Homoclinic points and topological entropy of a continuous mapping of a dendrite. (English. Russian original) Zbl 1190.37011

Makhrova, E. N.

J. Math. Sci., New York 158, No. 2, 241-248 (2009); translation from Sovrem. Mat. Prilozh. 54 (2008).

The author studies the sufficient and necessary conditions for homoclinic trajectories of a continuous dendrite mapping under which the topological entropy of the mapping is positive. In particular, sufficient conditions are obtained but it is also shown that the obtained conditions are not necessary conditions for the positivity of the topological entropy of continuous dendrite mappings.
A dendrite is a locally connected continuum containing no subspace homeomorphic to the circle. For \(x,y\in X\), \(\gamma [x,y]\) denotes the arc with endpoints \(x\) and \(y\) and \(\gamma (x,y)=\gamma [x,y]\setminus\{x,y\}\). Let \(X\) be a dendrite. If \(f:X\to X\) is a continuous mapping, then a point \(z\in X\) is called a point homoclinic to a periodic point \(x\in X\), if \(z\) belongs to the intersection of the stable and unstable manifolds of the point \(x\). A continuous map \(f:X\to X\) has a horseshoe \((\gamma [a,b],\gamma [c,d])\), if \(\gamma [a,b],\gamma [c,d]\subset X\), \(\gamma (a,b)\cap \gamma (c,d)=\emptyset\), and \(\gamma [a,b]\cup \gamma [c,d]\subset f(\gamma [a,b])\cap f(\gamma [c,d])\). The following are the main results of the paper.
1. Let \(X\) be a dendrite and \(f:X\to X\) a continuous mapping. Then (a) and (b) are equivalent:
(a) \(f^n\) has a horseshoe for some \(n\geq 1\)
(b) \(f\) has a point \(z_0\) homoclinic to a periodic point \(x\) of period \(m\geq 1\), satisfying the following condition: for a certain \(j\geq 1\) multiple of \(m\) there is a point \(z_{-j}\in f^{-j}(z_0)\) such that \(z_{-j}\in \gamma (x,z_0)\).
2. Let \(X\) be a dendrite and \(f:X\to X\) a continuous mapping. If there is a point \(z_0\) homoclinic to a periodic point \(x\) of period \(m\geq 1\) such that for a certain \(j\geq 1\) multiple of \(m\) there is a point \(z_{-j}\in f^{-j}(z_0)\) such that \(z_{-j}\in \gamma (x,z_0)\), then the topological entropy of \(f\) is positive.
3. A dendrite \(G\) and a continuous mapping \(f:G\to G\) are constructed such that the topological entropy of \(f\) is positive but none of the conditions (a) and (b) in 1. holds true.

Reviewer: Iztok Banič (Maribor)

MSC:

37B40	Topological entropy
37A35	Entropy and other invariants, isomorphism, classification in ergodic theory
37B45	Continua theory in dynamics

Keywords:

homoclinic points; dendrite; entropy; horsehoe

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