• Previous Article
    Non topologically weakly mixing interval exchanges
  • DCDS Home
  • This Issue
  • Next Article
    On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary
August  2010, 27(3): 1059-1078. doi: 10.3934/dcds.2010.27.1059

Countable inverse limits of postcritical $w$-limit sets of unimodal maps

1. 

School of Mathematics and Statistics, University of Birmingham, Birmingham, B15 2TT, United Kingdom

2. 

Mathematical Institute, University of Oxford, Oxford, OX1 3LB, United Kingdom

3. 

Department of Mathematics, Baylor University, Waco, TX 76798–7328, United States

Received  February 2009 Revised  February 2010 Published  March 2010

Let $f$ be a unimodal map of the interval with critical point $c$. If the orbit of $c$ is not dense then most points in lim{[0, 1], f} have neighborhoods that are homeomorphic with the product of a Cantor set and an open arc. The points without this property are called inhomogeneities, and the set, I, of inhomogeneities is equal to lim {w(c), f|w(c)}. In this paper we consider the relationship between the limit complexity of $w(c)$ and the limit complexity of I. We show that if $w(c)$ is more complicated than a finite collection of convergent sequences then I can have arbitrarily high limit complexity. We give a complete description of the limit complexity of I for any possible $\w(c)$.
Citation: Chris Good, Robin Knight, Brian Raines. Countable inverse limits of postcritical $w$-limit sets of unimodal maps. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1059-1078. doi: 10.3934/dcds.2010.27.1059
[1]

Sina Greenwood, Rolf Suabedissen. 2-manifolds and inverse limits of set-valued functions on intervals. Discrete and Continuous Dynamical Systems, 2017, 37 (11) : 5693-5706. doi: 10.3934/dcds.2017246

[2]

Henk Bruin, Sonja Štimac. On isotopy and unimodal inverse limit spaces. Discrete and Continuous Dynamical Systems, 2012, 32 (4) : 1245-1253. doi: 10.3934/dcds.2012.32.1245

[3]

Eduardo Liz, Gergely Röst. On the global attractor of delay differential equations with unimodal feedback. Discrete and Continuous Dynamical Systems, 2009, 24 (4) : 1215-1224. doi: 10.3934/dcds.2009.24.1215

[4]

Claudio Canuto, Anna Cattani. The derivation of continuum limits of neuronal networks with gap-junction couplings. Networks and Heterogeneous Media, 2014, 9 (1) : 111-133. doi: 10.3934/nhm.2014.9.111

[5]

Ana Anušić, Henk Bruin, Jernej Činč. Uncountably many planar embeddings of unimodal inverse limit spaces. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2285-2300. doi: 10.3934/dcds.2017100

[6]

Marco Cicalese, Antonio DeSimone, Caterina Ida Zeppieri. Discrete-to-continuum limits for strain-alignment-coupled systems: Magnetostrictive solids, ferroelectric crystals and nematic elastomers. Networks and Heterogeneous Media, 2009, 4 (4) : 667-708. doi: 10.3934/nhm.2009.4.667

[7]

Michihiro Hirayama. Periodic probability measures are dense in the set of invariant measures. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1185-1192. doi: 10.3934/dcds.2003.9.1185

[8]

Jawad Al-Khal, Henk Bruin, Michael Jakobson. New examples of S-unimodal maps with a sigma-finite absolutely continuous invariant measure. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 35-61. doi: 10.3934/dcds.2008.22.35

[9]

Kazuhiro Kawamura. Mean dimension of shifts of finite type and of generalized inverse limits. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4767-4775. doi: 10.3934/dcds.2020200

[10]

Manuel Fernández-Martínez. A real attractor non admitting a connected feasible open set. Discrete and Continuous Dynamical Systems - S, 2019, 12 (4&5) : 723-725. doi: 10.3934/dcdss.2019046

[11]

Tibor Krisztin. The unstable set of zero and the global attractor for delayed monotone positive feedback. Conference Publications, 2001, 2001 (Special) : 229-240. doi: 10.3934/proc.2001.2001.229

[12]

Zeqi Zhu, Caidi Zhao. Pullback attractor and invariant measures for the three-dimensional regularized MHD equations. Discrete and Continuous Dynamical Systems, 2018, 38 (3) : 1461-1477. doi: 10.3934/dcds.2018060

[13]

Nils Ackermann, Thomas Bartsch, Petr Kaplický. An invariant set generated by the domain topology for parabolic semiflows with small diffusion. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 613-626. doi: 10.3934/dcds.2007.18.613

[14]

Jiangfeng Huang, Zhiliang Deng, Liwei Xu. A Bayesian level set method for an inverse medium scattering problem in acoustics. Inverse Problems and Imaging, 2021, 15 (5) : 1077-1097. doi: 10.3934/ipi.2021029

[15]

Wangtao Lu, Shingyu Leung, Jianliang Qian. An improved fast local level set method for three-dimensional inverse gravimetry. Inverse Problems and Imaging, 2015, 9 (2) : 479-509. doi: 10.3934/ipi.2015.9.479

[16]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems and Imaging, 2021, 15 (3) : 387-413. doi: 10.3934/ipi.2020073

[17]

Tan Bui-Thanh, Quoc P. Nguyen. FEM-based discretization-invariant MCMC methods for PDE-constrained Bayesian inverse problems. Inverse Problems and Imaging, 2016, 10 (4) : 943-975. doi: 10.3934/ipi.2016028

[18]

Kaijen Cheng, Kenneth Palmer, Yuh-Jenn Wu. Period 3 and chaos for unimodal maps. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 1933-1949. doi: 10.3934/dcds.2014.34.1933

[19]

Christopher Cleveland. Rotation sets for unimodal maps of the interval. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 617-632. doi: 10.3934/dcds.2003.9.617

[20]

Andrei Fursikov. Local existence theorems with unbounded set of input data and unboundedness of stable invariant manifolds for 3D Navier-Stokes equations. Discrete and Continuous Dynamical Systems - S, 2010, 3 (2) : 269-289. doi: 10.3934/dcdss.2010.3.269

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (90)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]