American Institute of Mathematical Sciences

Advanced
  • Previous Article
    A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian
  • DCDS Home
  • This Issue
  • Next Article
    Variational properties and linear stabilities of spatial isosceles orbits in the equal-mass three-body problem
August  2017, 37(7): 4019-4034. doi: 10.3934/dcds.2017170

Typical points and families of expanding interval mappings

Tomas Persson

Centre for Mathematical Sciences, Lund University, Box 118,221 00 Lund, Sweden

Received  November 2015 Revised  March 2017 Published  April 2017

Fund Project: The author thanks D. Schnellmann for useful comments.

We study parametrised families of piecewise expanding interval mappings $T_a \colon [0,1] \to [0,1]$ with absolutely continuous invariant measures $\mu_a$ and give sufficient conditions for a point $X(a)$ to be typical with respect to $(T_a, \mu_a)$ for almost all parameters a. This is similar to a result by D.Schnellmann, but with different assumptions.

Keywords: Piecewise expanding interval mappings, typical points, invariant measures, families of mappings, β-transformations.
Mathematics Subject Classification: Primary: 37E05, 37A10; Secondary: 37D20.
Citation: Tomas Persson. Typical points and families of expanding interval mappings. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 4019-4034. doi: 10.3934/dcds.2017170
References:
[1]

M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  doi: 10.1016/S0019-3577(09)00017-2.  Google Scholar

[2]

Z. Kowalski, Invariant measure for piecewise monotonic transformation has a positive lower bound on its support, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 53-57.   Google Scholar

[3]

C. Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys., 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.  Google Scholar

[4]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.   Google Scholar

[5]

D. Schnellmann, Typical points for one-parameter families of piecewise expanding maps of the interval, Discrete Contin. Dyn. Syst., 31 (2011), 877-911.  doi: 10.3934/dcds.2011.31.877.  Google Scholar

[6]

D. Schnellmann, Law of iterated logarithm and invariance principle for one-parameter families of interval maps, Probab. Theory Related Fields, 162 (2015), 365-409.  doi: 10.1007/s00440-014-0575-7.  Google Scholar

[7]

G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324.  doi: 10.1007/BF00538119.  Google Scholar

[8]

S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc., 246 (1978), 493-500.  doi: 10.1090/S0002-9947-1978-0515555-9.  Google Scholar

show all references

References:
[1]

M. Björklund and D. Schnellmann, Almost sure equidistribution in expansive families, Indag. Math. (N.S.), 20 (2009), 167-177.  doi: 10.1016/S0019-3577(09)00017-2.  Google Scholar

[2]

Z. Kowalski, Invariant measure for piecewise monotonic transformation has a positive lower bound on its support, Bull. Acad. Polon. Sci. Sér. Sci. Math., 27 (1979), 53-57.   Google Scholar

[3]

C. Liverani, Decay of correlations for piecewise expanding maps, J. Statist. Phys., 78 (1995), 1111-1129.  doi: 10.1007/BF02183704.  Google Scholar

[4]

M. Rychlik, Bounded variation and invariant measures, Studia Math., 76 (1983), 69-80.   Google Scholar

[5]

D. Schnellmann, Typical points for one-parameter families of piecewise expanding maps of the interval, Discrete Contin. Dyn. Syst., 31 (2011), 877-911.  doi: 10.3934/dcds.2011.31.877.  Google Scholar

[6]

D. Schnellmann, Law of iterated logarithm and invariance principle for one-parameter families of interval maps, Probab. Theory Related Fields, 162 (2015), 365-409.  doi: 10.1007/s00440-014-0575-7.  Google Scholar

[7]

G. Wagner, The ergodic behaviour of piecewise monotonic transformations, Z. Wahrsch. Verw. Gebiete, 46 (1979), 317-324.  doi: 10.1007/BF00538119.  Google Scholar

[8]

S. Wong, Some metric properties of piecewise monotonic mappings of the unit interval, Trans. Amer. Math. Soc., 246 (1978), 493-500.  doi: 10.1090/S0002-9947-1978-0515555-9.  Google Scholar

Figure 1.  An example of a mapping T for which the assumptions in Corollary 1 are satisfied for $T_a (x) = T(ax)$, for all parameters in some interval $[1,a_1]$, $a_1 > 1$. Here we have taken $\delta = 2/5$. Assumption 6 is then that $\inf |T_a'| > 7/2$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  An illustration of the action of $E_s$ with $s = 1.2$. The dashed lines show the original graph.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Daniel Schnellmann. Typical points for one-parameter families of piecewise expanding maps of the interval. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 877-911. doi: 10.3934/dcds.2011.31.877
[2]

Jiu Ding, Aihui Zhou. Absolutely continuous invariant measures for piecewise $C^2$ and expanding mappings in higher dimensions. Discrete & Continuous Dynamical Systems, 2000, 6 (2) : 451-458. doi: 10.3934/dcds.2000.6.451
[3]

Salvador Addas-Zanata. Stability for the vertical rotation interval of twist mappings. Discrete & Continuous Dynamical Systems, 2006, 14 (4) : 631-642. doi: 10.3934/dcds.2006.14.631
[4]

Peng Huang, Xiong Li, Bin Liu. Invariant curves of smooth quasi-periodic mappings. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 131-154. doi: 10.3934/dcds.2018006
[5]

Jon Chaika. Hausdorff dimension for ergodic measures of interval exchange transformations. Journal of Modern Dynamics, 2008, 2 (3) : 457-464. doi: 10.3934/jmd.2008.2.457
[6]

Adrian Petruşel, Radu Precup, Marcel-Adrian Şerban. On the approximation of fixed points for non-self mappings on metric spaces. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 733-747. doi: 10.3934/dcdsb.2019264
[7]

Yun Yang. Horseshoes for $\mathcal{C}^{1+\alpha}$ mappings with hyperbolic measures. Discrete & Continuous Dynamical Systems, 2015, 35 (10) : 5133-5152. doi: 10.3934/dcds.2015.35.5133
[8]

Christopher Bose, Rua Murray. Minimum 'energy' approximations of invariant measures for nonsingular transformations. Discrete & Continuous Dynamical Systems, 2006, 14 (3) : 597-615. doi: 10.3934/dcds.2006.14.597
[9]

Lianpeng Yang, Xiong Li. Existence of periodically invariant tori on resonant surfaces for twist mappings. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1389-1409. doi: 10.3934/dcds.2020081
[10]

Xavier Bressaud. Expanding interval maps with intermittent behaviour, physical measures and time scales. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 517-546. doi: 10.3934/dcds.2004.11.517
[11]

Fawwaz Batayneh, Cecilia González-Tokman. On the number of invariant measures for random expanding maps in higher dimensions. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5887-5914. doi: 10.3934/dcds.2021100
[12]

Fengbo Hang, Fanghua Lin. Topology of Sobolev mappings IV. Discrete & Continuous Dynamical Systems, 2005, 13 (5) : 1097-1124. doi: 10.3934/dcds.2005.13.1097
[13]

Akhtam Dzhalilov, Isabelle Liousse, Dieter Mayer. Singular measures of piecewise smooth circle homeomorphisms with two break points. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 381-403. doi: 10.3934/dcds.2009.24.381
[14]

Prof. Dr.rer.nat Widodo. Topological entropy of shift function on the sequences space induced by expanding piecewise linear transformations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 191-208. doi: 10.3934/dcds.2002.8.191
[15]

Byung-Soo Lee. A convergence theorem of common fixed points of a countably infinite family of asymptotically quasi-$f_i$-expansive mappings in convex metric spaces. Numerical Algebra, Control & Optimization, 2013, 3 (3) : 557-565. doi: 10.3934/naco.2013.3.557
[16]

Huiyan Xue, Antonella Zanna. Generating functions and volume preserving mappings. Discrete & Continuous Dynamical Systems, 2014, 34 (3) : 1229-1249. doi: 10.3934/dcds.2014.34.1229
[17]

Valentin Afraimovich, Maurice Courbage, Lev Glebsky. Directional complexity and entropy for lift mappings. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3385-3401. doi: 10.3934/dcdsb.2015.20.3385
[18]

Dariusz Bugajewski, Piotr Kasprzak. On mappings of higher order and their applications to nonlinear equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 627-647. doi: 10.3934/cpaa.2012.11.627
[19]

Alexei Pokrovskii, Oleg Rasskazov. Structure of index sequences for mappings with an asymptotic derivative. Discrete & Continuous Dynamical Systems, 2007, 17 (3) : 653-670. doi: 10.3934/dcds.2007.17.653
[20]

Jiawei Chen, Guangmin Wang, Xiaoqing Ou, Wenyan Zhang. Continuity of solutions mappings of parametric set optimization problems. Journal of Industrial & Management Optimization, 2020, 16 (1) : 25-36. doi: 10.3934/jimo.2018138

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (55)
  • HTML views (50)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences