# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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.  Google Scholar [2] V. Baladi, M. 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.  Google Scholar [3] A. Blumental and Y. Yang, Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps, Preprint, (2018). Google Scholar [4] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116.   Google Scholar [5] T. Bogenschütz, Stochastic stability of equilibrium states, Random Comput. Dynam., 4 (1996), 85-98.   Google Scholar [6] T. Bogenschütz, Equilibrium States for Random Dynamical Systems, Institut für Dynamische Systeme, Universität Bremen, 1993. Google Scholar [7] T. Bogenschütz and V. M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random Comput. Dynam., 1 (1992/93), 219-227.   Google Scholar [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.  Google Scholar [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.  Google Scholar [10] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25.   Google Scholar [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.  Google Scholar [12] P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.  Google Scholar [13] H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.  Google Scholar [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.  Google Scholar [15] M. de Guzmán, Differentiation of Integrals in $\mathbb R^n$., Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin-New York, 1975.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Mathematische Annalen, 309 (1997), 593-609.  doi: 10.1007/s002080050129.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., (1981), 17–51.  Google Scholar [27] A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-442.  doi: 10.1090/S0002-9947-1947-0020618-0.  Google Scholar [28] F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fundamenta Mathematicae, 155 (1998), 189-199.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### References:
 [1] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. doi: 10.1007/978-3-662-12878-7.  Google Scholar [2] V. Baladi, M. 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.  Google Scholar [3] A. Blumental and Y. Yang, Positive Lyapunov exponent for random perturbations of predominantly expanding multimodal circle maps, Preprint, (2018). Google Scholar [4] T. Bogenschütz, Entropy, pressure, and a variational principle for random dynamical systems, Random Comput. Dynam., 1 (1992/93), 99-116.   Google Scholar [5] T. Bogenschütz, Stochastic stability of equilibrium states, Random Comput. Dynam., 4 (1996), 85-98.   Google Scholar [6] T. Bogenschütz, Equilibrium States for Random Dynamical Systems, Institut für Dynamische Systeme, Universität Bremen, 1993. Google Scholar [7] T. Bogenschütz and V. M. Gundlach, Symbolic dynamics for expanding random dynamical systems, Random Comput. Dynam., 1 (1992/93), 219-227.   Google Scholar [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.  Google Scholar [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.  Google Scholar [10] R. Bowen, Hausdorff dimension of quasicircles, Inst. Hautes Études Sci. Publ. Math., 50 (1979), 11-25.   Google Scholar [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.  Google Scholar [12] P. Collet and J.-P. Eckmann, Iterated Maps of the Interval as Dynamical Systems, Progress in Physics, 1. Birkhäuser, Boston, Mass., 1980.  Google Scholar [13] H. Crauel, Random Probability Measures on Polish Spaces, Stochastics Monographs, 11. Taylor & Francis, London, 2002.  Google Scholar [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.  Google Scholar [15] M. de Guzmán, Differentiation of Integrals in $\mathbb R^n$., Lecture Notes in Mathematics, Vol. 481. Springer-Verlag, Berlin-New York, 1975.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [19] Y. Kifer, Equilibrium states for random expanding transformations, Random Comput. Dynam., 1 (1992/93), 1-31.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [23] P. Koskela and S. Rohde, Hausdorff dimension and mean porosity, Mathematische Annalen, 309 (1997), 593-609.  doi: 10.1007/s002080050129.  Google Scholar [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.  Google Scholar [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.  Google Scholar [26] M. Misiurewicz, Absolutely continuous measures for certain maps of an interval, Inst. Hautes Études Sci. Publ. Math., (1981), 17–51.  Google Scholar [27] A. P. Morse, Perfect blankets, Trans. Amer. Math. Soc., 61 (1947), 418-442.  doi: 10.1090/S0002-9947-1947-0020618-0.  Google Scholar [28] F. Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fundamenta Mathematicae, 155 (1998), 189-199.   Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
A typical element of ${\mathcal M}( {\mathcal{G}};\kappa, A,{\gamma},\iota)$
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