# American Institute of Mathematical Sciences

December  2015, 20(10): 3415-3434. doi: 10.3934/dcdsb.2015.20.3415

## Formulas for the topological entropy of multimodal maps based on min-max symbols

 1 Universidad Miguel Hernández, Centro de Investigación Operativa, Avda. Universidad s/n, Elche (Alicante), 03202, Spain

Received  December 2014 Revised  March 2015 Published  September 2015

In this paper, a new formula for the topological entropy of a multimodal map $f$ is derived, and some basic properties are studied. By a formula we mean an analytical expression leading to a numerical algorithm; by a multimodal map we mean a continuous interval self-map which is strictly monotonic in a finite number of subintervals. The main feature of this formula is that it involves the min-max symbols of $f$, which are closely related to its kneading symbols. This way we continue our pursuit of finding expressions for the topological entropy of continuous multimodal maps based on min-max symbols. As in previous cases, which will be also reviewed, the main geometrical ingredients of the new formula are the numbers of transversal crossings of the graph of $f$ and its iterates with the so-called "critical lines". The theoretical and practical underpinnings are worked out with the family of logistic parabolas and numerical simulations.
Citation: José M. Amigó, Ángel Giménez. Formulas for the topological entropy of multimodal maps based on min-max symbols. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3415-3434. doi: 10.3934/dcdsb.2015.20.3415
##### References:

show all references

##### References:
 [1] Jinchuan Zhou, Changyu Wang, Naihua Xiu, Soonyi Wu. First-order optimality conditions for convex semi-infinite min-max programming with noncompact sets. Journal of Industrial & Management Optimization, 2009, 5 (4) : 851-866. doi: 10.3934/jimo.2009.5.851 [2] Roberto Livrea, Salvatore A. Marano. A min-max principle for non-differentiable functions with a weak compactness condition. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1019-1029. doi: 10.3934/cpaa.2009.8.1019 [3] Meixia Li, Changyu Wang, Biao Qu. Non-convex semi-infinite min-max optimization with noncompact sets. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1859-1881. doi: 10.3934/jimo.2017022 [4] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems - A, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 [5] Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems - A, 2011, 31 (2) : 545-556. doi: 10.3934/dcds.2011.31.545 [6] Fryderyk Falniowski, Marcin Kulczycki, Dominik Kwietniak, Jian Li. Two results on entropy, chaos and independence in symbolic dynamics. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3487-3505. doi: 10.3934/dcdsb.2015.20.3487 [7] Dante Carrasco-Olivera, Roger Metzger Alvan, Carlos Arnoldo Morales Rojas. Topological entropy for set-valued maps. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3461-3474. doi: 10.3934/dcdsb.2015.20.3461 [8] Lorenzo Sella, Pieter Collins. Computation of symbolic dynamics for two-dimensional piecewise-affine maps. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 739-767. doi: 10.3934/dcdsb.2011.15.739 [9] José S. Cánovas. Topological sequence entropy of $\omega$–limit sets of interval maps. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 781-786. doi: 10.3934/dcds.2001.7.781 [10] X. X. Huang, Xiaoqi Yang, K. L. Teo. A smoothing scheme for optimization problems with Max-Min constraints. Journal of Industrial & Management Optimization, 2007, 3 (2) : 209-222. doi: 10.3934/jimo.2007.3.209 [11] Baolan Yuan, Wanjun Zhang, Yubo Yuan. A Max-Min clustering method for $k$-means algorithm of data clustering. Journal of Industrial & Management Optimization, 2012, 8 (3) : 565-575. doi: 10.3934/jimo.2012.8.565 [12] Steven T. Piantadosi. Symbolic dynamics on free groups. Discrete & Continuous Dynamical Systems - A, 2008, 20 (3) : 725-738. doi: 10.3934/dcds.2008.20.725 [13] Xueting Tian, Paulo Varandas. Topological entropy of level sets of empirical measures for non-uniformly expanding maps. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5407-5431. doi: 10.3934/dcds.2017235 [14] Ghassen Askri. Li-Yorke chaos for dendrite maps with zero topological entropy and ω-limit sets. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 2957-2976. doi: 10.3934/dcds.2017127 [15] Steven M. Pederson. Non-turning Poincaré map and homoclinic tangencies in interval maps with non-constant topological entropy. Conference Publications, 2001, 2001 (Special) : 295-302. doi: 10.3934/proc.2001.2001.295 [16] Wacław Marzantowicz, Feliks Przytycki. Estimates of the topological entropy from below for continuous self-maps on some compact manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 21 (2) : 501-512. doi: 10.3934/dcds.2008.21.501 [17] Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 [18] Jaume Llibre. Brief survey on the topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3363-3374. doi: 10.3934/dcdsb.2015.20.3363 [19] Jim Wiseman. Symbolic dynamics from signed matrices. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 621-638. doi: 10.3934/dcds.2004.11.621 [20] George Osipenko, Stephen Campbell. Applied symbolic dynamics: attractors and filtrations. Discrete & Continuous Dynamical Systems - A, 1999, 5 (1) : 43-60. doi: 10.3934/dcds.1999.5.43

2018 Impact Factor: 1.008

## Metrics

• HTML views (0)
• Cited by (2)

• on AIMS