# American Institute of Mathematical Sciences

July  2009, 23(3): 937-955. doi: 10.3934/dcds.2009.23.937

## Entropy and variational principles for holonomic probabilities of IFS

 1 Instituto de Matemática, UFRGS, 91509-900, Porto Alegre, Brazil, Brazil

Received  September 2007 Revised  July 2008 Published  November 2008

An IFS ( iterated function system), $([0,1], \tau_{i})$, on the interval $[0,1]$, is a family of continuous functions $\tau_{0},\tau_{1}, ..., \tau_{d-1} : [0,1] \to [0,1]$. Associated to a IFS one can consider a continuous map $\hat{\sigma} : [0,1]\times \Sigma \to [0,1]\times \Sigma$, defined by $\hat{\sigma}(x,w)=(\tau_{X_{1}(w)}(x), \sigma(w))$ where $\Sigma=\{0,1, ..., d-1\}^{\mathbb{N}}$, $\sigma: \Sigma \to \Sigma$ is given by $\sigma(w_{1},w_{2},w_{3},...)=(w_{2},w_{3},w_{4}...)$ and $X_{k} : \Sigma \to \{0,1, ..., n-1\}$ is the projection on the coordinate $k$. A $\rho$-weighted system, $\rho \geq 0$, is a weighted system $([0,1], \tau_{i}, u_{i})$ such that there exists a positive bounded function $h : [0,1] \to \mathbb{R}$ and a probability $\nu$ on $[0,1]$ satisfying $P_{u}(h)=\rho h, \quad P_{u}^{*}(\nu)=\rho \nu$.
A probability $\hat{\nu}$ on $[0,1]\times \Sigma$ is called holonomic for $\hat{\sigma}$, if, $\int\, g \circ \hat{\sigma}\, d\hat{\nu}= \int \,g \,d\hat{\nu}, \, \forall g \in C([0,1])$. We denote the set of holonomic probabilities by $\mathcal H$.
For a holonomic probability $\hat{\nu}$ on $[0,1]\times \Sigma$ we define the entropy $h(\hat{\nu})$=inf$_f \in \mathbb{B}^{+} \int \ln(\frac{P_{\psi}f}{\psi f}) d\hat{\nu}\geq 0$, where, $\psi \in \mathbb{B}^{+}$ is a fixed (any one) positive potential.
Finally, we analyze the problem: given $\phi \in \mathbb{B}^{+}$, find solutions of the maximization problem $p(\phi)$= sup$_\hat{\nu} \in \mathcal{H} \{ \,h(\hat{\nu}) + \int \ln(\phi) d\hat{\nu} \,\}.$ We show an example where a holonomic not-$\hat{\sigma}$-invariant probability attains the supremum value. In the last section we consider maximizing probabilities, sub-actions and duality for potentials $A(x,w)$, $(x,w)\in [0,1]\times \Sigma$, for IFS.
Citation: Artur O. Lopes, Elismar R. Oliveira. Entropy and variational principles for holonomic probabilities of IFS. Discrete and Continuous Dynamical Systems, 2009, 23 (3) : 937-955. doi: 10.3934/dcds.2009.23.937
 [1] Zhanyou Ma, Wuyi Yue, Xiaoli Su. Performance analysis of a Geom/Geom/1 queueing system with variable input probability. Journal of Industrial and Management Optimization, 2011, 7 (3) : 641-653. doi: 10.3934/jimo.2011.7.641 [2] Lluís Alsedà, David Juher, Deborah M. King, Francesc Mañosas. Maximizing entropy of cycles on trees. Discrete and Continuous Dynamical Systems, 2013, 33 (8) : 3237-3276. doi: 10.3934/dcds.2013.33.3237 [3] H.T. Banks, Jimena L. Davis. Quantifying uncertainty in the estimation of probability distributions. Mathematical Biosciences & Engineering, 2008, 5 (4) : 647-667. doi: 10.3934/mbe.2008.5.647 [4] Welington Cordeiro, Manfred Denker, Michiko Yuri. A note on specification for iterated function systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3475-3485. doi: 10.3934/dcdsb.2015.20.3475 [5] Xiangxiang Huang, Xianping Guo, Jianping Peng. A probability criterion for zero-sum stochastic games. Journal of Dynamics and Games, 2017, 4 (4) : 369-383. doi: 10.3934/jdg.2017020 [6] Antonio Di Crescenzo, Maria Longobardi, Barbara Martinucci. On a spike train probability model with interacting neural units. Mathematical Biosciences & Engineering, 2014, 11 (2) : 217-231. doi: 10.3934/mbe.2014.11.217 [7] Eleonora Bardelli, Andrea Carlo Giuseppe Mennucci. Probability measures on infinite-dimensional Stiefel manifolds. Journal of Geometric Mechanics, 2017, 9 (3) : 291-316. doi: 10.3934/jgm.2017012 [8] Benedetto Piccoli, Francesco Rossi. Measure dynamics with Probability Vector Fields and sources. Discrete and Continuous Dynamical Systems, 2019, 39 (11) : 6207-6230. doi: 10.3934/dcds.2019270 [9] Noah H. Rhee, PaweŁ Góra, Majid Bani-Yaghoub. Predicting and estimating probability density functions of chaotic systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 297-319. doi: 10.3934/dcdsb.2017144 [10] 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 [11] Fabio A. C. C. Chalub. An asymptotic expression for the fixation probability of a mutant in star graphs. Journal of Dynamics and Games, 2016, 3 (3) : 217-223. doi: 10.3934/jdg.2016011 [12] Subhabrata Samajder, Palash Sarkar. Another look at success probability of linear cryptanalysis. Advances in Mathematics of Communications, 2019, 13 (4) : 645-688. doi: 10.3934/amc.2019040 [13] Saisai Shi, Bo Tan, Qinglong Zhou. Best approximation of orbits in iterated function systems. Discrete and Continuous Dynamical Systems, 2021, 41 (9) : 4085-4104. doi: 10.3934/dcds.2021029 [14] Anne-Sophie de Suzzoni. Continuity of the flow of the Benjamin-Bona-Mahony equation on probability measures. Discrete and Continuous Dynamical Systems, 2015, 35 (7) : 2905-2920. doi: 10.3934/dcds.2015.35.2905 [15] Simon Lloyd, Edson Vargas. Critical covering maps without absolutely continuous invariant probability measure. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2393-2412. doi: 10.3934/dcds.2019101 [16] Xavier Cabré. Elliptic PDE's in probability and geometry: Symmetry and regularity of solutions. Discrete and Continuous Dynamical Systems, 2008, 20 (3) : 425-457. doi: 10.3934/dcds.2008.20.425 [17] Xue Dong He, Roy Kouwenberg, Xun Yu Zhou. Inverse S-shaped probability weighting and its impact on investment. Mathematical Control and Related Fields, 2018, 8 (3&4) : 679-706. doi: 10.3934/mcrf.2018029 [18] Tapio Helin. On infinite-dimensional hierarchical probability models in statistical inverse problems. Inverse Problems and Imaging, 2009, 3 (4) : 567-597. doi: 10.3934/ipi.2009.3.567 [19] William Chad Young, Adrian E. Raftery, Ka Yee Yeung. A posterior probability approach for gene regulatory network inference in genetic perturbation data. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1241-1251. doi: 10.3934/mbe.2016041 [20] Andrea Tosin, Paolo Frasca. Existence and approximation of probability measure solutions to models of collective behaviors. Networks and Heterogeneous Media, 2011, 6 (3) : 561-596. doi: 10.3934/nhm.2011.6.561

2020 Impact Factor: 1.392