from the group Q. We show that when Q is "locally finite"—e.g. when Q = Aut (M) , where M is the Fraïssé limit of a Fraïssé class satisfying the disjoint amalgamation property—the corresponding symmetric model admits a theory of supports which is analogous to that in the basic Cohen model. [ABSTRACT FROM AUTHOR]

Published
2024
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9. Further Related Problems

Author

Duarte, Pedro, Klein, Silvius, Broer, Henk, Series editor, Hasselblatt, Boris, Series editor, Duarte, Pedro, and Klein, Silvius

Published
2016
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10. Introduction

Author

Duarte, Pedro, Klein, Silvius, Broer, Henk, Series editor, Hasselblatt, Boris, Series editor, Duarte, Pedro, and Klein, Silvius

Published
2016
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11. Stationary Measures

Author

Benoist, Yves, Quint, Jean-François, Ambrosio, Luigi, Series editor, Baladi, Viviane, Series editor, Greuel, Gert-Martin, Series editor, Gromov, Misha, Series editor, Huisken, Gerhard, Series editor, Jost, Jürgen, Series editor, Kollar, Janos, Series editor, Laumon, Gérard, Series editor, Tillmann, Ulrike, Series editor, Tits, Jacques, Series editor, Zagier, Don B., Series editor, Benoist, Yves, and Quint, Jean-François

Published
2016
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12. Computing Entropy

Author

Coudène, Yves, Axler, Sheldon, Series editor, Capasso, Vincenzo, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczyński, Wojbor A., Series editor, Coudène, Yves, and Erné, Reinie, Translated by

Published
2016
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13. Entropy

Author

Coudène, Yves, Axler, Sheldon, Series editor, Capasso, Vincenzo, Series editor, Casacuberta, Carles, Series editor, MacIntyre, Angus, Series editor, Ribet, Kenneth, Series editor, Sabbah, Claude, Series editor, Süli, Endre, Series editor, Woyczyński, Wojbor A., Series editor, Coudène, Yves, and Erné, Reinie, Translated by

Published
2016
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14. Introduction

Author

Capraro, Valerio, Lupini, Martino, Morel, J.-M., Editor-in-chief, Teissier, Bernard, Editor-in-chief, De Lellis, Camillo, Series editor, di Bernardo, Mario, Series editor, Figalli, Alessio, Series editor, Khoshnevisan, Davar, Series editor, Kontoyiannis, Ioannis, Series editor, Lugosi, Gabor, Series editor, Podolskij, Mark, Series editor, Serfaty, Sylvia, Series editor, Stroppel, Catharina, Series editor, Wienhard, Anna, Series editor, Capraro, Valerio, and Lupini, Martino

Published
2015
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16. Positive entropy actions of countable groups factor onto Bernoulli shifts.

Author

Seward, Brandon

Subjects
*TOPOLOGICAL entropy, *ENTROPY (Information theory), *INFINITE groups
Abstract

We prove that if a free ergodic action of a countably infinite group has positive Rokhlin entropy (or, less generally, positive sofic entropy), then it factors onto all Bernoulli shifts of lesser or equal entropy. This extends to all countably infinite groups the well-known Sinai factor theorem from classical entropy theory. [ABSTRACT FROM AUTHOR]

Published
2020
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28. Shift maps and their variants on inverse limits with set-valued functions.

Author

Kawamura, Kazuhiro and Kennedy, Judy

Subjects
*SET-valued maps, *BINOMIAL distribution, *HOMOTOPY groups, *TOPOLOGY, *MATHEMATICAL functions, *MATHEMATICAL mappings
Abstract

Inverse limit spaces of compacta with upper semi-continuous compact set-valued functions are studied via shift maps and their variants. We give a representation of such spaces as the limits of ordinary inverse sequences, which allows us to prove some known results and their extensions in a unified scheme. Next we present a scheme to construct shift dynamics on the inverse limit space with various dynamical features. In particular we construct an inverse sequence over [ 0 , 1 ] with a single upper semi-continuous function f as its bonding function such that (i) the inverse limit space [ 0 , 1 ] f is homeomorphic to the Cantor set and (ii) the shift map σ f : [ 0 , 1 ] f → [ 0 , 1 ] f is topologically conjugate to a minimal subshift of a Bernoulli full shift. Also we study local/global connectivity of the inverse limit space over a compactum with a single upper semi-continuous bonding function in terms of homotopy/(co)homology groups, again via shift maps and their variants. [ABSTRACT FROM AUTHOR]

Published
2018
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29. Dynamical Entropy and Information

Author

Benatti, Fabio, Beiglböck, W., editor, Eckmann, J.-P., editor, Grosse, H., editor, Loss, M., editor, Smirnov, S., editor, Takhtajan, L., editor, Yngvason, J., editor, and Benatti, Fabio

Published
2009
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30. Donsker Principles

Author

Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, Prieur, Clémentine, Bickel, P., editor, Diggle, P., editor, Fienberg, S., editor, Gather, U., editor, Olkin, I., editor, Zeger, S., editor, Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, and Prieur, Clémentine

Published
2007
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31. Models

Author

Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, Prieur, Clémentine, Bickel, P., editor, Diggle, P., editor, Fienberg, S., editor, Gather, U., editor, Olkin, I., editor, Zeger, S., editor, Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, and Prieur, Clémentine

Published
2007
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32. Introduction

Author

Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, Prieur, Clémentine, Bickel, P., editor, Diggle, P., editor, Fienberg, S., editor, Gather, U., editor, Olkin, I., editor, Zeger, S., editor, Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, and Prieur, Clémentine

Published
2007
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33. Econometric applications and resampling

Author

Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, Prieur, Clémentine, Bickel, P., editor, Diggle, P., editor, Fienberg, S., editor, Gather, U., editor, Olkin, I., editor, Zeger, S., editor, Dedecker, Jérôme, Doukhan, Paul, Lang, Gabriel, José Rafael, León R., Louhichi, Sana, and Prieur, Clémentine

Published
2007
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36. Systems of Algebraic Origin

Author

Gromov, M., editor, Jost, J., editor, Kollár, J., editor, Laumon, G., editor, Lenstra, H. W., Jr., editor, Tits, J., editor, Zagier, D. B., editor, Ziegler, G. M., editor, Remmert, R., editor, Neshveyev, Sergey, and Størmer, Erling

Published
2006
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37. Binary Shifts

Author

Gromov, M., editor, Jost, J., editor, Kollár, J., editor, Laumon, G., editor, Lenstra, H. W., Jr., editor, Tits, J., editor, Zagier, D. B., editor, Ziegler, G. M., editor, Remmert, R., editor, Neshveyev, Sergey, and Størmer, Erling

Published
2006
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38. Maximality of Entropy and Commutativity

Author

Gromov, M., editor, Jost, J., editor, Kollár, J., editor, Laumon, G., editor, Lenstra, H. W., Jr., editor, Tits, J., editor, Zagier, D. B., editor, Ziegler, G. M., editor, Remmert, R., editor, Neshveyev, Sergey, and Størmer, Erling

Published
2006
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39. Dynamical Abelian Models

Author

Gromov, M., editor, Jost, J., editor, Kollár, J., editor, Laumon, G., editor, Lenstra, H. W., Jr., editor, Tits, J., editor, Zagier, D. B., editor, Ziegler, G. M., editor, Remmert, R., editor, Neshveyev, Sergey, and Størmer, Erling

Published
2006
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40. Dynamical Entropy

Author

Gromov, M., editor, Jost, J., editor, Kollár, J., editor, Laumon, G., editor, Lenstra, H. W., Jr., editor, Tits, J., editor, Zagier, D. B., editor, Ziegler, G. M., editor, Remmert, R., editor, Neshveyev, Sergey, and Størmer, Erling

Published
2006
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41. Dynamical Systems in the 1960s: The Hyperbolic Revolution

Author

Anosov, D. V., Bolibruch, A. A., editor, Osipov, Yu. S., editor, Sinai, Ya. G., editor, Arnold, V. I., editor, Bolibruch, A. A., editor, Vershik, A. M., editor, Manin, Yu. I., editor, Sinai, Ya.G., editor, Tikhomirov, V.M., editor, Faddeev, L.D., editor, and Philippov, V. B., editor

Published
2006
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42. Entropy

Author

Bronstein, Manuel, editor, Cohen, Arjeh M., editor, Cohen, Henri, editor, Eisenbud, David, editor, Sturmfels, Bernd, editor, and Choe, Geon Ho

Published
2005
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43. Recurrence and Entropy

Author

Bronstein, Manuel, editor, Cohen, Arjeh M., editor, Cohen, Henri, editor, Eisenbud, David, editor, Sturmfels, Bernd, editor, and Choe, Geon Ho

Published
2005
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45. Random Dynamics

Author

Khrennikov, Andrei Yu., Nilson, Marcus, Hazewinkel, M., editor, Khrennikov, Andrei Yu., and Nilson, Marcus

Published
2004
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47. Entropy and Complexity

Author

Gallavotti, Giovanni, Bonetto, Federico, Gentile, Guido, Balian, R., editor, Beiglböck, W., editor, Grosse, H., editor, Lieb, E. H., editor, Reshetikhin, N., editor, Spohn, H., editor, Thirring, W., editor, Gallavotti, Giovanni, Bonetto, Federico, and Gentile, Guido

Published
2004
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48. Quantum Entropies in a Classical Context

Author

Fannes, Mark, Haegeman, Bart, Beig, R., editor, Englert, B. -G., editor, Frisch, U., editor, Hänggi, P., editor, Hepp, K., editor, Hillebrandt, W., editor, Imboden, D., editor, Jaffe, R. L., editor, Lipowsky, R., editor, v. Löhneysen, H., editor, Ojima, I., editor, Sornette, D., editor, Theisen, S., editor, Weise, W., editor, Wess, J., editor, Zittartz, J., editor, Benatti, Fabio, editor, and Floreanini, Roberto, editor

Published
2003
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