# American Institute of Mathematical Sciences

• Previous Article
Ground state solutions for Hamiltonian elliptic system with inverse square potential
• DCDS Home
• This Issue
• Next Article
Non-autonomous stochastic evolution equations in Banach spaces of martingale type 2: Strict solutions and maximal regularity
August  2017, 37(8): 4543-4563. doi: 10.3934/dcds.2017194

## Measurable sensitivity via Furstenberg families

 Department of Mathematics, University of Science and Technology of China, Hefei, Anhui 230026, China

* Corresponding author: Tao Yu

Received  September 2016 Revised  March 2017 Published  April 2017

Fund Project: The author was supported by NNSF of China (11371339,11431012,11571335)

Let $(X, T)$ be a topological dynamical system, and $\mu$ be a $T$-invariant Borel probability measure on $X$. Let $\mathcal{F}$ be a family of subsets of $\mathbb{Z}_+$. We introduce notions of $\mathcal{F}$-sensitivity for $\mu$ and block $\mathcal{F}$-sensitivity for $\mu$.

Let $\mathcal{F}_t$ (resp. $\mathcal{F}_{ip}$) be the families consisting of thick sets (resp. IP-sets). The following Auslander-Yorke's type dichotomy theorems are obtained: (1) a minimal system is either $\mathcal{F}_{t}$-sensitive for $\mu$ or an almost one-to-one extension of its maximal equicontinous factor. (2) a minimal system is either block $\mathcal{F}_{t}$-sensitive for $\mu$ or a proximal extension of its maximal equicontinous factor. (3) a minimal system is either block $\mathcal{F}_{ip}$-sensitive for $\mu$ or an almost one-to-one extension of its $\infty$-step nilfactor.

We also introduce the notion of topological $l$-sensitivity, and construct a minimal system which is $l$-sensitive but not $(l+1)$-sensitive for $l\in\mathbb{N}$.

Citation: Tao Yu. Measurable sensitivity via Furstenberg families. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4543-4563. doi: 10.3934/dcds.2017194
##### References:

show all references

##### References:
$B^{(1)}, B^{(2)}$
 [1] Jisang Yoo. Decomposition of infinite-to-one factor codes and uniqueness of relative equilibrium states. Journal of Modern Dynamics, 2018, 13: 271-284. doi: 10.3934/jmd.2018021 [2] Alexander Vladimirov. Equicontinuous sweeping processes. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 565-573. doi: 10.3934/dcdsb.2013.18.565 [3] H. Thomas Banks, Kidist Bekele-Maxwell, Lorena Bociu, Marcella Noorman, Giovanna Guidoboni. Local sensitivity via the complex-step derivative approximation for 1D Poro-Elastic and Poro-Visco-Elastic models. Mathematical Control & Related Fields, 2019, 9 (4) : 623-642. doi: 10.3934/mcrf.2019044 [4] Peng Sun. Minimality and gluing orbit property. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4041-4056. doi: 10.3934/dcds.2019162 [5] Gabriel Ponce, Ali Tahzibi, Régis Varão. Minimal yet measurable foliations. Journal of Modern Dynamics, 2014, 8 (1) : 93-107. doi: 10.3934/jmd.2014.8.93 [6] Jie Li. How chaotic is an almost mean equicontinuous system?. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4727-4744. doi: 10.3934/dcds.2018208 [7] Ivan Dynnikov, Alexandra Skripchenko. Minimality of interval exchange transformations with restrictions. Journal of Modern Dynamics, 2017, 11: 219-248. doi: 10.3934/jmd.2017010 [8] Gabriel Núñez, Jana Rodriguez Hertz. Minimality and stable Bernoulliness in dimension 3. Discrete & Continuous Dynamical Systems - A, 2020, 40 (3) : 1879-1887. doi: 10.3934/dcds.2020097 [9] Kamil Rajdl, Petr Lansky. Fano factor estimation. Mathematical Biosciences & Engineering, 2014, 11 (1) : 105-123. doi: 10.3934/mbe.2014.11.105 [10] Felipe García-Ramos, Brian Marcus. Mean sensitive, mean equicontinuous and almost periodic functions for dynamical systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (2) : 729-746. doi: 10.3934/dcds.2019030 [11] Alba Málaga Sabogal, Serge Troubetzkoy. Minimality of the Ehrenfest wind-tree model. Journal of Modern Dynamics, 2016, 10: 209-228. doi: 10.3934/jmd.2016.10.209 [12] Petr Kůrka. Minimality in iterative systems of Möbius transformations. Conference Publications, 2011, 2011 (Special) : 903-912. doi: 10.3934/proc.2011.2011.903 [13] Piotr Oprocha. Double minimality, entropy and disjointness with all minimal systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 263-275. doi: 10.3934/dcds.2019011 [14] Marc Rauch. Variational principles for the topological pressure of measurable potentials. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 367-394. doi: 10.3934/dcdss.2017018 [15] Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098 [16] Xiaoyu Xing, Hailiang Yang. American type geometric step options. Journal of Industrial & Management Optimization, 2013, 9 (3) : 549-560. doi: 10.3934/jimo.2013.9.549 [17] Kai-Uwe Schmidt, Jonathan Jedwab, Matthew G. Parker. Two binary sequence families with large merit factor. Advances in Mathematics of Communications, 2009, 3 (2) : 135-156. doi: 10.3934/amc.2009.3.135 [18] Ryusuke Kon. Dynamics of competitive systems with a single common limiting factor. Mathematical Biosciences & Engineering, 2015, 12 (1) : 71-81. doi: 10.3934/mbe.2015.12.71 [19] Ke Ruan, Masao Fukushima. Robust portfolio selection with a combined WCVaR and factor model. Journal of Industrial & Management Optimization, 2012, 8 (2) : 343-362. doi: 10.3934/jimo.2012.8.343 [20] Yanming Ge. Analysis of airline seat control with region factor. Journal of Industrial & Management Optimization, 2012, 8 (2) : 363-378. doi: 10.3934/jimo.2012.8.363

2018 Impact Factor: 1.143