Dendrite Map Research Articles

A note on periodic points and recurrent points of maps of dendrites

Let f: X → X be a map of a continuum X. Let P(f) denote the set of all periodic points of f and R(f) denote the set of all recurrent points of f. In [2], Coven and Hedlund proved that if f: I → I is a map of the unit interval I = [0, 1], then CI(P(f)) = CI(R(f)). In [7], Ye generalised this result to maps of a tree. It is natural to ask whether the result generalises to maps of a dendrite. (A dendrite is a locally connected continuum which contains no simple closed curve.) The aim of this paper is to show that the answer is negative.

Depositional patterns of Pennsylvanian sediments in Sullivan County, West-Central Indiana, U.S.A.

Individual Pennsylvanian clastic sediment intervals above the Seelyville Coal were examined on electric logs from Sullivan County, Indiana, U.S.A. for vertical sequence, interval thickness, and sand content. This information was used to evaluate local variability of this area of the Illinois Basin depositional system. Each clastic interval is composed of a lateral association of vertical sequences of sediments, bounded above and below by a thin association of coal, limestone, black shale, and/or underclay. An “average” constructive clastic unit is characterized by the following thickness parameters: mean ( x) = 14 m , standard deviation ( s) = 3.4 m, minimum = 6.1 m, and maximum = 25 m. Mean ( x) sand content, as measured by electric log resistivity, is 37%. Clastic sediment intervals characterized as “deep water” sediments tend to be clayrich, have the greatest lateral continuity, are composed of 50% to 90% coarsening upward sequences, are comprised of less than 16% fining upward sequences, and are composed of less than 10% of sequences of facies which are transitional between coarsening and fining upward. Sandstone channels are linear to coarsely dendritic and probably postdate lower portions of the coarsening upward sequences. One example of longshore shoaling between the Springfield and Houchin Creek coals was discovered. This is the youngest stratigraphic evidence of longshore currents in Illinois Basin clastic sediments. This milieu probably represents a distal delta position. Shallow-water sediments are sand-rich, complex, and gradational. They tend to be interfingered and to display poor lateral continuity. Coarsening upward sequences comprise less than 20% of the data sites. Fining upward fluvial sequences are represented by well-developed dendritic map patterns and constitute 20–30% of the sites. Transitional sequences between coarsening and fining upward log profiles are represented by both interfingered and gradational sequences and constitute 20–55% of the sites. Each constructive vertical sequence represents only a portion of the overall progradational deltatic environment. Destructive linear erosional channels are oriented downdip into the basin. The first occurrence in Indiana of the Trivoli Sandstone channel facies, located between the Ditney and West Franklin horizons, was delineated. Destructive channel sands are found commonly between the Houchin Creek and Colchester coals. Clastic subintervals locally began to develop when compacted unit thickness exceeded 18 m. Average regional wedging contributes 0.13 m/km to local sediment interval thickness variability. The average compacted clastic interval thickness has a local range of 19 m owing to local depositional environments. This variation is related closely to the overall clastic ratio of end-member sequences which are situated in close proximity. The average clastic unit varies in thickness of ± 1.4 m owing to the influence of compaction of the underlying Pennsylvanian sediment filling the Mississippian unconformity valleys. Compactional effects between clastic units become negligible as composite interval thickness exceeds 30 m.