# American Institute of Mathematical Sciences

March  2017, 37(3): 1227-1246. doi: 10.3934/dcds.2017051

## Functional envelopes relative to the point-open topology on a subset

 1 Department of Mathematics, Nanjing University, Nanjing 210093, China 2 School of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

* Corresponding author

Received  March 2016 Revised  October 2016 Published  December 2016

Fund Project: Project was supported partly by National Natural Science Foundation of China (Grants No. 11371380).

If $(X, f)$ is a dynamical system given by a locally compact separable metric space $X$ without isolated points and a continuous map $f : X\to X$ , and $A$ is a countable dense subset of $X$ , then by the functional envelope of $(X, f)$ relative to $\mathcal{P}_A$ we mean the dynamical system $(S_A(X), F_f)$ whose phase space $S_A(X)$ is the space of all continuous selfmaps of $X$ endowed with the point-open topology on $A$ and the map $F_f : S_A(X)\to S_A(X)$ is defined by $F_f (\varphi)=fo\varphi$ for any $\varphi∈ S_A(X)$ .

In this paper, we mainly deal with the connection between the properties of a system and the properties of its functional envelope. We show that:(1) $(X, f)$ is weakly mixing if and only if there exists a countable dense subset $A$ of $X$ so that $\big(S_A(X), F_f\big)$ has a transitive point $φ∈ S(X)$ which is surjective; (2) $(X, f)$ is sensitive if and only if $\big(S_A(X), F_f\big)$ is sensitive for every countable dense subset $A$ of $X$ . Moreover, if $(X, f)$ is weakly mixing, then $\big(S_A(X), F_f\big)$ is Auslander-Yorke chaotic for many countable dense subsets $A$ of $X$ . As an application, we consider a class of one-dimensional wave equations with van der Pol boundary condition and show that if the boundary condition is weakly mixing, then there exists an initial condition such that the solutions of the equations exhibit complicated behaviours.

Citation: Zhijing Chen, Yu Huang. Functional envelopes relative to the point-open topology on a subset. Discrete and Continuous Dynamical Systems, 2017, 37 (3) : 1227-1246. doi: 10.3934/dcds.2017051
##### References:
 [1] J. Auslander, S. Kolyada and L. Snoha, Functional envelope of a dynamical system, Nonlinearity, 20 (2007), 2245-2269.  doi: 10.1088/0951-7715/20/9/012. [2] J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, T$\widehat{o}$hoku Math. J., 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634. [3] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅰ, controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4. [4] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅱ, energy injection, period doubling and homoclinic orbits, Int. J. Bifur. Chaos, 8 (1998), 423-445.  doi: 10.1142/S0218127498000280. [5] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅲ, natural hysteresis memory effects, Int. J. Bifur. Chaos, 8 (1998), 447-470.  doi: 10.1142/S0218127498000292. [6] G. Chen, S. B. Hsu and J. Zhou, Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the spane, J. Math. Phys., 39 (1998), 6459-6489.  doi: 10.1063/1.532670. [7] G. Chen, S. B. Hsu and J. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Int. J. Bifur. Chaos, 12 (2002), 535-559.  doi: 10.1142/S0218127402004504. [8] G. Chen, T. Huang and Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Int. J. Bifur. Chaos, 14 (2004), 2161-2186.  doi: 10.1142/S0218127404010540. [9] G. Chen and Y. Huang, Chaotic Maps: Dynamics, Fractals and Rapid Fluctuations, Synthesis, Lectures on Mathematics and Statistics, ed. Steven G. Krantz, (Morgan & Claypool Publishers, Williston, VT), 2011. [10] X. Dai, Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258 (2015), 2794-2805.  doi: 10.1016/j.jde.2014.12.027. [11] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory, 28 (1992), 93-103. [12] W. Huang, S. Shao and X. Ye, Mixing and proximal cells along sequences, Nonlinearity, 17 (2004), 1245-1260.  doi: 10.1088/0951-7715/17/4/006. [13] Y. Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Int. J. Bifur. Chaos, 13 (2003), 1183-1195.  doi: 10.1142/S0218127403007138. [14] Y. Huang, J. Luo and Z. Zhou, Geometrical features for dynamics of interval mappings, preprint. [15] Y. Huang, J. Luo and Z. Zhou, Rapid fluctuations of snapshots of one dimensional linear wave equation with a Van der Pol nonlinear boundary condition, Int. J. Bifur. Chaos, 15 (2005), 567-580.  doi: 10.1142/S0218127405012223. [16] Y. Huang and Z. S. Feng, Infinite-dimensional dynamical systems induced by interval maps, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 13 (2006), 509-524. [17] S. Kolyada and L. Snoha, Some aspects of topological transitivity-a survey, Grazer Math. Ber., 334 (1997), 3-35. [18] L. Li and Y. Huang, Growth rates of total variations of snapshots of 1D linear wave equations with nonlinear right-end boundary conditions, J. Math. Anal. Appl., 361 (2010), 69-85.  doi: 10.1016/j.jmaa.2009.09.011. [19] J. Munkres, Topology 2$^{nd}$ edition, Pearson, 2000. [20] P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825.  doi: 10.3934/dcds.2011.31.797. [21] S. Ruette, Chaos for continuous interval maps: a survey of relationship between the various kinds of chaos, preprint arXiv: 1504.03001v1. [22] L. Snoha and V. Špitalský, A quantitative approach to transitivity and mixing, Chaos Solitons Fractals, 40 (2009), 958-965.  doi: 10.1016/j.chaos.2007.08.052. [23] J. Xiong and Z. Yang, Chaos caused by a topologically mixing map, Dynamical systems and related topics (Nagoya, 1990), 550–572, Adv. Ser. Dynam. Systems, 9, World Sci. Publ. , River Edge, NJ, 1991.

show all references

##### References:
 [1] J. Auslander, S. Kolyada and L. Snoha, Functional envelope of a dynamical system, Nonlinearity, 20 (2007), 2245-2269.  doi: 10.1088/0951-7715/20/9/012. [2] J. Auslander and J. A. Yorke, Interval maps, factors of maps, and chaos, T$\widehat{o}$hoku Math. J., 32 (1980), 177-188.  doi: 10.2748/tmj/1178229634. [3] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅰ, controlled hysteresis, Trans. Amer. Math. Soc., 350 (1998), 4265-4311.  doi: 10.1090/S0002-9947-98-02022-4. [4] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅱ, energy injection, period doubling and homoclinic orbits, Int. J. Bifur. Chaos, 8 (1998), 423-445.  doi: 10.1142/S0218127498000280. [5] G. Chen, S. B. Hsu and J. Zhou, Chaotic vibrations of the one-dimensional wave equation due to a self-excitation boundary condition, Part Ⅲ, natural hysteresis memory effects, Int. J. Bifur. Chaos, 8 (1998), 447-470.  doi: 10.1142/S0218127498000292. [6] G. Chen, S. B. Hsu and J. Zhou, Snapback repellers as a cause of chaotic vibration of the wave equation with a van der Pol boundary condition and energy injection at the middle of the spane, J. Math. Phys., 39 (1998), 6459-6489.  doi: 10.1063/1.532670. [7] G. Chen, S. B. Hsu and J. Zhou, Nonisotropic spatiotemporal chaotic vibration of the wave equation due to mixing energy transport and a van der Pol boundary condition, Int. J. Bifur. Chaos, 12 (2002), 535-559.  doi: 10.1142/S0218127402004504. [8] G. Chen, T. Huang and Y. Huang, Chaotic behavior of interval maps and total variations of iterates, Int. J. Bifur. Chaos, 14 (2004), 2161-2186.  doi: 10.1142/S0218127404010540. [9] G. Chen and Y. Huang, Chaotic Maps: Dynamics, Fractals and Rapid Fluctuations, Synthesis, Lectures on Mathematics and Statistics, ed. Steven G. Krantz, (Morgan & Claypool Publishers, Williston, VT), 2011. [10] X. Dai, Chaotic dynamics of continuous-time topological semiflow on Polish spaces, J. Differential Equations, 258 (2015), 2794-2805.  doi: 10.1016/j.jde.2014.12.027. [11] D. A. Herrero, Hypercyclic operators and chaos, J. Operator Theory, 28 (1992), 93-103. [12] W. Huang, S. Shao and X. Ye, Mixing and proximal cells along sequences, Nonlinearity, 17 (2004), 1245-1260.  doi: 10.1088/0951-7715/17/4/006. [13] Y. Huang, Growth rates of total variations of snapshots of the 1D linear wave equation with composite nonlinear boundary reflection relations, Int. J. Bifur. Chaos, 13 (2003), 1183-1195.  doi: 10.1142/S0218127403007138. [14] Y. Huang, J. Luo and Z. Zhou, Geometrical features for dynamics of interval mappings, preprint. [15] Y. Huang, J. Luo and Z. Zhou, Rapid fluctuations of snapshots of one dimensional linear wave equation with a Van der Pol nonlinear boundary condition, Int. J. Bifur. Chaos, 15 (2005), 567-580.  doi: 10.1142/S0218127405012223. [16] Y. Huang and Z. S. Feng, Infinite-dimensional dynamical systems induced by interval maps, Dyn. Contin. Discrete Impuls. Syst. Ser. A, 13 (2006), 509-524. [17] S. Kolyada and L. Snoha, Some aspects of topological transitivity-a survey, Grazer Math. Ber., 334 (1997), 3-35. [18] L. Li and Y. Huang, Growth rates of total variations of snapshots of 1D linear wave equations with nonlinear right-end boundary conditions, J. Math. Anal. Appl., 361 (2010), 69-85.  doi: 10.1016/j.jmaa.2009.09.011. [19] J. Munkres, Topology 2$^{nd}$ edition, Pearson, 2000. [20] P. Oprocha, Coherent lists and chaotic sets, Discrete Contin. Dyn. Syst., 31 (2011), 797-825.  doi: 10.3934/dcds.2011.31.797. [21] S. Ruette, Chaos for continuous interval maps: a survey of relationship between the various kinds of chaos, preprint arXiv: 1504.03001v1. [22] L. Snoha and V. Špitalský, A quantitative approach to transitivity and mixing, Chaos Solitons Fractals, 40 (2009), 958-965.  doi: 10.1016/j.chaos.2007.08.052. [23] J. Xiong and Z. Yang, Chaos caused by a topologically mixing map, Dynamical systems and related topics (Nagoya, 1990), 550–572, Adv. Ser. Dynam. Systems, 9, World Sci. Publ. , River Edge, NJ, 1991.
 [1] Chris Good, Robert Leek, Joel Mitchell. Equicontinuity, transitivity and sensitivity: The Auslander-Yorke dichotomy revisited. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2441-2474. doi: 10.3934/dcds.2020121 [2] Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082 [3] Davit Martirosyan. Exponential mixing for the white-forced damped nonlinear wave equation. Evolution Equations and Control Theory, 2014, 3 (4) : 645-670. doi: 10.3934/eect.2014.3.645 [4] Jean René Chazottes, F. Durand. Local rates of Poincaré recurrence for rotations and weak mixing. Discrete and Continuous Dynamical Systems, 2005, 12 (1) : 175-183. doi: 10.3934/dcds.2005.12.175 [5] Oliver Knill. Singular continuous spectrum and quantitative rates of weak mixing. Discrete and Continuous Dynamical Systems, 1998, 4 (1) : 33-42. doi: 10.3934/dcds.1998.4.33 [6] Ethan M. Ackelsberg. Rigidity, weak mixing, and recurrence in abelian groups. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1669-1705. doi: 10.3934/dcds.2021168 [7] Krzysztof Frączek, Leonid Polterovich. Growth and mixing. Journal of Modern Dynamics, 2008, 2 (2) : 315-338. doi: 10.3934/jmd.2008.2.315 [8] Anthony Quas, Terry Soo. Weak mixing suspension flows over shifts of finite type are universal. Journal of Modern Dynamics, 2012, 6 (4) : 427-449. doi: 10.3934/jmd.2012.6.427 [9] Corinna Ulcigrai. Weak mixing for logarithmic flows over interval exchange transformations. Journal of Modern Dynamics, 2009, 3 (1) : 35-49. doi: 10.3934/jmd.2009.3.35 [10] Sanchit Chaturvedi, Jonathan Luk. Phase mixing for solutions to 1D transport equation in a confining potential. Kinetic and Related Models, 2022, 15 (3) : 403-416. doi: 10.3934/krm.2022002 [11] Piotr Oprocha. Specification properties and dense distributional chaos. Discrete and Continuous Dynamical Systems, 2007, 17 (4) : 821-833. doi: 10.3934/dcds.2007.17.821 [12] Asaf Katz. On mixing and sparse ergodic theorems. Journal of Modern Dynamics, 2021, 17: 1-32. doi: 10.3934/jmd.2021001 [13] Daniel Gonçalves, Bruno Brogni Uggioni. Li-Yorke Chaos for ultragraph shift spaces. Discrete and Continuous Dynamical Systems, 2020, 40 (4) : 2347-2365. doi: 10.3934/dcds.2020117 [14] A. Crannell. A chaotic, non-mixing subshift. Conference Publications, 1998, 1998 (Special) : 195-202. doi: 10.3934/proc.1998.1998.195 [15] Zhi Lin, Katarína Boďová, Charles R. Doering. Models & measures of mixing & effective diffusion. Discrete and Continuous Dynamical Systems, 2010, 28 (1) : 259-274. doi: 10.3934/dcds.2010.28.259 [16] Rui Kuang, Xiangdong Ye. The return times set and mixing for measure preserving transformations. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 817-827. doi: 10.3934/dcds.2007.18.817 [17] Nir Avni. Spectral and mixing properties of actions of amenable groups. Electronic Research Announcements, 2005, 11: 57-63. [18] Richard Miles, Thomas Ward. A directional uniformity of periodic point distribution and mixing. Discrete and Continuous Dynamical Systems, 2011, 30 (4) : 1181-1189. doi: 10.3934/dcds.2011.30.1181 [19] Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 [20] Ralf Spatzier, Lei Yang. Exponential mixing and smooth classification of commuting expanding maps. Journal of Modern Dynamics, 2017, 11: 263-312. doi: 10.3934/jmd.2017012

2021 Impact Factor: 1.588