## Topological Methods in Nonlinear Analysis

### Three zutot

#### Abstract

Three topics in dynamical systems are discussed. First we deal with cascades and solve two open problems concerning, respectively, product recurrence, and uniformly rigid actions. Next we provide a new example that displays some unexpected properties of strictly ergodic actions of non-amenable groups.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 49, Number 1 (2017), 351-358.

Dates
First available in Project Euclid: 11 April 2017

https://projecteuclid.org/euclid.tmna/1491876033

Digital Object Identifier
doi:10.12775/TMNA.2016.084

Mathematical Reviews number (MathSciNet)
MR3635649

Zentralblatt MATH identifier
1379.37020

#### Citation

Glasner, Eli; Weiss, Benjamin. Three zutot. Topol. Methods Nonlinear Anal. 49 (2017), no. 1, 351--358. doi:10.12775/TMNA.2016.084. https://projecteuclid.org/euclid.tmna/1491876033

#### References

• E. Akin and S. Kolyada, Li–Yorke sensitivity, Nonlinearity 16 (2003), 1421–1433.
• J. Auslander and H. Furstenberg, Product recurrence and distal points, Trans. Amer. Math. Soc. 343, (1994), no. 1, 221–232.
• P. Dong, S. Shao and X. Ye, Product recurrent properties, disjointness and weak disjointness, Israel J. Math. 188 (2012), 463–507.
• H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, 1981.
• H. Furstenberg, H. Keynes and L. Shapiro, Prime flows in topological dynamics, Israel J. Math. 14 (1973), 26–38.
• H. Furstenberg and B. Weiss, On almost $1$–$1$ extensions, Israel J. Math. 65 (1989), 311–322.
• E. Glasner, Ergodic Theory via Joinings, Math. Surveys Monogr. 101, Amer. Math. Soc., Providence, 2003.
• E. Glasner and D. Maon, Rigidity in topological dynamics, Ergodic Theory Dynam. Systems 9 (1989), 309–320.
• E. Glasner and B. Weiss, On the construction of minimal skew products, Israel J. Math. 34 (1979), 321–336.
• ––––, A weakly mixing upside-down tower of isometric extensions, Ergodic Theory Dynam. Systems 1 (1981), 151–157.
• K. Haddad and W. Ott, Recurrence in pairs, Ergodic Theory Dynam. Systems 28 (2008), 1135–1143.
• J. James, T. Koberda, K. Lindsey, C.E. Silva and P. Speh, On ergodic transformations that are both weakly mixing and uniformly rigid, New York J. Math. 15 (2009), 393–403.
• J.L. King, A map with topological minimal self-joinings in the sense of del Junco, Ergodic Theory Dynam. Systems 10 (1990), 745–761.
• B. Weiss, Multiple recurrence and doubly minimal systems, Topological Dynamics and Applications (Minneapolis, 1995), 189–196, Contemp. Math. 215, Amer. Math. Soc., Providence, 1998.
• ––––, Minimal models for free actions, Dynamical Systems and Group Actions, 249–264, Contemp. Math. 567, Amer. Math. Soc., Providence, 2012.