August  2020, 40(8): 4839-4906. doi: 10.3934/dcds.2020204

Critically finite random maps of an interval

1. 

School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

2. 

Department of Mathematics, University of North Texas, Denton, TX 76203-1430, USA

* Corresponding author: Mariusz Urbański

Received  August 2019 Revised  February 2020 Published  May 2020

Fund Project: The research of the first author was supported by an ARC Discovery Project. The research of the second named author was funded in part by the Simons Foundation 581668

Given a finite collection
$ {\mathcal{G}} $
of closed subintervals of the unit interval
$ [0,1] $
with mutually empty interiors, we consider random multimodal
$ C^3 $
maps with negative Schwarzian derivative, mapping each interval of
$ {\mathcal{G}} $
onto the unit interval
$ [0,1] $
. The randomness is governed by an invertible ergodic map
$ {\theta}:{\Omega}\to{\Omega} $
preserving a probability measure
$ m $
on some probability space
$ {\Omega} $
. We denote the corresponding skew product map by
$ T $
and call it a critically finite random map of an interval. We prove that there exists a subset
$ AA(T) $
, defined in Definition 9.1, of
$ [0,1] $
with the following properties:
1. For each
$ t\in AA(T) $
a
$ t $
–conformal random measure
$ \nu_t $
exists. We denote by
$ {\lambda}_{t,\nu_t,{\omega}} $
the corresponding generalized eigenvalues of the corresponding dual operators
$ {\mathcal{L}}_{t,{\omega}}^* $
,
$ {\omega}\in{\Omega} $
.
2. Given
$ t\ge 0 $
any two
$ t $
–conformal random measures are equivalent.
3. The expected topological pressure of the parameter
$ t $
:
$ \begin{align*} { {\mathcal{E}}{{\rm{P}}}}(t): = \int_{{\Omega}}\log {\lambda}_{t,\nu,{\omega}}dm({\omega}). \end{align*} $
is independent of the choice of a
$ t $
–conformal random measure
$ \nu $
.
4. The function
$ AA(T)\ni t\longmapsto { {\mathcal{E}}{{\rm{P}}}}(t)\in\mathbb R $
is monotone decreasing and Lipschitz continuous.
5. With
$ b_T $
being defined as the supremum of such parameters
$ t\in AA(T) $
that
$ { {\mathcal{E}}{{\rm{P}}}}(t)\ge 0 $
, it holds that
$ { {\mathcal{E}}{{\rm{P}}}}(b_T) = 0 \ \ \ {\rm and} \ \ \ [0,b_T]{\subset} {{{\rm{Int}}}}(AA(T)). $
6.
$ {\rm{HD}}( {\mathcal{J}}_{\omega}(T)) = b_T $
for
$ m $
–a.e
$ {\omega}\in{\Omega} $
, where
$ {\mathcal{J}}_{\omega}(T) $
,
$ {\omega}\in{\Omega} $
, form the random closed set generated by the skew product map
$ T $
.
7.
$ b_T = 1 $
if and only if
$ {\bigcup}_{ {\Delta}\in {\mathcal{G}}} {\Delta} = [0,1] $
, and then
$ {\mathcal{J}}_{\omega}(T) = [0,1] $
for all
$ {\omega}\in{\Omega} $
.
Citation: Jason Atnip, Mariusz Urbański. Critically finite random maps of an interval. Discrete and Continuous Dynamical Systems, 2020, 40 (8) : 4839-4906. doi: 10.3934/dcds.2020204
References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

V. BaladiM. Benedicks and V. Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps, Ann. Sci. École Norm. Sup. (4), 35 (2002), 77-126.  doi: 10.1016/S0012-9593(01)01083-7.

[3]

A. Blumental and Y. Yang, Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps, Preprint, (2018).

[4]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116. 

[5]

T. Bogenschütz, Stochastic stability of equilibrium states, Random Comput. Dynam., 4 (1996), 85-98. 

[6]

T. Bogenschütz, Equilibrium States for Random Dynamical Systems, Institut für Dynamische Systeme, Universität Bremen, 1993.

[7]

T. Bogenschütz and V. M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random Comput. Dynam., 1 (1992/93), 219-227. 

[8]

T. Bogenschütz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergodic Theory and Dynamical Systems, 15 (1995), 413-447.  doi: 10.1017/S0143385700008464.

[9]

T. Bogenschütz and G. Ochs, The Hausdorff dimension of conformal repellers under random perturbation, Nonlinearity, 12 (1999), 1323-1338.  doi: 10.1088/0951-7715/12/5/307.

[10]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. 

[11]

D. L. Cohn, Measure Theory, Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6956-8.

[12]

P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.

[13]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.

[14]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, Journal of Dynamics and Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.

[15]

M. de Guzmán, Differentiation of Integrals in $\mathbb R^n$., Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin-New York, 1975.

[16]

K. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinaǐ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 28 (1996), 107–140. doi: 10.1090/trans2/171/10.

[17]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[18]

Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16. Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.

[19]

Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31. 

[20]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions, Fractal Geometry and Stochastics (Finsterbergen, 1994), Progr. Probab., Birkhäuser, Basel, 37 (1995), 145-164.  doi: 10.1007/978-3-0348-7755-8_7.

[21]

Y. Kifer, Fractal dimensions and random transformations, Transactions of the American Mathematical Society, 348 (1996), 2003-2038.  doi: 10.1090/S0002-9947-96-01608-X.

[22]

Y. Kifer and P.-D. Liu, Random dynamics, Handbook of Dynamical Systems, Elsevier B. V., Amsterdam, 1B (2006), 379-499.  doi: 10.1016/S1874-575X(06)80030-5.

[23]

P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Mathematische Annalen, 309 (1997), 593-609.  doi: 10.1007/s002080050129.

[24]

V. Mayer, B. Skorulski and M. Urbański, Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics, 2036. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-23650-1.

[25]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 25. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.

[26]

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

[27]

A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-442.  doi: 10.1090/S0002-9947-1947-0020618-0.

[28]

F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fundamenta Mathematicae, 155 (1998), 189-199. 

[29]

M. Roy and M. Urbański, Random graph directed Markov Systems, Discrete Contin. Dyn. Syst., 30 (2011), 261-298.  doi: 10.3934/dcds.2011.30.261.

[30]

H. H. Rugh, On the dimension of conformal repellors. Randomness and parameter dependency, Annals of Mathematics, 168 (2008), 695-748.  doi: 10.4007/annals.2008.168.695.

[31]

H. H. Rugh, Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory, Annals of Mathematics, 171 (2010), 1707-1752.  doi: 10.4007/annals.2010.171.1707.

[32]

D. Simmons and M. Urbański, Relative equilibrium states and dimensions of fiberwise invariant measures for distance expanding random maps, Stochastics and Dynamics, 14 (2014), 1350015, 25 pp. doi: 10.1142/S0219493713500159.

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.

[2]

V. BaladiM. Benedicks and V. Maume-Deschamps, Almost sure rates of mixing for i.i.d. unimodal maps, Ann. Sci. École Norm. Sup. (4), 35 (2002), 77-126.  doi: 10.1016/S0012-9593(01)01083-7.

[3]

A. Blumental and Y. Yang, Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps, Preprint, (2018).

[4]

T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116. 

[5]

T. Bogenschütz, Stochastic stability of equilibrium states, Random Comput. Dynam., 4 (1996), 85-98. 

[6]

T. Bogenschütz, Equilibrium States for Random Dynamical Systems, Institut für Dynamische Systeme, Universität Bremen, 1993.

[7]

T. Bogenschütz and V. M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random Comput. Dynam., 1 (1992/93), 219-227. 

[8]

T. Bogenschütz and V. M. Gundlach, Ruelle's transfer operator for random subshifts of finite type, Ergodic Theory and Dynamical Systems, 15 (1995), 413-447.  doi: 10.1017/S0143385700008464.

[9]

T. Bogenschütz and G. Ochs, The Hausdorff dimension of conformal repellers under random perturbation, Nonlinearity, 12 (1999), 1323-1338.  doi: 10.1088/0951-7715/12/5/307.

[10]

R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25. 

[11]

D. L. Cohn, Measure Theory, Second edition, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser/Springer, New York, 2013. doi: 10.1007/978-1-4614-6956-8.

[12]

P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.

[13]

H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.

[14]

H. Crauel and F. Flandoli, Hausdorff dimension of invariant sets for random dynamical systems, Journal of Dynamics and Differential Equations, 10 (1998), 449-474.  doi: 10.1023/A:1022605313961.

[15]

M. de Guzmán, Differentiation of Integrals in $\mathbb R^n$., Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin-New York, 1975.

[16]

K. Khanin and Y. Kifer, Thermodynamic formalism for random transformations and statistical mechanics, Sinaǐ's Moscow Seminar on Dynamical Systems, Amer. Math. Soc. Transl. Ser. 2, Amer. Math. Soc., Providence, RI, 28 (1996), 107–140. doi: 10.1090/trans2/171/10.

[17]

Y. Kifer, Ergodic Theory of Random Transformations, Progress in Probability and Statistics, 10. Birkhäuser Boston, Inc., Boston, MA, 1986. doi: 10.1007/978-1-4684-9175-3.

[18]

Y. Kifer, Random Perturbations of Dynamical Systems, Progress in Probability and Statistics, 16. Birkhäuser Boston, Inc., Boston, MA, 1988. doi: 10.1007/978-1-4615-8181-9.

[19]

Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31. 

[20]

Y. Kifer, Fractals via random iterated function systems and random geometric constructions, Fractal Geometry and Stochastics (Finsterbergen, 1994), Progr. Probab., Birkhäuser, Basel, 37 (1995), 145-164.  doi: 10.1007/978-3-0348-7755-8_7.

[21]

Y. Kifer, Fractal dimensions and random transformations, Transactions of the American Mathematical Society, 348 (1996), 2003-2038.  doi: 10.1090/S0002-9947-96-01608-X.

[22]

Y. Kifer and P.-D. Liu, Random dynamics, Handbook of Dynamical Systems, Elsevier B. V., Amsterdam, 1B (2006), 379-499.  doi: 10.1016/S1874-575X(06)80030-5.

[23]

P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Mathematische Annalen, 309 (1997), 593-609.  doi: 10.1007/s002080050129.

[24]

V. Mayer, B. Skorulski and M. Urbański, Distance Expanding Random Mappings, Thermodynamical Formalism, Gibbs Measures and Fractal Geometry, Lecture Notes in Mathematics, 2036. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-23650-1.

[25]

W. de Melo and S. van Strien, One-Dimensional Dynamics, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 25. Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-78043-1.

[26]

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

[27]

A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-442.  doi: 10.1090/S0002-9947-1947-0020618-0.

[28]

F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fundamenta Mathematicae, 155 (1998), 189-199. 

[29]

M. Roy and M. Urbański, Random graph directed Markov Systems, Discrete Contin. Dyn. Syst., 30 (2011), 261-298.  doi: 10.3934/dcds.2011.30.261.

[30]

H. H. Rugh, On the dimension of conformal repellors. Randomness and parameter dependency, Annals of Mathematics, 168 (2008), 695-748.  doi: 10.4007/annals.2008.168.695.

[31]

H. H. Rugh, Cones and gauges in complex spaces: Spectral gaps and complex Perron-Frobenius theory, Annals of Mathematics, 171 (2010), 1707-1752.  doi: 10.4007/annals.2010.171.1707.

[32]

D. Simmons and M. Urbański, Relative equilibrium states and dimensions of fiberwise invariant measures for distance expanding random maps, Stochastics and Dynamics, 14 (2014), 1350015, 25 pp. doi: 10.1142/S0219493713500159.

Figure 1.  A typical element of $ {\mathcal M}( {\mathcal{G}};\kappa, A,{\gamma},\iota) $
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