Constant slope, entropy, and horseshoes for a map on a tame graph

@article{49220,
   author = {Bartoš, Adam and Bobok, Jozef and Pyrih, Pavel and Roth, Samuel Joshua and Vejnar, Benjamin},
   article_location = {New York},
   article_number = {11},
   doi = {http://dx.doi.org/10.1017/etds.2019.29},
   keywords = {Markov map; tame graph; constant slope; conjugacy; entropy},
   language = {eng},
   issn = {0143-3857},
   journal = {Ergodic Theory and Dynamical Systems},
   title = {Constant slope, entropy, and horseshoes for a map on a tame graph},
   url = {https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/constant-slope-entropy-and-horseshoes-for-a-map-on-a-tame-graph/92747064767B5DC745355B2ED02C6071},
   volume = {40},
   year = {2020}
}

TY  - JOUR
ID  - 49220
AU  - Bartoš, Adam - Bobok, Jozef - Pyrih, Pavel - Roth, Samuel Joshua - Vejnar, Benjamin
PY  - 2020
TI  - Constant slope, entropy, and horseshoes for a map on a tame graph
JF  - Ergodic Theory and Dynamical Systems
VL  - 40
IS  - 11
SP  - 2970-2994
EP  - 2970-2994
PB  - Cambridge University Press
SN  - 01433857
KW  - Markov map
KW  - tame graph
KW  - constant slope
KW  - conjugacy
KW  - entropy
UR  - https://www.cambridge.org/core/journals/ergodic-theory-and-dynamical-systems/article/constant-slope-entropy-and-horseshoes-for-a-map-on-a-tame-graph/92747064767B5DC745355B2ED02C6071
N2  - We study continuous countably (strictly) monotone maps defined on a tame graph, i.e. a special Peano continuum for which the set containing branch points and end points has countable closure. In our investigation we confine ourselves to the countable Markov case. We show a necessary and sufficient condition under which a locally eventually onto, countably Markov map f of a tame graph G is conjugate to a map g of constant slope. In particular, we show that in the case of a Markov map f that corresponds to a recurrent transition matrix, the condition is satisfied for a constant slope e(htop(f)), where e(htop(f))is the topological entropy of f. Moreover, we show that in our class the topological entropy e(htop(f)) is achievable through horseshoes of the map f.
ER  - 

BARTOŠ, Adam, Jozef BOBOK, Pavel PYRIH, Samuel Joshua ROTH and Benjamin VEJNAR. Constant slope, entropy, and horseshoes for a map on a tame graph. \textit{Ergodic Theory and Dynamical Systems}. New York: Cambridge University Press, 2020, vol.~40, No~11, p.~2970-2994. ISSN~0143-3857. Available from: https://dx.doi.org/10.1017/etds.2019.29.