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The tent map is an elementary example of an interval map possessing manyinteresting properties, such as dense periodicity, exactness, Lipschitzness anda kind of length-expansiveness. It is often used in constructions of dynamicalsystems on the interval/trees/graphs. The purpose of the present paper is toconstruct, on totally regular continua (i.e. on topologically rectifiablecurves), maps sharing some typical properties with the tent map. These mapswill be called length-expanding Lipschitz maps, briefly LEL maps. We show thatevery totally regular continuum endowed with a suitable metric admits a LELmap. As an application we obtain that every totally regular continuum admits anexactly Devaney chaotic map with finite entropy and the specification property.

Length-expanding Lipschitz maps on totally regular continua