# American Institute of Mathematical Sciences

2013, 2013(special): 227-236. doi: 10.3934/proc.2013.2013.227

## A reinjected cuspidal horseshoe

 1 Department of Mathematical Sciences, Florida Atlantic University, 777 Glades Road, 33431 Boca Raton, United States, United States

Received  September 2012 Revised  July 2013 Published  November 2013

Horseshoes play a central role in dynamical systems and are observed in many chaotic systems. However most points in a neighborhood of the horseshoe escape after finite iterations. In this work we construct a model that possesses an attracting set that contains a cuspidal horseshoe with positive entropy. This model is obtained by reinjecting the points that escape the horseshoe and can be realized in a 3-dimensional vector field.
Citation: Marcus Fontaine, William D. Kalies, Vincent Naudot. A reinjected cuspidal horseshoe. Conference Publications, 2013, 2013 (special) : 227-236. doi: 10.3934/proc.2013.2013.227
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##### References:
 [1] 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 [2] Zhiqin Qiao, Deming Zhu, Qiuying Lu. Bifurcation of a heterodimensional cycle with weak inclination flip. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 1009-1025. doi: 10.3934/dcdsb.2012.17.1009 [3] Dan Liu, Shigui Ruan, Deming Zhu. Nongeneric bifurcations near heterodimensional cycles with inclination flip in $\mathbb{R}^4$. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1511-1532. doi: 10.3934/dcdss.2011.4.1511 [4] Christian Bonatti, Shaobo Gan, Dawei Yang. On the hyperbolicity of homoclinic classes. Discrete & Continuous Dynamical Systems, 2009, 25 (4) : 1143-1162. doi: 10.3934/dcds.2009.25.1143 [5] Eric Bedford, Serge Cantat, Kyounghee Kim. Pseudo-automorphisms with no invariant foliation. Journal of Modern Dynamics, 2014, 8 (2) : 221-250. doi: 10.3934/jmd.2014.8.221 [6] Andrus Giraldo, Bernd Krauskopf, Hinke M. Osinga. Computing connecting orbits to infinity associated with a homoclinic flip bifurcation. Journal of Computational Dynamics, 2020, 7 (2) : 489-510. doi: 10.3934/jcd.2020020 [7] Andres del Junco, Daniel J. Rudolph, Benjamin Weiss. Measured topological orbit and Kakutani equivalence. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 221-238. doi: 10.3934/dcdss.2009.2.221 [8] Keonhee Lee, Manseob Lee. Hyperbolicity of $C^1$-stably expansive homoclinic classes. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 1133-1145. doi: 10.3934/dcds.2010.27.1133 [9] Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339 [10] Katrin Gelfert. Lower bounds for the topological entropy. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 555-565. doi: 10.3934/dcds.2005.12.555 [11] 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 [12] Yong Fang, Patrick Foulon, Boris Hasselblatt. Longitudinal foliation rigidity and Lipschitz-continuous invariant forms for hyperbolic flows. Electronic Research Announcements, 2010, 17: 80-89. doi: 10.3934/era.2010.17.80 [13] Christian Wolf. A shift map with a discontinuous entropy function. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 319-329. doi: 10.3934/dcds.2020012 [14] Eleonora Catsigeras, Xueting Tian. Dominated splitting, partial hyperbolicity and positive entropy. Discrete & Continuous Dynamical Systems, 2016, 36 (9) : 4739-4759. doi: 10.3934/dcds.2016006 [15] Dongkui Ma, Min Wu. Topological pressure and topological entropy of a semigroup of maps. Discrete & Continuous Dynamical Systems, 2011, 31 (2) : 545-557 . doi: 10.3934/dcds.2011.31.545 [16] Piotr Oprocha, Paweł Potorski. Topological mixing, knot points and bounds of topological entropy. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3547-3564. doi: 10.3934/dcdsb.2015.20.3547 [17] Boris Hasselblatt, Zbigniew Nitecki, James Propp. Topological entropy for nonuniformly continuous maps. Discrete & Continuous Dynamical Systems, 2008, 22 (1&2) : 201-213. doi: 10.3934/dcds.2008.22.201 [18] Michał Misiurewicz. On Bowen's definition of topological entropy. Discrete & Continuous Dynamical Systems, 2004, 10 (3) : 827-833. doi: 10.3934/dcds.2004.10.827 [19] Lluís Alsedà, David Juher, Francesc Mañosas. Forward triplets and topological entropy on trees. Discrete & Continuous Dynamical Systems, 2022, 42 (2) : 623-641. doi: 10.3934/dcds.2021131 [20] Enrique R. Pujals. On the density of hyperbolicity and homoclinic bifurcations for 3D-diffeomorphisms in attracting regions. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 179-226. doi: 10.3934/dcds.2006.16.179

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