May  2019, 39(5): 2393-2412. doi: 10.3934/dcds.2019101

Critical covering maps without absolutely continuous invariant probability measure

1. 

Department of Mathematical Sciences, Xi'an Jiaotong–Liverpool University, 111 Ren'ai Road, Suzhou 215123, China

2. 

Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, São Paulo CEP 05508-090, Brazil

* Corresponding author: Simon Lloyd

Received  October 2017 Revised  October 2018 Published  January 2019

Fund Project: The first author is supported by Fundação de Amparo à Pesquisa do Estado de São Paulo grant numbers 2011/01482-3 and 2017/10106-1. The second author is supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico grant number 310749/2015-8.

We consider the dynamics of smooth covering maps of the circle with a single critical point of order greater than $ 1 $. By directly specifying the combinatorics of the critical orbit, we show that for an uncountable number of combinatorial equivalence classes of such maps, there is no periodic attractor nor an ergodic absolutely continuous invariant probability measure.

Citation: Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete & Continuous Dynamical Systems - A, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101
References:
[1]

V. I. Arnol'd, Small denominators I. Mapping the circle onto itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86.   Google Scholar

[2]

A. AvilaM. Lyubich and W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math., 154 (2003), 451-550.  doi: 10.1007/s00222-003-0307-6.  Google Scholar

[3]

H. BruinJ. Rivera-LetelierW. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.  doi: 10.1007/s00222-007-0108-4.  Google Scholar

[4]

H. BruinW. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306.  doi: 10.1007/s00220-003-0928-z.  Google Scholar

[5]

P. Collet and J. P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46.  doi: 10.1017/S0143385700001802.  Google Scholar

[6]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.E.S., 49 (1979), 5-233.   Google Scholar

[7]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys., 127 (1990), 319-337.  doi: 10.1007/BF02096761.  Google Scholar

[8]

M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81 (1981), 39-88.  doi: 10.1007/BF01941800.  Google Scholar

[9]

S. D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 173-348.  doi: 10.1007/BF01207362.  Google Scholar

[10]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.  Google Scholar

[11]

M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Math., 156 (2002), 1-78.  doi: 10.2307/3597183.  Google Scholar

[12]

R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100 (1985), 495-524.  doi: 10.1007/BF01217727.  Google Scholar

[13]

W. de Melo and S. van Strien, One-dimensional Dynamics. Ergebnisse Der Mathematik und Ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[14]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.   Google Scholar

[15]

T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math., 105 (1991), 123-136.  doi: 10.1007/BF01232258.  Google Scholar

[16]

S. van Strien and E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc., 17 (2004), 749-782.  doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Small denominators I. Mapping the circle onto itself, Izv. Akad. Nauk. Math., 25 (1961), 21-86.   Google Scholar

[2]

A. AvilaM. Lyubich and W. de Melo, Regular or stochastic dynamics in real analytic families of unimodal maps, Invent. Math., 154 (2003), 451-550.  doi: 10.1007/s00222-003-0307-6.  Google Scholar

[3]

H. BruinJ. Rivera-LetelierW. Shen and S. van Strien, Large derivatives, backward contraction and invariant densities for interval maps, Invent. Math., 172 (2008), 509-533.  doi: 10.1007/s00222-007-0108-4.  Google Scholar

[4]

H. BruinW. Shen and S. van Strien, Invariant measures exist without a growth condition, Comm. Math. Phys., 241 (2003), 287-306.  doi: 10.1007/s00220-003-0928-z.  Google Scholar

[5]

P. Collet and J. P. Eckmann, Positive Liapunov exponents and absolute continuity for maps of the interval, Ergodic Theory Dynam. Systems, 3 (1983), 13-46.  doi: 10.1017/S0143385700001802.  Google Scholar

[6]

M. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.E.S., 49 (1979), 5-233.   Google Scholar

[7]

F. Hofbauer and G. Keller, Quadratic maps without asymptotic measure, Comm. Math. Phys., 127 (1990), 319-337.  doi: 10.1007/BF02096761.  Google Scholar

[8]

M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys., 81 (1981), 39-88.  doi: 10.1007/BF01941800.  Google Scholar

[9]

S. D. Johnson, Singular measures without restrictive intervals, Comm. Math. Phys., 110 (1987), 173-348.  doi: 10.1007/BF01207362.  Google Scholar

[10]

G. Keller, Exponents, attractors and Hopf decompositions for interval maps, Ergodic Theory Dynam. Systems, 10 (1990), 717-744.  doi: 10.1017/S0143385700005861.  Google Scholar

[11]

M. Lyubich, Almost every real quadratic map is either regular or stochastic, Annals of Math., 156 (2002), 1-78.  doi: 10.2307/3597183.  Google Scholar

[12]

R. Mañé, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys., 100 (1985), 495-524.  doi: 10.1007/BF01217727.  Google Scholar

[13]

W. de Melo and S. van Strien, One-dimensional Dynamics. Ergebnisse Der Mathematik und Ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.  Google Scholar

[14]

M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., 53 (1981), 17-51.   Google Scholar

[15]

T. Nowicki and S. van Strien, Invariant measures exist under a summability condition for unimodal maps, Invent. Math., 105 (1991), 123-136.  doi: 10.1007/BF01232258.  Google Scholar

[16]

S. van Strien and E. Vargas, Real bounds, ergodicity and negative Schwarzian for multimodal maps, J. Amer. Math. Soc., 17 (2004), 749-782.  doi: 10.1090/S0894-0347-04-00463-1.  Google Scholar

Figure 1.  The graph of the critical covering map $f_1$
Figure 2.  The first entry map to the interval $V_{2n-1}^+$, showing the critical branch $\phi_n$ and some of the branches $\sigma_{k,\ell_k}$, with $k \geq n$
Figure 3.  Comparing graphs of the functions $\mathcal{T}_1$ (left) and $\mathcal{T}_2$ (right)
Figure 4.  The graph of $\mathcal{T}$
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